aberration

**is:** distortion in an imageoptical image produced by the optical system forming the image and composed of contributions arising from a number of well known causes that include spherical aberration, coma, and chromatic aberration. [PHYS 6.2, PHYS 6.4]

**is also:** those features of a lens or mirror which cause such distortions. [PHYS 6.4]

absolute error

**is:** the absolute value (i.e. modulus) of an error or uncertainty in a quantity. [PHYS 1.2]

**has:** the same dimensions as the quantity itself. [PHYS 1.1]

absolute maximum

See global maximum.

absolute minimum

See global minimum.

absolute temperature

**is:** a temperature expressed in kelvin (K) on a temperature scale that starts at absolute zero. Such scales include the ideal gas absolute scale, the thermodynamic Kelvin temperature scale and the International Practical Temperature Scale 1990. [PHYS 7.2]

absolute value

See modulus.

absolute zero

**is:** the lowest possible temperature. [PHYS 7.2]

**is defined:** as 0 K (i.e. 0 kelvin). [PHYS 7.2]

**corresponds:** to −273.15 °C (i.e. −273.15 degrees Celsius). [PHYS 7.2]

absorbed dose

**is:** the amount of energy from ionizing radiation absorbed per unit mass by a body. [PHYS 9.3]

**has as its SI unit:** the gray (Gy), where 1 Gy = 1 J kg^{−1} (i.e. 1 joule per kilogram). [PHYS 9.3]

absorption

**of:** electromagnetic radiation

**is:** the outcome of any process whereby the energy carried by electromagnetic radiation is transformed and added to the internal energy of the medium through which the electromagnetic radiation is travelling.

**should be contrasted:** with emission, and reflection.

**more generally is:** the outcome of any process in which an entity or agency is partly or wholly assimilated into another.

absorption line spectra

**is:** an absorption spectrum that exhibits absorption lines. [PHYS 8.2]

absorption line

**in:** the absorption spectrum of a medium (especially a gas, vapour or plasma)

**are:** characteristic narrow ranges of frequency or wavelength (often treated as single frequencies or wavelengths) at which the spectral brightness is significantly less than the (average) spectral brightness in neighbouring parts of the relevant spectrum. [PHYS 8.2]

**correspond individually:** to a transition between two bound states of a particular kind of atom, molecule or ion (or to any other process) that causes the absorption of electromagnetic radiation at particular frequencies or wavelengths. [PHYS 8.2]

absorption spectrum

**of:** electromagnetic radiation, often produced from a continuous emission spectrum (e.g. a source of white light) which has been passed through a specified absorbing medium.

**is:** the distribution of (relative) spectral brightness with respect to frequency or wavelength. [PHYS 8.2]

**may be displayed:** as a graph of the (relative) spectral brightness plotted against wavelength or frequency, or (photographically) as a band of varying levels of brightness and darkness. [PHYS 8.2]

**may exhibit:** (especially for a gas, a vapour or a plasma) characteristic absorption lines, in which case it is often referred to as an absorption line spectrum, or (especially in the case of a solid or a liquid) smoothly varying absorption across a broad range of frequencies or wavelengths. [PHYS 8.2]

absorption transition

**in:** an atom, molecule or ion

**is:** a transition in which the atom, molecule or ion absorbs energy from incoming electromagnetic radiation and is thereby excited from one bound state to another bound state of higher energy. Each absorption transition gives rise to an absorption line in an appropriate absorption spectrum. [PHYS 8.2]

**usually:** involves the ground state as the lower energy state.

AC circuit, a.c. circuit

**is:** an electrical circuit in which an alternating current flows, or may be presumed to flow. [PHYS 5.4]

AC, a.c.

See alternating current.

acceleration

**is:** a vector quantity a which specifies the rate of change of velocity with time. [MATH 2.1, MATH 2.4, PHYS 2.1]

**in one dimension is:** *a*_{x} = *υ*_{x} − *u*_{x} for a particle moving in a straight line with *t* uniform acceleration *a*_{x} along the *x*–axis, where *u*_{x} and *υ*_{x} are the initial and final velocities respectively and *t* is the time taken for the change in velocity. [PHYS 2.1]

**is defined generally:** as a = *dυ*_{ }/*dt*, the derivative of velocity with respect to time. [MATH 4.1, MATH 5.1]

**is often specified:** in terms of its scalar components, *a*_{x}, *a*_{y}, *a*_{z} by a = (*a*_{x}, *a*_{y}, *a*_{z}) = (*dυ*_{x}/*dt*, *dυ*_{y}/*dt*, *dυ*_{z}/*dt*). [MATH 4.1, MATH 5.1]

**has as its SI unit:** m s^{−2} (i.e. metre per second squared). [MATH 4.1]

**is given graphically:** at any particular time, by the gradient of the tangent_to_a_curvetangent to the velocity–time graph of the motion at that time. [MATH 4.1, PHYS 2.1]

See also instantaneous acceleration.

acceleration due to gravity

**is:** the acceleration with which an object falls near to the surface of the Earth, due to the gravitational force that acts upon it. The magnitude of this acceleration is given the symbol, *g* and has the approximate value 9.8 m s^{−2}. [PHYS 2.1]

**is equal:** to the gravitational field at the Earth’s surface. [PHYS 3.2]

**may be regarded:** as the free fall acceleration of an object at the Earth’s surface, or the surface gravity. [PHYS 3.2]

See magnitude of the acceleration due to gravity.

accommodated (eye)

**is:** an eye in which the ciliary muscles (which control the lens) are not fully relaxed. [PHYS 6.4]

**is focused:** somewhere closer than at its far point (usually infinity). [PHYS 6.4]

Contrast with unaccommodated (eye). [PHYS 6.4]

accuracy

**of:** a measurement or value

**is:** a measure of the extent to which the measurement (or value) differs from the true value. [PHYS 1.1]

**is also:** a measure of the extent to which the measurement (or value) is free of systematic error. [PHYS 1.1]

**linguistically is:** perverse. The greater the accuracy, the smaller is its numerical value. A clearer way of expressing it is to say that a quantity is ‘accurate to within plus–or–minus so–much’. [PHYS 1.1]

Compare with precision.

achromatic doublet

**is:** a combination of two lenses (glued together), designed to minimize chromatic aberration at two predetermined wavelengths. [PHYS 6.4]

**traditionally consists:** of a converging lens of crown glass with low dispersion and a weaker diverging lens of flint glass with high dispersion. [PHYS 6.4]

acoustic energy

**is:** the energy transported by sound. [PHYS 5.7]

acoustic wave

See sound wave.

acoustics

**is:** the branch of physics concerned with the study of sound.

actinides

**are:** the fourteen chemical elements with atomic numbers in the range 89-102 inclusive (i.e. from actinium to nobelium). [PHYS 8.4]

**are all:** radioactive. [PHYS 9.3]

**include:** uranium and plutonium. [PHYS 8.4]

**occur:** in a part of the periodic table where the 5f subshell of atoms in their ground state is being progressively filled. [PHYS 8.4]

activity

**is:** the rate *R*_{ }(*t*) at which the nuclei of a radioactive substance disintegrate due to radioactive decay. [PHYS 9.2]

**is also:** a measure of the rate of emission of α–particles, β–particles or γ–radiation from a radioactive isotope. [PHYS 9.2]

**is related:** to the number *N*_{ }(*t*) of unstable nuclei of decay constant *λ* in a pure sample (containing only a single type of radionuclide) by *R*_{ }(*t*) = −*dN*_{ }/*dt* = *λ*N_{ }(*t*). [PHYS 9.2]

**has as its SI unit:** the becquerel (Bq). 1 Bq = 1 decay per second. The non-SI unit of activity, the curie (Ci, 1 Ci = 3.70 × 10^{10} Bq) is also in common use. [PHYS 9.2]

See activity law.

activity law

**is:** the law which governs the activity *R*_{ }(*t*) of a sample of a radioactive isotope, which will remain after a given time *t* has elapsed. The law is exponential: *R*_{ }(*t*) = *R*_{0}_{ }e^{−λt}, where *R*_{0} is the initial activity and *λ* is the decay constant. [PHYS 9.2]

See radioactive decay and radioactive decay law.

acute angle

**is:** an angle of less than 90°. [MATH 2.1]

Contrast with obtuse angle and reflex angle.

addition (of vectors)

See vector addition.

addition formulae

**are:** a class of trigonometric identities. [MATH 1.6]

See trigonometric functions in the Maths For Science handbook.

addition identities

**are:** a class of hyperbolic function identities. [MATH 4.6]

See hyperbolic functions in the Maths For Science handbook.

adiabat

**is:** a path representing a quasistatic adiabatic process, usually on a *PVT*–surface (or some similar surface) or on one of its projections. [PHYS 7.4]

adiabatic

**describes:** a situation in which no heat enters or leaves a system, so that Δ*Q* = 0. [PHYS 7.3]

adiabatic condition

**for:** a fixed quantity of ideal gas

**states:** that *PV*^{γ} = constant, where the constant is characteristic of the process, and *γ* the ratio of specific heats of the gas (*C*_{P}/*C*_{V}), is approximately constant for the gas. [PHYS 7.4]

**characterizes:** an adiabatic process. [PHYS 7.4]

adiabatic process

**takes place:** without heat entering or leaving the system, so Δ*Q* = 0. [PHYS 7.3]

See adiabat and adiabatic condition.

adjacent side

**of:** a right–angled triangle

**is:** the side (not the hypotenuse) that is adjacent to any specified one of the acute angles. [MATH 1.6]

See trigonometric functions in the Maths For Science handbook.

air friction

**is:** air resistance. [PHYS 5.2]

air resistance

**is:** a force that opposes motion through air. [PHYS 2.3]

**has magnitude:** proportional to the square of the object’s speed, for objects of moderate size and speed, moving through the Earth’s atmosphere close to the Earth’s surface. [PHYS 2.3, PHYS 5.2]

Airy disc

**is:** the central circular region of an Airy pattern, extending as far as the first minimum. [PHYS 6.4]

Airy pattern

**is:** the (angular) distribution of radiation diffracted by a circular aperture. [PHYS 6.4]

alcohol–in–glass thermometer

**is:** a glass capillary with a bulb containing alcohol. Changes in temperature cause the glass and alcohol to expand (or contract) by different amounts, and the result is that the meniscus moves to different positions in the capillary. [PHYS 7.2]

**can be calibrated:** by marking meniscus positions corresponding to fixed points such as the boiling_pointboiling and freezing points of water, and then interpolating between them. [PHYS 7.2]

algebra

**is:** the branch of mathematics concerned with symbols and their manipulation according to defined rules.

algebraic

**pertains:** to algebra, the branch of mathematics concerned with symbols and their manipulation. [MATH 1.1]

algebraic division

**is:** the application of division to an algebraic expression. [MATH 1.4]

algebraic expression

**is:** an expression that contains algebraic symbols as well as numbers.

algebraic sum

**is:** a process of addition that respects a sign convention. [PHYS 2.7]

alkali metals

**are:** the metallic chemical elementselements lithium, sodium, potassium, rubidium, caesium and francium. [PHYS 8.4]

**are so named:** because the metals dissolve in water to give solutions that contain significant concentrations of aqueous hydroxide (OH^{−}) ions. Materials generating such solutions are said to be alkalis (essentially the opposite of acids). [PHYS 8.4]

**occur:** in Group I of the periodic table. [PHYS 8.4]

alloy

**is:** a material with characteristically metallic properties, formed from a combination of element_chemicalelements, of which at least one major constituent is itself a metal. Although specified by a chemical formula, its constituents do not form molecules that correspond to the chemical formula.

α–decay

**is:** the process in which a nucleus undergoes radioactive decay to form a less massive nucleus with the ejection of an alpha–particleα–particle, e.g. ^{238}_{92}U → ^{234}_{90}Th + ^{4}_{2}He (where ^{4}_{2}He denotes the α–particle). [PHYS 9.2]

**is a type:** of radioactive decay. [PHYS 9.2]

α–particle

**is:** a helium nucleus with positive charge 2*e* and relative atomic mass 4.0026. [PHYS 8.1, PHYS 9.1, PHYS 9.2]

**is ejected:** in radioactive α–decay. [PHYS 9.2]

**is denoted:** α or ^{4}_{2}He (or ^{4}_{2}He^{2+} since it is a helium atom stripped of its two electrons). [PHYS 8.1, PHYS 9.1, PHYS 9.2]

alternate angles

See transversal.

alternating current, a.c.

**is:** an electric current which changes magnitude and direction_of_a_vectordirection in a regular periodic way. [PHYS 5.4, PHYS 5.5]

**often is:** sinusoidal, i.e. it may be described by the formula *I*_{ }(*t*) = *I*_{0}_{ }sin(*ω* *t* + *ϕ*), where *I*_{0} is the peak value or amplitude of the current, *ω* is the angular frequency, *ϕ* is the phase constant and (*ωt* + *ϕ*) is called the phase of the current. [PHYS 5.4, PHYS 5.5]

**may also be described:** using complex quantities, so in the sinusoidal case $I(t) = {\rm Re}[\mathscr{I}_0\exp(\omega t + \phi)]$. [PHYS 5.5]

**more generally refers:** to other associated electrical quantities whose direction varies with time, e.g. a.c. voltage. [PHYS 5.4]

**is abbreviated:** AC at the beginning of a sentence, and a.c. elsewhere. [PHYS 5.4]

alternator

**is:** a device that generates an induced voltage of changing polarity by rotating a coil within a magnetic field. [PHYS 4.4]

**is also known:** as an alternating current (aca.c.) dynamo. [PHYS 4.4]

ammeter

**is:** an instrument for measuring electric current that is placed in series (connection)series with other circuit components through which the current to be measured flows. [PHYS 4.1]

**ideally has:** zero resistance, so that it does not affect the circuit to which it is connected. [PHYS 4.1]

amount of substance

**is:** a measure of the quantity of substance in a sample, expressed in terms of the number of basic entities (atoms, molecules, etc.) of the substance that are present in the sample.

**has as its SI unit:** the mole (mol).

ampere, A

**is:** the SI unit of electric current (i.e. rate of flow of electric charge), one of the seven base units. [PHYS 4.1]

**is defined:** as that constant current which, if maintained in each of two infinitely long, straight, parallel wires of negligible cross section, placed 1 metre apart, in a vacuum, will cause each wire to experience a force of magnitude 2 × 10^{−7} newton per metre of its length. [PHYS 4.3]

**is equivalent:** to the transfer of one coulomb per second, so 1 A = 1 C s^{−1}. [PHYS 4.1]

amplitude

**of:** an oscillation or a wave

**is:** the maximum magnitude of displacement from an equilibrium value. [PHYS 5.1, PHYS 5.5, PHYS 6.1]

**is represented:** by the constant *A* that appears in the general solution of the simple harmonic motion equation when written in the form *y* = *A*_{ }sin(*ωt* + *ϕ*). [MATH 5.1, PHYS 5.5, PHYS 5.6]

**also appears:** in similar equations such as that describing damped driven harmonic motion. [MATH 6.3, MATH 6.4]

**is exemplified:** by the maximum value of the pressure change caused by the passage of a sound wave. [PHYS 5.7]

angle

**is:** the inclination of one line with respect to another or, equivalently, the amount by which one line must be rotated about a point in order to align it with another line passing through the same point. [MATH 1.6]

**is commonly measured:** in degrees or radians. [MATH 1.6]

**is also called:** plane angle.

angle of contact

**in:** capillarity

**is:** the angle between a meniscus and a solid surface at their point of contact. [PHYS 7.6]

angle of deviation

**in:** geometrical optics and acoustics.

**is:** the angle through which a ray is turned, often by refraction on entering a different material or medium. [PHYS 6.3]

angle of dip

See angle of inclination.

angle of incidence

**in:** geometrical optics and acoustics.

**is:** the angle between the incident ray and the normal to the surface or interface at the point of incidence. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

angle of inclination

**is:** the angle between the (local) Earth’s magnetic field and the horizontal. [PHYS 4.2]

**is also called:** angle of dip. [PHYS 4.2]

angle of reflection

**in:** geometrical optics and acoustics.

**is:** the angle between the reflected ray and the normal to the surface or interface at the point of incidence. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

**is equal:** to the angle of incidence, according to the law of reflection. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

angle of refraction

**in:** geometrical optics and acoustics.

**is:** the angle between the refracted ray and the normal to the surface or interface at the point of incidence. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

**is related:** to the angle of incidence via Snell’s law (the law of refraction). [PHYS 5.7, PHYS 6.1, PHYS 6.2]

angular acceleration

**is:** the rate of change of angular velocity, either in magnitude or in direction or in both. [PHYS 2.7]

**is defined:** as **a**_{θ} = *d***ω**_{ }/*dt*. [PHYS 2.7]

**has as its SI unit:** rad s^{−2} (i.e. radian per second squared. [PHYS 2.7]

**can be represented:** if the direction of the angular velocity does not change, by the scalar quantity *a*_{θ} = *dω*_{ }/*dt*. [PHYS 2.7]

**can be determined:** if the angular acceleration *a*_{θ} is uniform, by *a*_{θ} = (*ω*_{2} − *ω*_{1})/*t*, where *ω*_{2} and *ω*_{1} are the angular speeds at the end and beginning respectively of the time interval *t*. [PHYS 2.7]

angular frequency

**of:** oscillatory motion

**is:** a measure of the rate at which complete oscillations are executed. [MATH 5.1, PHYS 5.5]

**is related:** to the frequency *f* of the oscillation by *ω* = 2*πf*. [PHYS 5.1, PHYS 5.4, PHYS 5.5, PHYS 5.6]

**is also related:** to the period *T* of the oscillation by *T* = 2*π*_{ }/*ω*. [PHYS 5.1, PHYS 5.5]

**has as its SI unit:** the hertz (Hz), where 1 Hz = 1 s^{−1} (i.e. per second). [PHYS 5.1, PHYS 5.4, PHYS 5.5]

**is represented:** by the parameter *ω* in the formula *y* = *A*_{ }sin(*ωt* + *ϕ*) that describes simple harmonic motion. [MATH 6.3, MATH 6.4, PHYS 5.1]

Compare with angular speed.

angular limit of resolution

See angular resolving power.

angular magnification

**is:** the ratio of the angle subtended at an observerobserver’s eye by an optical image, to the angle subtended by the object from which it is derived. [PHYS 6.4]

Compare with magnifying power.

angular momentum

**of:** a particle

**about:** a chosen origin O, from which the position of the particle is specified by the position vector **r**,

**is:** **L** = **r** **×** **p**, where **p** is the momentum of the particle. [PHYS 2.8]

**is also known as:** the moment of momentum of the particle about O. [MATH 2.7]

**of:** a collection of particles

**about:** a given point P

**is:** the (vector) sum of all the moments of momenta of each of the particles about P. [PHYS 2.8, PHYS 11.3]

**of:** a rigid body

**with:** angular velocity **ω** about a single fixed axis of rotation, and moment of inertia *I* about that axis

**is given by:** **L** = *I***ω**. [PHYS 2.8]

**has as its SI unit:** kg m^{2} s^{−1} (i.e. kilogram metre squared per second).

See conservation of angular momentum.

angular position

**of:** a particle in a plane, with respect to a point O, taken to be the origin of Cartesian coordinates in the plane

**is:** the angle *θ* between the particle’s position vector and the positive *x*–axis. [PHYS 2.7]

**equivalently is:** the polar angle of the point at which the particle is located, measured in a system of polar coordinates with origin at O.

angular probability density

**in:** Schrödinger modelSchrödinger’s model of the hydrogen atom

**is:** the factor |_{ }*Y*_{lm}_{ }(*θ*, *ϕ*)_{ }|^{2}, that arises in calculating the probability density |_{ }*Ψ*_{ }(*r*, *θ*, *ϕ*)_{ }|^{2}, where *Y*_{lm}_{ }(*θ*, *ϕ*) is the angular part of the wavefunction *Ψ*_{ }(*r*, *θ*, *ϕ*). [PHYS 11.3]

angular resolving power

**of:** an optical system

**is:** a measure of the systemsystem’s ability to produce or distinguish two separate images of two point–like objects which are, or appear to be, very close together. [PHYS 6.4]

**is defined:** as the minimum angular separation that the objects must have if their images are to satisfy the Rayleigh criterion. [PHYS 6.4]

**is limited:** by the ‘diffraction limit’ of the aperture of the optical system, which for a circular aperture of diameter *d* admitting light of wavelength *λ* is (1.22 radian)*λ*_{ }/*d*. [PHYS 6.4]

**is also known:** as the angular limit of resolution. [PHYS 6.4]

angular speed

**of:** a particle moving in a plane (taken to be the (*x*, *y*) plane) around a point O (taken to be the origin of the (*x*, *y*) plane) with an instantaneous angular position *θ* (measured between the particle’s position vector **r** and the positive *x*–axis)

**is:** the magnitude of the rate of change of *θ* with respect to time. That is, *ω* = |_{ }*d**θ*_{ }/*dt*_{ }|, where *θ* is normally measured in radians. [PHYS 2.6, PHYS 3.2]

**has as its SI unit:** rad1s^{−1} (i.e. radian per second). [PHYS 2.7, PHYS 5.1]

**is exemplified:** in the case of uniform circular motion about O, at constant speed *υ*, by the relation *ω* = *υ*_{ }/*r*. [PHYS 2.6]

**of:** a rigid body in uni–axial rotation about a single axis of rotation that is fixed in relation to the body

**is:** the (positive) angle swept out per second by a line drawn from the axis of rotation to any point in the body that is not on the axis. [PHYS 2.7]

See also instantaneous angular speed.

angular velocity

**of:** a particle (or rigid body) in uni–axial rotation about a single fixed axis of rotation

**is:** a vector, usually represented by the symbol **ω**, whose magnitude is the angular speed *ω*, about that axis, and whose direction is along the axis, in the sense given by the right–hand grip rule (i.e. if the fingers of the right hand are curled in the direction of rotation of the body, then the extended thumb points in the direction of the angular velocity). [PHYS 2.7, PHYS 2.8]

**satisfies:** the relation **υ** = **ω** **×** **r**, where **υ** is the velocity and **r** the position vector of the particle (or of any point in the rigid body) measured from an origin O on the axis of rotation. [PHYS 2.7]

**has as its SI unit:** rad s^{−1} (i.e. radian per second).

angular wavenumber

**for:** a periodic wave of wavelength *λ*

**is defined:** as *k* = 2*π*_{ }/*λ*. [MATH 6.4, PHYS 5.6]

**has as its SI unit:** m^{−1} (i.e. per metre). [MATH 6.4, PHYS 5.6]

**is widely referred to:** as the wavenumber, though this latter term is more properly reserved for *σ* = 1/*λ*, i.e. *k*_{ }/2*π*. [MATH 6.4, PHYS 5.6]

See also angular wave vector and compare with angular frequency. [MATH 6.4, PHYS 5.6]

angular wave vector

**is:** the generalization of the scalar angular wavenumber to a vector quantity which characterizes waves propagating in two or three dimensions. [PHYS 5.6]

**is equal:** in magnitude to 2*π*_{ }/*λ*. [PHYS 5.6]

**has direction:** parallel to the direction of propagation of the wave. [PHYS 5.6]

**usually is denoted:** by the symbol k. [PHYS 5.6]

**more commonly is referred to:** as the wave vector, or the propagation vector. [PHYS 5.6]

anharmonic oscillations

**are:** oscillations which are not simple harmonic motionsimple harmonic. [PHYS 5.1, PHYS 5.3]

**are characterized:** by a restoring force which is not proportional to the displacement, and a period which depends on amplitude. [PHYS 5.1, PHYS 5.3]

anharmonic oscillator

**is:** an oscillator which displays anharmonic oscillations. [PHYS 5.3]

anion

**is:** a negatively charged ion. [PHYS 8.4]

annulus

**is:** a region of a plane lying between two concentric circles. [MATH 2.1]

anode

**is:** an electrode connected to the positive terminal of a supply of electric current. (The term is used especially in the context of a discharge tube or a similar device.) [PHYS 8.1]

antidifferentiation

See inverse differentiation.

antilog

See antilogarithmic function.

antilogarithmic function

**is:** the function which undoes the effect of the log function *f*_{ }(*x*) = log_{a}(*x*), i.e. is the inverse function of log_{a}(*x*). [MATH 1.5]

**is given:** by *g*_{ }(*x*) = *a*^{x} (where *a* > 0), since *g*_{ }(*f*_{ }(*x*)) = *a*^{loga(x)} which by definition is simply equal to *x*. [MATH 1.5]

See also exponential function.

antinode

**in:** a standing wavestanding (stationary) wave

**is:** one of the positions where the maximum displacement from equilibrium occurs. [PHYS 5.6]

antiparallel (vectors)

**are:** two vectors which point in exactly opposite directions. [MATH 2.4, PHYS 2.2]

See also parallel (vectors)

antiparticle

**is:** a particle having the same rest mass as its partner particle but with its other attributes having the opposite sign. For example, the electron (a particle) and the positron (its antiparticle) have equal masses and opposite charges. [PHYS 9.2]

anti–phase

**is:** the condition in which two oscillations or waves of the same frequency have a phase difference of *π* (often referred to as *π* rad or 180°). The maxima of one disturbance then coincide with the minima of the other and vice versa, and the two oscillations or waves are totally out of step. [PHYS 6.1]

**is equivalently:** the condition of being totally out of phase. [PHYS 5.1]

anti–reflection coating

**is:** a thin transparent film applied to the surface of an optical component such as a lens in order to reduce (via interference) the amount of light which the surface reflects. [PHYS 6.1]

antisymmetric function

See odd function.

aperture

**of:** a lens or mirror

**is:** its effective size (usually expressed as a circular diameter).

**more generally is:** an opening or gap.

aperture stop

**in:** an optical system

**is:** the size of aperture which defines the amount of light entering the system. [PHYS 6.4]

apparatus

**is:** equipment used in a scientific experiment or investigation.

apparent depth

**of:** an object viewed by refraction at a plane surface

**is:** the depth below the surface at which the image appears to be. For near normal viewing the ratio of real depth to apparent depth is equal to the refractive index. [PHYS 6.2]

approximation

**of:** a number or quantity *y* by another number or quantity *x*

**is obtained:** when *x* and *y* have ’similar’ values. The term ’similar’ is not precisely defined but generally means that |_{ }*x* − *y*_{ }|/|_{ }*x*_{ }| is much less than 1. Such a relationship is shown by writing *x* ≈ *y*. [MATH 1.2]

**also refers:** to approximation of a real situation by a model. [PHYS 1.1]

**occurs:** in ‘orders’, as in a crude ‘first (order) approximation’ or a more accurate ’second (order) approximation’. [PHYS 8.3, PHYS 8.4]

**also occurs:** in ‘degrees’, as in the approximation of a function by a polynomial (such as a Taylor polynomial of degree *n*). [MATH 1.7, MATH 4.5]

See also numerical integration and numerical procedures for information on the approximation of definite integrals and roots of equations.

aqueous humour

**is:** the clear, watery fluid between the cornea and the lens of the eye. [PHYS 6.4]

arbitrary constants

**are:** constants that arise (as constants of integration) in the solution of differential equations. The general solution of an *n*^{th}–order linear, ordinary differential equation contains *n* independent arbitrary constants (which are also known as essential constants). [PHYS 5.5]

arc

**is:** a part of a curve.

**often specifically means:** a part of the circumference of a circle, though this should more properly be called a circular arc. [MATH 1.6, MATH 2.1]

arc length

**is:** a length measured along an arc. [MATH 1.6]

arccos

See inverse trigonometric functions, and the Maths For Science handbook.

arccosh

See inverse hyperbolic functions, and the Maths For Science handbook.

arccosec

See inverse trigonometric functions, and the Maths For Science handbook.

arccosech

See inverse hyperbolic functions, and the Maths For Science handbook.

arccot

See inverse trigonometric functions, and the Maths For Science handbook.

arccoth

See inverse hyperbolic functions, and the Maths For Science handbook.

Archimedes’ principle

**states:** that an object immersed in a fluid will experience a force due to the fluid which acts upward through the object’s centre of gravity, with a magnitude equal to the weight of the fluid which has been displaced by the object. [PHYS 7.6]

arcsec

See inverse trigonometric functions, and the Maths For Science handbook.

arcsech

See inverse hyperbolic functions, and the Maths For Science handbook.

arcsin

See inverse trigonometric functions, and the Maths For Science handbook.

arcsinh

See inverse hyperbolic functions, and the Maths For Science handbook.

arctan

See inverse trigonometric functions, and the Maths For Science handbook.

arctanh

See inverse hyperbolic functions, and the Maths For Science handbook.

area

**is:** a measure of the amount of surface within given closed boundaries.

See Table 11 in Section 2 of the Maths For Science handbook for the areas of particular shapes.

area between two graphs

**of:** the functions *f*_{ }(*x*) and *g*_{ }(*x*) which intersect at the points *x* = *a* and *x* = *b* (where *a* < *b*), and for which any other points of intersection lie between *x* = *a* and *x* = *b*

**is given:** by the integral $\displaystyle \int_a^b \left\lvert\,f(x)-g(x)\,\right\rvert\,dx$.

area under a graph

**of:** the function *f*_{ }(*x*) between *a* and *b*

**is:** a synonym, used in *FLAP*, for the definite integral of *f*_{ }(*x*) from *a* to *b*, namely $\displaystyle \int_a^b f(x)\,dx$ where *b* > *a*. [MATH 5.1,MATH 5.2]

**sometimes is referred to:** as the signed area since, for *b* > *a* it will be negative in any region where *f*_{ }(*x*) < 0. [MATH 5.2]

**can be identified:** in graphical terms, with the physical area enclosed by the curve representing *f*_{ }(*x*) the *x*–axis, and the lines *x* = *a* and *x* = *b* provided that *a* < *x* < *b* for all *x* satisfying *a* < *x* < *b* and the area is measured in the scale units that are appropriate to the graph in question. [MATH 5.1, MATH 5.2]

Argand diagram

**is:** a plane making use of Cartesian coordinates in which the *x*–axis represents the real part of a complex number and the *y*–axis represents the imaginary part. [MATH 3.1, PHYS 5.5]

argument (of a function)

**of:** a function (e.g. *f*_{ }(*x*))

**is:** the independent variableindependent variable(s) e.g. *x* whose value(s) determines the value of the function. [MATH 1.3]

argument (of a complex number)

**of:** a complex number in the polar form *z* = *r*_{ }(cos_{ }*θ* + *i*_{ }sin_{ }*θ*), or the exponential form *z* = *r*_{ }e^{i θ}

**is:** the value of *θ*. [MATH 3.2, PHYS 5.5]

**of:** a complex number in the Cartesian form *z* = *a* + *i*_{ }*b*

**may be:** any value of *θ* that satisfies the equations

$\sin\theta = \dfrac{b}{\sqrt{\smash[b]{a^2+b^2}}}$

$\cos\theta = \dfrac{a}{\sqrt{\smash[b]{a^2+b^2}}}$ [MATH 3.2, PHYS 5.5]

**usually is:** the particular value of *θ* (the principal value) that also satisfies the additional requirement −*π* < *θ* ≤ *π*. [MATH 3.2]

**is denoted:** by arg(*z*), though some authors use Arg(*z*) to indicate the principal value of arg(*z*). [MATH 3.2]

arithmetic

**pertains:** to the branch of mathematics concerned with numbers and their manipulation. [MATH 1.1]

arithmetic progression

**is:** a series of the form:

$\displaystyle \sum_{k=0}^{n-1}(a + kh) = a + (a + h) + (a + 2h) + \dots + [a + (n - 1)h]
= na + \dfrac{(n - 1)}{2}h$ where the constant, *h* is known as the common difference. [MATH 1.7]

arithmetic series

See arithmetic progression.

articulated body

**is:** a body of several defined, jointed parts, which otherwise can be treated as a rigid body. [PHYS 2.8]

aspheric lens

**is:** a lens whose surfaces are non-spherical. [PHYS 6.4]

aspheric surface

**is:** a non-spherical surface of a lens or mirror. [PHYS 6.4]

astronomical telescope

**is:** a telescope which produces a final image that is inverted. [PHYS 6.4]

asymptote

**is:** a straight line which a curve approaches but does not meet. [MATH 1.3]

**more precisely is:** a straight line related to a curve in such a way that there is at least one direction of travel along the curve in which the shortest distance between them decreases progressively as the distance from the origin to the point becomes very large. [MATH 4.4]

**more formally is:** a straight line which is the limit of the tangent to a curvetangents to a curve as the point at which those tangent to a curvetangents touch the curve tends to infinity. [MATH 2.2, MATH 2.3]

asymptotically

**means:** in the way that a curve approaches, but never meets, its asymptote. [MATH 1.3]

atmosphere, atm

**is:** a non-SI unit of pressure.

**is defined:** by 1 atm = 1.013 25 × 10^{5} N m^{−2}. [PHYS 7.2]

**is more properly called:** standard atmosphere.

**more generally is:** the layer of air above the Earth’s surface which exerts atmospheric pressure.

atmospheric pressure

**is:** the pressure due to the weight of the atmosphere. [PHYS 7.2]

**is not:** a constant, but varies with time and position. [PHYS 7.2]

**has a value:** at the Earth’s surface varying only by relatively small amounts. [PHYS 7.2]

**has as a useful unit:** the standard atmosphere (see atmosphereatmosphere, atm) defined by 1 atm = 1.013 25 × 10^{5} N m^{−2}. [PHYS 7.2]

atom

**is:** the basic building block of all normal solid, liquid or gas matter. [PHYS 7.1]

**is:** the smallest part of a chemical element that retains the fundamental chemical and physical properties of that chemical elementelement. [PHYS 8.1]

**extends:** over a diameter of approximately 10^{−10} m. [PHYS 7.1, PHYS 8.1]

**has:** a dense, positively charged central nucleus with a diameter of order 10^{−14} m composed of neutrons and positively charged protons, surrounded by a cloud of negatively charged electrons equal in number to the number of protons, according to nuclear modelRutherford’s nuclear model and all ‘realistic’ models ever since. [PHYS 8.1]

**has:** zero electrical charge overall. [PHYS 8.1]

Contrast with ion.

atomic force microscope

**is:** an instrument that measures the vertical displacement of a probe tip, with a diameter of a few nanometers, as it is moved across the surface of a material in such a way that the force it experiences remains constant. [PHYS 7.1]

**measures:** the profile of the surface with an approximate resolution of 10^{−10} m. [PHYS 7.1]

**can be used:** to build a three–dimensional representation of the distribution of atoms on the surface of a material. [PHYS 7.1]

atomic mass

**is:** the mass of an atom of a chemical element expressed in atomic mass units. It is approximately equivalent to the number of protons and neutrons in the atom (the mass number) or to the average number allowing for the relative abundances of different isotopes.

atomic mass unit, u

**is:** a non-SI unit of mass. [PHYS 8.1]

**is defined:** as one twelfth of the mass of one atom of the commonest carbon isotope ^{12}_{6}C, so the mass of one carbon–12 atom is exactly 12 u. According to current measurements, 1 u = 1.660154 × 10^{−27} kg (to six significant figures), or approximately 931 MeV/*c*^{2}. [PHYS 8.1, PHYS 9.1]

atomic number

**is:** the number of protons within the nucleus of an atom, usually denoted by the symbol *Z* [PHYS 7.1, PHYS 9.1]

**characterizes:** each chemical element uniquely, since the nuclear charge of each atom of a chemical element with atomic number *Z* is simply *Ze*. [PHYS 7.1, PHYS 8.1]

**also represents:** the number of electrons required to balance the nuclear charge in an atom, and therefore determines the chemical behaviour of the atom. [PHYS 8.1]

attenuation coefficient

**is:** a quantity *μ* that measures the rate of exponential decrease in intensity, *I*, of γ–radiation with distance, *x*, travelled through a material. [PHYS 9.2]

**is defined:** by *I* = *I*_{0}_{ }e^{−μx}. [PHYS 9.2]

**depends for its value:** on the material and on the energy of the γ–radiation photons. [PHYS 9.2]

auxiliary equation

**of:** the differential equation

$a\dfrac{d^2y}{dt^2}+b\dfrac{dy}{dt}+cy+0$

**is:** the quadratic equation *ap*^{2} + *bp* + *c* = 0. [MATH 5.5, MATH 6.3]

**has the significance:** that its root_of_an_equationroots, *p*_{1} and *p*_{2} appear in the general solution *B*_{ }exp(*p*_{1}*t*) + *C*_{ }exp(*p*_{2}*t*) of the differential equation. [MATH 5.5, MATH 6.3]

**may be generalized:** (with changed significance) to other differential equations with constant coefficients.

average

**means:** typical or representative, often describing a condition which, if it persisted, would have the same effect over a specified range as that of which it is an average.

**is often used:** as a synonym for mean.

average a.c. power

**of:** an a.c. circuit, or a part of such a circuit,

**is:** the total energy dissipated in one period of oscillation divided by the duration of that period. [PHYS 5.4]

**is given by:** $\langle P\rangle = V_{\rm rms}I_{\rm rms}\cos\phi$, where *V*_{rms} and *I*_{rms} are the root–mean-square values of the current *I* and potential difference *V* and *ϕ* is the phase difference between *I* and *V*.

**has as its SI unit:** the watt (W). [PHYS 5.4]

average acceleration

**over:** a time interval Δ*t*

**of:** a body moving in one dimension, along the *x*–axis

**is given most simply:** by the change of velocity Δ*υ*_{x} divided by the time interval Δ*t*. That is, $\langle a_x\rangle = \Delta\upsilon_x/\Delta t$. [PHYS 2.1]

**is given more specifically:** for a body moving with velocity *υ*_{x1} at time *t*_{1} and velocity *υ*_{x2} at time *t*_{2}, by

$\langle a_x\rangle = \dfrac{\upsilon_{x2}-\upsilon_{x1}}{t_2-t_1}$ [PHYS 2.1]

average angular speed

**over:** a time interval Δ*t*

**of:** a particle moving in a circle (whose centre is taken to be the origin)

**is:** the (positive) angle Δ*θ* swept out by the position vector of the particle divided by the time interval Δ*t*, i.e. $\langle\omega\rangle = \Delta\theta/\Delta t$. [PHYS 2.6]

average speed

**of:** the molecules in a gas with speed distribution *f*_{ }(*υ*)

**is obtained:** by dividing the sum of the speeds of all the molecules by the total number of molecules. [PHYS 7.5]

**is also obtained:** by evaluating the integral $\displaystyle \langle\upsilon\rangle = \int_0^\infty \upsilon f(\upsilon)\,d\upsilon$. [PHYS 7.5]

See applications of integration in the Maths For Science handbook.

average value of a function

**over:** the interval from *a* to *b*

**is defined:** as $\displaystyle f_{\rm av} = \dfrac{1}{(b-a)}\int_a^b f(x)\,dx$. [MATH 5.4]

average velocity

**over:** a time interval Δ*t*

**of:** a body moving in one dimension, along the *x*–axis

**is given most simply:** by the change of position Δ*x* divided by the time interval Δ*t*, i.e. $\langle\upsilon_x = \Delta x/\Delta t$. [MATH 4.1, PHYS 2.1]

**is given more specifically:** for a body with position *x*_{1} at time *t*_{1} and position *x*_{2} at time *t*_{2} by

$\langle\upsilon_x\rangle = \dfrac{x_2-x_1}{t_2-t_1}$ [MATH 4.1, PHYS 2.1]

**may be similarly expressed:** in terms of the displacement *s*_{x} from a fixed point, rather than the position *x*. [PHYS 2.1]

**of:** a body moving in three dimensions

**is given most simply:** by the change of position Δ**r** divided by the time interval Δ*t*, i.e. $\langle\upsilon\rangle = \Delta {\boldsymbol r}/\Delta t$. [PHYS 2.2]

**is given more specifically:** if the particle has position **r**_{1} at time *t*_{1} and position **r**_{2} at time *t*_{2}, by

$\langle{\boldsymbol\upsilon}\rangle = \dfrac{{\boldsymbol r}_2-{\boldsymbol r}_1}{t_2-t_1}$

So, writing **r**_{2} − **r**_{1} = Δ**r** = (Δ*x*, Δ*y*, Δ*z*),

$\langle{\boldsymbol\upsilon}\rangle = \left(\langle\upsilon_x\rangle,\langle\upsilon_y\rangle,\langle\upsilon_z\rangle\right) = \dfrac{\Delta{\boldsymbol r}}{\Delta t} = \left(\dfrac{\Delta x}{\Delta t},\,\dfrac{\Delta y}{\Delta t},\,\dfrac{\Delta z}{\Delta t}\right)$ [PHYS 2.2]

Avogadro’s constant

**is:** the physical constant *N*_{A} that represents the number of basic entities (atoms, molecules, ions etc.) per mole of any substance, [PHYS 7.1, PHYS 7.2]

**has:** the value *N*_{A} = 6.0223 × 10^{23} mol^{−1} (to five significant figures). [PHYS 7.1, PHYS 7.2]

Compare with Avogadro’s number (which has no units).

Avogadro’s hypothesis

**states:** that equal volumes of all gases at the same temperature and pressure contain the same number of atoms or molecules. [PHYS 7.1]

Avogadro’s number

**is:** the number of basic entities (atoms, molecules, ions, etc.) in one mole of any substance, namely 6.0223 × 10^{23} (to five significant figures). [PHYS 7.1, PHYS 7.2]

Compare with Avogadro’s constant (which is defined per mole, and consequently has units mol^{−1}).

axes

**are:** straight lines at an angle to one another, along which and from which we can measure the coordinates of a point. [PHYS 1.3]

**usually are:** Cartesian coordinate axes, which are at right angles to one another and which intersect at a common point called the origin. [MATH 1.3]

axial ray

**is:** a light ray which is coincident with the optical axis, before and after refraction or reflection. [PHYS 6.4]

axis of rotation

**of:** a rotating rigid body

**is:** the straight line connecting all parts of the body which are at rest. [PHYS 2.8]

bac cab rule

**is:** a mnemonic reference to the vector identity **a** **×** (**b** **×** **c**) = **b**_{ }(**a** **⋅** **c**) − **c**_{ }(**a** **⋅** **b**). [MATH 2.7]

balanced bridge

**is:** a bridge circuit whose electrical components are arranged so that there is no voltage between its output terminals. [PHYS 4.1]

balanced forces

**are:** two or more forces whose magnitude_of_a_vector_or_vector_quantitymagnitudes and directions are such that their net force or resultant force is zero. [PHYS 2.3]

ballistic galvanometer

**is usually:** a type of moving–coil galvanometer. [PHYS 4.4]

**is designed:** with a weak restoring force and a weak damping force, so that a transient current produces an initial swing whose amplitude is proportional to the total charge passed.

**is used:** to measure quantities of electric charge, and (in conjunction with a search coil) magnetic fields. [PHYS 4.4]

Balmer series

**is:** the set of visible lines in the spectrum of atomic hydrogen, whose wavelengths are given by Balmer’s formula. [PHYS 8.2]

Balmer’s formula

**is:** the formula discovered by Johann Balmer (1825–1898) which gives, to a very high accuracy, the wavelengths of the visible spectral lines emitted by atomic hydrogen:

$\lambda = 364.56\left[\dfrac{n^2}{n^2-4}\right]\,{\rm nanometres}$. [PHYS 8.2]

back e.m.f.

See induced voltage.

band theory

**is:** the proposal that the energy levels of electrons in (crystalline) solids are distributed in energy bands. [PHYS 11.4]

**is also:** the theoretical study of energy bands and their consequences. [PHYS 11.4]

bar

**is:** a non-SI unit of pressure.

**is defined:** as 1 bar = 10^{5} Pa (i.e. 10^{5} N m^{−2}). [PHYS 7.2]

**is slightly smaller:** than another non-SI unit of pressure, the standard atmosphere; 1 atm = 1·013 25 bar. [PHYS 7.2]

barrier penetration

See quantum tunnelling.

base (of a number system)

**of:** a system for specifying numbers

**is:** a number that takes on the role that 10 plays in the specification of decimal numbers. A base *n* system uses *n* digits and is based on power_mathematicalpowers of *n*. [MATH 1.2]

base (of a logarithm)

**is defined:** as the value of *a* in the identity *a*^{loga(x)} = *x*. [MATH 1.5]

**must be:** positive. [MATH 1.5]

**is most commonly:** e (the base of natural logarithms) or 10 (the base of common logarithms). [MATH 1.5]

basic differentiation

**is:** an informal term used to denote a range of mathematical skills in the area of differentiation. [MATH 4.2]

**includes:** the ability to differentiate ’standard’ functions such as sin(*kx*), cos(*kx*), exp(*kx*) and log_{e}(*kx*), together with constant multiples, sums, products and quotients of such functions. [MATH 4.2]

basic identities

**are:** a class of trigonometric identities. [MATH 1.6]

See trigonometric functions in the Maths For Science handbook for further details.

basic units

**are:** seven SI units. [PHYS 1.1]

**comprise:** the metre, kilogram, second, ampere, kelvin, mole and candela. [PHYS 1.1]

battery

**consists:** of two or more electric cells connected together to act as a single current source. (Colloquially, a single cell is also called a battery.) [PHYS 4.5]

beam

**is:** a collection of waves or particles travelling along closely parallel paths.

**is also:** a bundle of closely parallel rays.

beat frequency

**between:** two oscillations or waves of similar frequency that are superposed

**is:** the frequency of the (modulated) amplitude of the superposed waves. [PHYS 5.1]

**is equal:** to the difference between the frequencies of the two oscillations or waves. [PHYS 5.1, PHYS 5.3, PHYS 5.7]

**is also equal:** to the reciprocal of the beat period. [PHYS 5.7]

beat period

**is:** the time interval between successive beats in situations where two waves (e.g. sound waves) of slightly different frequency are superposed. [PHYS 5.7]

**is equal:** to the reciprocal of the beat frequency. [PHYS 5.7]

beating

**between:** two oscillations or waves of similar frequency that are superposed

**is:** the periodic variation of the total amplitude that gives rise to beats. [PHYS 5.1]

**occurs:** at the beat frequency. [PHYS 5.1]

beats

**are:** periodic variations in intensity due to beating. [PHYS 5.1, PHYS 5.7]

**are produced:** when two waves of nearly equal frequency and similar amplitude are superposed. [PHYS 5.1, PHYS 5.7]

becquerel, Bq

**is:** the SI unit of activity.

**is defined:** as an activity of 1 decay per second. **is related**: to a common non-SI unit, the curie (Ci), by 1 Ci = 3.70 × 10^{10} Bq. [PHYS 9.2]

β–decay (beta–decay)

**is:** the process in which a nucleus undergoes radioactive decay to form a less massive nucleus of a different element_chemicalelement with the emission of a β–particle. [PHYS 9.2]

**is classified:** in two types: β^{−}–decay and β^{+}–decay. [PHYS 9.2]

**if β**^{−}–decay, is: radioactive decay with the ejection of an electron (a β^{−}–particle) and an electron antineutrino, e.g. ^{15}_{6}C → ^{15}_{7}N + ^{ 0}_{−1}e + ν^{_}_{e}. A neutron in the original nucleus is transformed into a proton, an electron and an electron antineutrino: n → p + e^{−} + ν^{_}_{e}. [PHYS 9.2]

**if β**^{+}–decay, is: radioactive decay with the ejection of a positron and an electron neutrino, e.g. ^{11}_{6}C → ^{15}_{5}B + ^{ 0}_{+1}e + ν_{e}. A proton in the original nucleus is transformed into a neutron, a positron and an electron neutrino: p → n + e^{+} + ν_{e}. [PHYS 9.2]

β–particle

**is:** a particle that is emitted in β–decay. [PHYS 9.2]

**is classified:** in two types: the β^{−}–particle (an electron) which is emitted in beta_decayβ^{−}–decay, and the β^{+}–particle (a positron) which is emitted in beta_decayβ^{+}–decay. [PHYS 9.2]

biconcave lens

**is:** a lens having two surfaces which curve inwards into the material. The centre is thinner than the edges. [PHYS 6.3]

**often is called simply:** a concave lens. [PHYS 6.3]

biconvex lens

**is:** a lens having two surfaces which curve outwards from the material. The centre is thicker than the edges. [PHYS 6.3]

**often is called simply:** a convex lens. [PHYS 6.3]

bimetallic strip

**is:** a thermally sensitive device consisting of two thin strips of different metals soldered, or otherwise attached, face to face. [PHYS 7.2]

**bends:** with any change in temperature, since the extent to which the two metals expand in response to a given change of temperature will generally differ. [PHYS 7.2]

**can be used:** to measure temperature or as a means of temperature–sensitive control. [PHYS 7.2]

binding energy (of a nucleus)

**of:** a nucleus

**is:** the minimum energy required to break a nucleus apart into its free constituent nucleons. [PHYS 9.1]

**more generally is:** the minimum energy required to separate any system into appropriately specified components.

binding energy (of an electron)

**of:** an electron

**in:** an atom

**is:** the minimum energy required to remove the electron from the atom. [PHYS 8.1]

binomial coefficient

**is:** any one of the coefficients, ^{n}*C*_{r}, that arise in the binomial expansion. [MATH 1.7]

**is defined:** as

${}^nC_r = \dfrac{n!}{r!(n-r)!} = \dfrac{n(n-1)(n-2)\dots(n-r+2)(n-r+1)}{r(r-1)(r-2)\dots2\times1}$

(See factorial for the definition of *n*!) [MATH 1.7]

See summations and series in the Maths For Science handbook. where *n* ≥ *r*.

binomial expansion

**is:** a polynomial expression for (*a* + *b*)^{n}, where *n* is a positive integer:

$\displaystyle (a+b)^n = \sum_{k=0}^n {}^nC_{n-k}a^{n-k}b^k$

where ^{n}*C*_{r} is a binomial coefficient and Σ is the summation symbol. [MATH 1.7]

See binomial series, binomial theorem.

See also summations and series in the Maths For Science handbook.

binomial series

**is:** an infinite series for (1 + *x*)^{r}, where *r* is any real number and −1 < *x* < 1:

$(1+x)^r = 1 + \dfrac{rx}{1!} + \dfrac{r(r-1)x^2}{2!} + \dfrac{r(r-1)(r-2)x^3}{3!} + \dots$ [MATH 1.7]

**is equivalent:** to the corresponding binomial expansion when *r* is an integer.

See summations and series in the Maths For Science handbook.

binomial theorem

**is:** an alternative expression for the binomial expansion or the binomial series. [MATH 1.7]

bisection method

**for:** locating a root_of_an_equationroot of an equation

**works:** by constructing a sequence of intervals of decreasing length, such that the associated function changes sign on each interval. [MATH 1.4]

bisector

**is:** a line drawn in such a way that it cuts a specified angle into two equal parts. [MATH 2.1]

black body

**is:** an idealized object that absorbs all the electromagnetic radiation that falls upon it. [PHYS 8.2, PHYS 10.1]

**reflects:** absolutely no radiation. [PHYS 8.2, PHYS 10.1]

**is also:** an ideal emitter of radiation. [PHYS 8.2, PHYS 10.1]

**emits:** black–body radiation - which has a spectrum that depends only on the temperature of the black body. [PHYS 8.2, PHYS 10.1]

**has:** spectral brightness which is given by Planck’s function. [PHYS 7.3]

**is approximated roughly:** by a matt black surface. [PHYS 7.3]

**is approximated well:** by a cavity maintained at a well–defined temperature and connected to its environment by a small aperture. The spectrum of radiation inside such a cavity is described quite accurately by Planck’s function, and the radiation emitted from the small hole closely approximates black–body radiation irrespective of the material of the container or the state of its inner surface. [PHYS 7.3, PHYS 8.2, PHYS 10.1]

black–body radiation

**is:** electromagnetic radiation emitted by a black–body. [PHYS 8.2, PHYS 10.1]

**has:** a characteristic spectrum whose spectral brightness at wavelength *λ* is given by Planck’s function:

$R = \dfrac{2hc^2}{\lambda^5(\exp(hc/\lambda kT)-1)}$

with an overall shape, a wavelength for peak emission, and a total radiation_thermalradiated power per unit surface area all determined entirely by the temperature of the black–body. [PHYS 7.3, PHYS 8.2, PHYS 10.1]

**is also found:** within a cavity in thermodynamic equilibrium. [PHYS 8.2, PHYS 10.1]

**therefore can be realized in practice:** by using a cavity with a small aperture. [PHYS 8.2, PHYS 10.1]

black–body spectrum

See black–body radiation.

body

**is:** a collection of interacting particles which extends throughout a particular region of space.

Bohr model

**of:** the hydrogen atom

**is:** now supplanted but remains historically important as the first theoretical account of atomic structure to make use of quantum physics. [PHYS 8.2]

**was formulated:** by Niels Bohr (1885–1962) in 1913. [PHYS 8.2]

**postulates:** (1) that the negatively charged electron is held in a circular orbit around the positively charged nucleus by the Coulomb force between them; (2) that the range of allowable orbits is restricted by the requirement that the angular momentum of the orbiting electron is quantized in units of *h*_{ }/2*π*, where *h* is Planck’s constant; (3) that, contrary to classical physics, the orbiting electron does not continuously lose energy through the emission of electromagnetic radiation; (4) that electromagnetic radiation is emitted when the electron makes a transition from an initial orbit of energy *E*_{i} to a final orbit of energy *E*_{f} and that the frequency of that radiation is given by the Planck–Einstein formula as *f* = (*E*_{i} − *E*_{f})/*h*. [PHYS 8.2]

**explains:** many features of the spectrum of atomic hydrogen, including Balmer’s formula. [PHYS 8.2, PHYS 11.3]

**may be:** extended to atoms other than hydrogen, but only with limited success.

See Bohr orbit, Bohr radius, Bohr’s quantization conditionBohr’s quantization, Bohr’s quantum number.

Bohr orbit

**in:** the Bohr model of the hydrogen atom

**is:** one of the allowed orbits for the electron. An electron in such an orbit moves with a definite speed and has a constant energy; contrary to classical physics, it does not continuously emit electromagnetic radiation.

**corresponds:** to a definite energy level of the atom. [PHYS 8.2] [PHYS 8.2]

Bohr radius

**in:** the Bohr model of the hydrogen atom

**is:** the radius of the smallest Bohr orbit for the electron. [PHYS 8.2, PHYS 11.3]

**is given:** by $a_0 = \dfrac{\varepsilon_0h^2}{\pi m_{\rm e}e^2}$ where *ε*_{0} is the permittivity of free space, *m*_{e} is the mass of the electron, *e* the charge on the proton, and *h* is Planck’s constant. [PHYS 8.2]

**therefore is:** quantized by Bohr’s quantum number *n*. [PHYS 8.2]

Bohr’s quantization condition

**in:** the Bohr model of the hydrogen atom

**states:** that the magnitude_of_a_vector_or_vector_quantitymagnitude *L* of the angular momentum of the electron as it orbits the nucleus must be a positive integer multiple of Planck’s constant *h* divided by 2*π*. Thus:

$L = \dfrac{nh}{2\pi}$ for *n* = 1, 2, 3, ...

where *n* is called Bohr’s quantum number. [PHYS 8.2]

Bohr’s quantum number

**in:** the Bohr model of the hydrogen atom

**is:** an integer *n* that may take any positive value starting from 1, and which determines the (quantized) angular momentum magnitude_of_a_vector_or_vector_quantitymagnitude *L* of the electron in the *n*^{th} Bohr orbit around the nucleus (see Bohr’s quantization). [PHYS 8.2]

**also determines:** the radius of the *n*^{th} Bohr orbit and the associated energy level: *E*_{n} = −(13.6 eV)/*n*^{2} for *n* = 1, 2, 3, ... [PHYS 8.2]

boiling point

**of:** a liquid subjected to a specified external pressure (usually standard atmospheric pressure)

**is:** the temperature at which the saturated vapour pressure of the liquid is equal to the external pressure.

Boltzmann’s constant

**is:** the physical constant *k* that has the value *k* = 1.380 × 10^{−23} J K^{−1} (to four significant figures). [PHYS 7.5]

**is expressible:** in terms of two other physical constants, the molar gas constant *R* and Avogadro’s constant *N*_{A}, by *k* = *R*_{ }/*N*_{A}. From this, Boltzmann’s constant is seen to act as the gas constant per molecule. [PHYS 7.5]

**appears:** in equations which relate microscopic properties to macroscopic parameters of physical systems; e.g. in an ideal gas at temperature *T*, the mean translational kinetic energy of a particle is 3*kT*_{ }/2. [PHYS 7.5]

bond

**between:** atoms in a molecule or molecules in a solid (more particularly between specified states of those atoms or molecules)

**is:** a quantum physical phenomenon associated with a specific bonding energy that causes the atoms or molecules that have bonded to act as a single entity. [PHYS 11.4]

**is fundamentally explained:** by electromagnetic forces between the atoms or molecules. [PHYS 11.4]

**may be classified:** according to the number of electrons involved in maintaining the bond.

bonding electron

**is:** an electron involved in forming or maintaining a bond between atoms or molecules.

bonding energy

**is:** the minimum energy required to break a specific bond.

Born probability interpretation (hypothesis)

**is:** the association between the wavefunction of a particle and the probability of finding that particle in a given region of space at a particular time.

**states:** that if a particle moving in one dimension has the wavefunction Ψ(*x*,*t*), then the probability of finding the particle in a small region Δ*x* around *x* at time *t* is proportional to |_{ }*Ψ*_{ }(*x*,*t*)_{ }|^{2}_{ }Δ*x*. If the wavefunction is normalized, then the probability is equal to |_{ }*Ψ*_{ }(*x*,*t*)_{ }|^{2}_{ }Δ*x*. [PHYS 10.4, PHYS 11.1]

bound state

**of:** a quantum system

**is:** a state of a composite system in which a finite amount of energy is required to separate the components of the system.

**is:** in Schrödinger modelSchrödinger’s model of the hydrogen atom, a state in which the probability that the electron will escape infinitely far from the proton, is zero.

**is:** in the Bohr model of the hydrogen atom, any one of an infinite number of possible states, corresponding to the allowed Bohr orbits for the electron, each with its own definite energy corresponding to one of the energy levels. [PHYS 8.2, PHYS 11.3]

boundary conditions

**for:** differential equations

**are:** conditions which specify the value of the dependent variable or its derivatives, for specific values of the independent variable. [MATH 6.1, PHYS 11.1, PHYS 11.2]

**can be used:** to determine (or help to determine) any arbitrary constants that arise in the general solution of a differential equation. [PHYS 5.4, PHYS 5.5, PHYS 11.1, PHYS 11.2]

**often arise as:** conditions imposed on a wave at the boundary of a medium, usually involving the value of either the displacement of the medium or the derivative of the displacement with respect to position. [PHYS 5.6, PHYS 10.3]

Boyle’s law

**states:** that at constant temperature, the pressure *P* and volume *V* of a fixed amount of ideal gas are related by *PV* = constant. [PHYS 7.2]

See ideal gas equation of state.

brackets

**take the form:** (),[],or{}. [MATH 1.1]

**have a hierarchy:** {[( )]}. [MATH 1.1]

**are used:** to separate one part of an expression from the rest. In a calculation, the part of an expression enclosed in brackets must be evaluated before being combined with other terms. [MATH 1.1]

Bragg’s law

**for:** diffraction of monochromatic electromagnetic radiation of wavelength *λ*

**from:** parallel planes of atoms separated by a distance *d* in an orderly array of atoms (such as a crystal)

**determines:** the values of the angle *θ*, (measured between the incident ray and the plane of atoms) at which local maxima of intensity are formed in the diffraction pattern by constructive interference of the reflected rays from adjacent planes of atoms. [PHYS 7.1]

**is normally expressed:** as *nλ* = 2*d*_{ }sin_{ }*θ*, where *n* is an integer. [PHYS 7.1]

branchies

**of:** a hyperbola

**refers:** to the two separate parts of a hyperbola that are produced when a plane intersects a double cone. [MATH 2.3]

breaking point

**of:** a given material

**is:** the point on the loading curve of the material at which the material breaks apart. [PHYS 7.6]

**corresponds:** to the maximum tensile stress that the material can sustain. [PHYS 7.6]

breeder reactor

**is:** a nuclear fission reactor whose reaction products include material that can be used as fuel for further reactions. [PHYS 9.3]

bremsstrahlung

**is:** the electromagnetic radiation emitted when an electrically charged particle is accelerated, in particular, when it is slowed down. For example, when high- energy electrons collide with a target, X–rays are produced with a continuous spectrum. [PHYS 8.3]

**linguistically is:** German for ‘braking radiation’. [PHYS 8.3, PHYS 10.1]

bridge circuit

**is:** a circuit consisting of four electrical components (generally resistors) connecting four points (A, B, C, D, say) to form a closed loop. [PHYS 4.1]

**produces:** an output voltage between two non–adjacent points (A and C say) when a voltage source is connected across the other two points (B and D). [PHYS 4.1]

**is used:** to compare resistances. [PHYS 4.1]

**is balanced:** when the resistances are such that the output voltage is zero.

bridge circuit balance condition

**is:** the equation which relates the four resistances in a balanced bridge circuit. [PHYS 4.1]

brittleness

**is:** the property of a material which causes it to fracture without appreciable plasticity, before or soon after the elastic limit is reached. [PHYS 7.6]

Brownian motion

**is:** the microscopic random motion of pollen grains and other small particles suspended in gases or liquids. [PHYS 7.1, PHYS 8.1]

**was first observed:** by the botanist Robert Brown (1773–1858). [PHYS 7.1, PHYS 8.1]

**was explained:** as the result of numerous unseen molecular collisions, by Albert Einstein (1879–1955) in 1905. [PHYS 7.1, PHYS 8.1]

bulk modulus

**of:** a material (solid, liquid or gas)

**is:** an elastic modulus, conventionally denoted *K*. [PHYS 5.7, PHYS 7.6]

**is defined:** as the ratio of the applied volume stress *σ*_{vol} to the resulting volume strain *ε*_{vol}:

$K = \dfrac{\sigma_{\rm vol}}{\varepsilon_{\rm vol}} =
\dfrac{-\text{pressure change}}{\text{fractional volume change}}$ (note the − sign) [PHYS 5.7]

**has as its SI unit:** N m^{−2} or Pa (i.e. pascal). [PHYS 5.7, PHYS 7.6]

buoyancy

**is:** the phenomenon by which a fluid tends to reduce the apparent weight of a body through the buoyancy force. [PHYS 7.6]

buoyancy force

**is:** the vertical upward force exerted on a body by a static fluid in which it is submerged or floating. [PHYS 7.6]

**is quantified:** by Archimedes’ principle. [PHYS 7.6]

**is also called:** the upthrust. [PHYS 7.6]

caesium atomic clock

**is:** a device that uses an atomic resonance in caesium of very narrow resonance absorption bandwidth and very high *Q*–factor, to provide a time or frequency standard. [PHYS 5.3]

**is used:** to establish the SI unit of time, the second. [PHYS 5.3]

calculation

**is:** a sequence of mathematical operations performed with the objective of determining the answer to a question.

calculus

**is:** a branch of mathematics which is concerned with the way in which (small) changes in one quantity determine or are determined by changes in related quantities. [MATH 4.1, PHYS 2.1]

**is more properly called:** infinitesimal calculus. [MATH 4.1, PHYS 2.1]

**includes:** differentiation and integration. [MATH 4.1, PHYS 2.1]

calibration

**is:** the process of checking one measuring instrument against another, more accurate one. [PHYS 1.1]

calibration points

**for:** a thermometer

**are:** two or more fixed points which can be used to calibrate the scale of the thermometer. [PHYS 7.2]

**are usually:** triple points or boiling_pointboiling or freezing points. In the case of the Kelvin temperature scale one of the two points is the unattainable absolute zero, the other is the triple point of H_{2}O. [PHYS 7.2]

**permit between or beyond them:** interpolations or extrapolations, often using polynomial thermometric relations. [PHYS 7.2]

calorimeter

**is:** a container of known heat capacity used in calorimetry experiments. [PHYS 7.4]

calorimetry

**is:** the branch of physics concerned with the measurement of heat and its effects. [PHYS 7.4]

camera

**is:** a device for producing a record of an image, either on photographic film or via some other means (e.g. electronic). [PHYS 6.4]

See also pinhole camera. [PHYS 6.4]

cancelling

**is:** a term used to describe the mathematical process in which (a) a factor appearing on both sides of an equation is eliminated by dividing both sides of the equation by that factor; or (b) a factor appearing in both the numerator and denominator of a fraction (arithmetic or algebraic) is eliminated by dividing both the numerator and the denominator by that factor. [MATH 1.1]

candela, cd

**is:** the SI unit of luminous intensity, one of the seven base units. (Not used in *FLAP .*)

capacitance

**of:** an isolated electrical conductor

**is:** the ratio of the charge *q* stored on the electrical_conductorconductor, to the potential difference *V* between it and some selected reference point. [PHYS 4.5]

**is given:** by *C* = *q*_{ }/*V*. [PHYS 4.5, PHYS 5.5]

**more generally is:** the charge stored between two points per unit potential difference between those points.

**is exemplified:** by the capacitance between the terminals of a capacitor, which for parallel plates of area *A* separated by a dielectric with permittivity *ε* and thickness *d* is *C* = *εA*_{ }/*d*. [PHYS 4.5]

**has as its SI unit:** the farad, (F), where 1 F = 1 C V^{−1}. [PHYS 4.5, PHYS 5.5]

capacitive reactance

**of:** a capacitor with capacitance *C*, when passing alternating current of angular frequency *ω*

**is:** the ratio of the peak voltage to the peak current *V*_{0}/*I*_{0}.

**is given:** by *X*_{C} = 1/*ωC*. [PHYS 5.4, PHYS 5.5]

**has as its SI unit:** the ohm (Ω). [PHYS 5.4, PHYS 5.5]

See complex capacitive reactance, impedance, reactance. [PHYS 5.4, PHYS 5.5]

capacitive time constant

**is:** the time for the current, charge or voltage across a capacitor to decay exponentially by a factor e. [PHYS 4.5]

**is given:** for a circuit in which a capacitor of capacitance *C* discharges through a resistance *R*, by *τ* = *RC*. [PHYS 4.5]

capacitor

**is:** a device for storing electric charge. [PHYS 4.5, PHYS 5.5]

**usually consists:** of two parallel metal surfaces (not necessarily flat) separated by a dielectric. [PHYS 4.5]

**generally has:** in practical electronic circuits, a capacitance very much less than 1 farad (1 F) so its capacitance might well be expressed in microfarad (μF) or picofarad (pF).

capillarity

**is:** the elevation or depression of the surface of a liquid in contact with a solid due to the relative attraction of the liquid molecules for each other as compared to their attraction to those of the solid. [PHYS 7.6]

See meniscus.

capillary

**is:** a tube of narrow internal diameter. [PHYS 7.6]

See capillarity.

Carnot cycle

**in:** thermodynamics

**is:** a reversible closed cycle consisting of four steps, two isothermal processes linked by two adiabatic processes. [PHYS 7.4]

See Carnot engine.

Carnot engine

**is:** a reversible heat engine that utilizes the Carnot cycle. [PHYS 7.4]

**has:** efficiency *η* = 1 − *T*_{cold}/*T*_{hot} when operating between temperatures *T*_{hot} and *T*_{cold}. (Any reversible heat engine operating between those temperatures must have the same efficiency.) [PHYS 7.4]

**is:** the most efficient possible heat engine operating between two fixed temperatures. [PHYS 7.4]

Cartesian axes

See Cartesian coordinate system.

Cartesian component vectors

**of:** a vector *υ*, with respect to a given Cartesian coordinate system

**are:** the vectors *υ*_{x}, *υ*_{y}, *υ*_{z} directed along the Cartesian axes such that *υ*_{x} + *υ*_{y} + *υ*_{z} = *υ*. [MATH 2.5]

**are therefore:** individually proportional to the corresponding Cartesian unit vectors **i**, **j** and **k**. [MATH 2.5]

**should not be confused with:** the Cartesian scalar components (*υ*_{x}, *υ*_{y}, *υ*_{z}) of **υ** which are individually the scalars by which a Cartesian unit vector must be scaling_of_a_vectorscaled to produce the corresponding component vector (e.g. **υ**_{x} = *υ*_{x}_{ }**i**). [MATH 2.5]

Cartesian coordinate system

**is:** a coordinate system that uses Cartesian coordinates, measured along mutually perpendicular axes from a point of common intersection called the origin. In three dimensions, the three axes are conventionally labelled as the *x*–axis, *y*–axis and *z*–axis, and it is conventional to perform the labelling so as to produce a right–handed coordinate system rather than a left–handed coordinate system.

**can be generalized:** to any number of dimensions.

Cartesian coordinates

**are:** coordinates measured from a common origin along axes that intersect (at the origin) at right angles. The horizontal axis normally is used to represent values of *x*. In two dimensions, the vertical axis is used to represent values of *y*. In three dimensions, the second horizontal axis is used to represent values of *y*, and the vertical axis to represent values of *z*. The convention is to refer to these as the *x*–axis, *y*–axis and *z*–axis. [MATH 1.3, MATH 2.2]

**can be used:** for any number of dimensions. [MATH 1.3, MATH 2.2]

Cartesian form (of a complex number)

**represents:** a complex number as *a* + *i*_{ }*b* with *a* and *b* real. [MATH 3.2]

Compare and contrast with exponential form and polar_form_of_a_complex_numberpolar form, and see complex numbers in the Maths For Science handbook for the relationship between these forms.

Cartesian form (of a vector)

**is:** the form in which a vector **υ** is represented as a vector sum of Cartesian component vectors: **υ** = **υ**_{x} + **υ**_{y} + **υ**_{z} or of scaled Cartesian unit vectors: **υ** = **υ**_{x}_{ }**i** + **υ**_{y}_{ }**j** + **υ**_{z}_{ }**k** or, equivalently, as an ordered triple of Cartesian scalar components: **υ** = (*υ*_{x}, *υ*_{y}, *υ*_{z}). [MATH 2.5]

See scalars and vectors in the Maths For Science handbook.

Cartesian representation (of a complex number)

See cartesian_form_of_a_complex_numberCartesian form.

Cartesian scalar components

**of:** a vector **υ**

**are:** the scalar quantities *υ*_{x}, *υ*_{y} and *υ*_{z} which appear in the expression for **υ** when given in the cartesian_form_of_a_vectorCartesian form *υ*_{x}_{ }**i** + *υ*_{y}_{ }**j** + *υ*_{z}_{ }**k**. [MATH 2.5]

**are individually equal:** to the projection of **υ** onto the corresponding Cartesian unit vector, so *υ*_{x} = **υ** **⋅** **i**, etc. [MATH 2.6]

Cartesian sign convention

**in:** optics

**is:** a sign convention which takes the pole of a surface or the centre of a lens as the origin of a Cartesian coordinate system, ascribing positive signs to positions measured to the right or upwards, and negative signs to positions measured to the left or downwards of this origin. [PHYS 6.3]

Cartesian unit vectors

**are:** unit vectors in the mutually perpendicular directions of the Cartesian coordinate axes. Two such vectors are required in two dimensions, usually denoted by **i** and **j** in the directions of the *x*–axis and *y*–axis respectively. In three dimensions the three unit vectors are usually denoted **i**, **j** and **k**. [MATH 2.5]

cathode

**of:** a discharge tube or a similar device

**is:** an electrode connected to the negative terminal of a supply of electric current. [PHYS 8.1]

cathode rays

**are:** the ‘rays’ emanating from the cathode of a discharge tube containing gas at a sufficiently low pressure. [PHYS 8.1]

**are in fact:** high–speed flows of electrons, as shown by J.J. Thomson (1856–1940). [PHYS 8.1]

cation

**is:** a positively charged ion. [PHYS 8.4]

caustic curve

**is:** the curve formed by the superposition of rays from a lens or mirror which suffers from spherical aberration. [PHYS 6.4]

Cavendish’s experiment

**is:** an experiment to determine Newton’s universal gravitational constant, *G*, first performed by Henry Cavendish (1731–1810) in 1798. [PHYS 3.2]

cell

See electric cell, and (in the context of crystals) unit cell.

Celsius temperature scale

**is:** a nearly centigrade temperature scale which tracks the Kelvin temperature scale precisely. [PHYS 7.2]

**is defined:** in terms of the Kelvin temperature scale by *T*_{C}/°C = *T*_{ }/K − 273.15, where *T* is an absolute temperature and *T*_{C} is the corresponding Celsius temperature. [PHYS 7.2]

**is named:** after Anders Celsius (1701–1744). [PHYS 7.2]

centigrade

**is:** the description given to a temperature which is measured on a centigrade temperature scale. [PHYS 7.2]

centigrade temperature scale

**is:** any temperature scale based on a thermometric property *X* that uses a thermometric relation of the form

$T_{\rm cen} = \dfrac{X - X_0}{X_{100}-X_0} \times 100°{\rm centigrade}$ [PHYS 7.2]

**will agree:** with another centigrade scale at any fixed points (normally the freezing_pointfreezing and boiling points of water) that are common to both scales, but will not necessarily agree at any other points because different physical properties *X* may vary differently with temperature. [PHYS 7.2]

central force

**is:** a force that is always directed towards a fixed point (sometimes called the force centre) and which has the property that its magnitude_of_a_vector_or_vector_quantitymagnitude depends only on the distance from that point. [PHYS 2.4, PHYS 2.7, PHYS 2.8]

centre

**of:** a circle (or sphere)

**is:** the unique point that is at the same distance from every point on the circumference (or surface). [MATH 2.1, MATH 2.3]

**is also:** the unique point at which any two different diameters intersect. [MATH 2.1]

**is more generally:** the mid–point of a body or system.

centre of gravity

**of:** a rigid body in a gravitational field

**only exists:** if the gravitational field is uniform, or if the body has a sufficiently high degree of symmetry.

**is:** the point (fixed with respect to the body, but not necessarily within the body) at which the entire mass of the body can be considered to be concentrated for the purpose of determining the effect of gravitational forces on the body. [PHYS 2.3, PHYS 2.7]

**is therefore:** the point about which the gravitational forces produce no resultant torque irrespective of the orientation of the body. [PHYS 2.7]

**is determined:** by the gravitational forces acting on the body as well as the distribution of mass within the body, but will always coincide with the centre of mass for a body in a uniform gravitational field.

centre of mass

**of:** a rigid body

**is:** the point (not necessarily within the body) at which the entire mass of the body can be considered to be concentrated for the purpose of determining the translational motion of the body under an applied force. If the body is entirely free to move and the line of action of the force passes through the centre of mass, that force will cause translation of the centre of mass but not rotation about the centre of mass. [PHYS 2.3, PHYS 2.7, PHYS 2.8]

**is determined:** by considering the body to consist of (infinitesimal) mass elements Δ*m*_{i} at positions **r**_{i} and then finding the point specified by **r**_{c}, such that

$\displaystyle {\boldsymbol r}_c = \dfrac{\sum_i \Delta m_i{\boldsymbol r}_i}{\sum_i \Delta m_i}$ [MATH 5.4]

**should not be confused:** with the centre of gravity.

centrifugal force

**is:** a fictitious force with no physical basis in fact, invented to allow Newton’s laws of motion to be applied in a rotating frame of reference, which is a non-inertial frame of reference where Newton’s laws are otherwise invalid. [PHYS 2.3]

centripetal acceleration

**of:** a particle in uniform circular motion

**is:** an acceleration directed towards the centre of the circle

**has magnitude:** *rω*^{2}, where *r* is the radius of the circle and *ω* is the particleparticle’s angular speed. [PHYS 2.6, PHYS 3.2]

centripetal force

**on:** a particle in uniform circular motion

**is:** the force which is necessary to maintain the uniform circular motion. [PHYS 2.3]

**is directed:** towards the centre of the circle. [PHYS 2.3]

**has magnitude:** *mrω*^{2}, where *r* is the radius of the circle, *m* is the mass of the particle and *ω* is its angular speed. [PHYS 2.6, PHYS 3.2]

chain rule

**is:** a rule used for differentiating a function of a function, such as *f*_{ }(*g*_{ }(*x*)). The rule states that if *u* = *g*_{ }(*x*) and *y* = *f*_{ }(*u*) so that *y* = *f*_{ }(*g*_{ }(*x*)) then

$\dfrac{dy}{dx} = \dfrac{dy}{du}\times\dfrac{du}{dx} = f'(u)\times g'(x)$ [MATH 4.3]

See the chain rule and its uses in the Maths For Science handbook.

change of phase

**is:** a process in which a substance changes from the solid phase, liquid phase or gaseous phase to one of the others. [PHYS 7.3]

changing the (dependent) variable

**is:** a technique used to transform a first–order differential equation into one that can be solved by a standard method such as separation of variables or use of an integrating factor. A new dependent variable is defined as an appropriate function of the old dependent variable and the independent variable. [MATH 6.2]

chaos

**is:** a property exhibited by deterministic systems which are described by certain non–linear differential equations (or sets of non–linear equations).

**occurs:** when two systems governed by the same non–linear differential equation but with slightly different initial states subsequently develop in completely dissimilar ways. [MATH 6.1]

characteristic emission spectrum

**of:** a chemical element

**is:** the emission spectrum from that chemical element and is unique to that chemical element. [PHYS 8.2]

**often contains:** prominent emission lines and is then referred to as the emission line spectrum of the chemical element. [PHYS 8.2]

characteristic X–ray spectrum

**of:** a heavy atom

**is:** the characteristic emission spectrum in the X–ray wavelength range from that atom and is unique to that kind of atom. [PHYS 8.3]

**is produced:** when an ejected inner shell (comparatively low energy) electron is replaced by an outer shell (comparatively high energy) electron, provided that the spacing between the energy levels is at least several thousand electronvolts. [PHYS 8.3]

charge

See electric charge.

charge carriers

**are:** mobile charged particles (e.g. electrons and ions) that can move within a material. [PHYS 4.1]

See hole.

charge sharing

**is:** the process by which a body can be charged by receiving some of the charge from another charged object with which it makes contact. [PHYS 3.3]

charge–to–mass ratio

**for:** a particle of charge *q* and mass *m*

**is equal:** to *q*_{ }/*m*. In the case of the electron the quantity *e*_{ }/*m*. is often referred to as the charge–to–mass ratio, even though the charge of the electron is actually *q* = −*e*. [PHYS 8.1]

Charles’ law

**states:** that at constant pressure, the volume *V* and absolute temperature *T* of a fixed quantity of ideal gas are related by *V*_{ }/*T* = constant. [PHYS 7.2]

chemical bonding

**is:** the binding together of chemical elements by forces that are fundamentally electromagnetic_forceelectromagnetic. [PHYS 8.4]

See covalent bonding, ionic bonding, metallic bonding.

chemical compound

**is:** a substance that consists of more than one element_chemicalelement, the atoms being bound together in a fixed ratio that is characteristic of the substance. [PHYS 8.1]

**is also:** a substance which can be broken down into more elementary substances by a process such as heating or the passing of an electric current. [PHYS 7.1]

chemical element

**traditionally is:** a substance which cannot be divided or separated by chemical means, including heating and passing of electrical current. [PHYS 8.1]

**currently, is more appropriately defined:** as matter consisting of atoms characterized by a single atomic number *Z*, which consequently contain a definite number of protons. (The atoms may be bound together to form molecules, as in the case of the diatomic oxygen molecule O_{2}.) [PHYS 7.1, PHYS 8.1]

chemical formula

**is:** a formula such as H_{2}O which uses chemical symbols to indicate the chemical elements involved in a chemical compound and subscripts to show the relative numbers of atoms of those chemical elements. [PHYS 8.1]

chemical reaction

**is:** a process in which bonds between atoms and molecules are made or broken with the result that materials are transformed.

chemical symbol

**is:** a symbol consisting of one or two letters that may be used to represent the name of a chemical element. The first letter is always upper case while the second, if there is one, is lower case. [PHYS 8.1]

**is exemplified:** by H for hydrogen, He for helium and Na for sodium. (See a copy of the periodic table for a complete list.) [PHYS 8.1]

chord

**is:** a straight line that cuts a curve at two points. [MATH 2.1, MATH 4.1, MATH 4.2]

chromatic aberration

**is:** aberration caused by the variation of the focal length of a lens with wavelength, as a result of dispersion. [PHYS 6.4]

**appears as:** coloured fringes seen on images. [PHYS 6.4]

ciliary muscles

**make up:** the ring of muscles surrounding the lens of the human eye. The focal length of the lens is changed as these muscles contract or relax. [PHYS 6.4]

circle

**of:** radius *R*

**centred:** on a point P

**is:** the locus of all points in a plane that are located at a distance *R* from P. [MATH 2.1, MATH 2.3, PHYS 3.2]

See equation of a circle.

circle of least confusion

**is:** the minimum but finite image size of a point object, which results from spherical aberration − in which the focal length of the lens varies with the radial distance of rays from the optical axis. [PHYS 6.4]

circuit

**is:** a continuous closed pathway, or network of pathways, along which electric charge may flow. [PHYS 4.1, PHYS 5.5]

circuit components

**is:** a general term for any of the many devices (e.g. capacitors, inductors, resistors) that might form part of a circuit.

circular

**in:** geometry

**means:** pertaining to a circle. [MATH 2.1]

circumference (of a circle)

**is (1):** the distance 2*πR* around a circle of radius *R*. [MATH 2.1]

**is (2):** the circle itself (as in ‘a point on the circumference’). [MATH 2.1]

See also perimeter.

classical mechanics

See Newtonian mechanics.

classical physics

**is:** that part of physics which includes and builds on Newtonian mechanics, Maxwell’s theory of electromagnetism, the laws of thermodynamics and (usually) relativity, but which specifically excludes quantum physics. [MATH 6.4, PHYS 10.2]

Clausius–Clapeyron equation

**is:** an equation that relates the slope (*dP*_{ }/*dT*) of the boundary curve between two phases on a *P*−*T* diagram to the latent heat (*ml*) and change of volume (Δ*V*) involved in an isothermal crossing of that boundary at temperature *T*:

$\dfrac{dP}{dT} = \dfrac{ml}{T\Delta V}$ [PHYS 7.4]

closed cycle

**is:** any succession of processes (which may be reversible or irreversible) which restores a system to its initial state. [PHYS 7.4]

codomain (of a function)

**of:** a function

**is:** that set within which can be found the range of values of the dependent variable which are generated by the function over its domain. [MATH 1.3]

coefficient

**is:** any one of the constants, *a*_{0}, *a*_{1}, *a*_{2}, ... *a*_{n−1} and *a*_{n}, that appear in a polynomial expression of the form *a*_{0} + *a*_{1}*x* + *a*_{2}*x*^{2} + ... + *a*_{n−1}*x*^{n−1} + *a*_{n}*x*^{n}. [MATH 1.3, MATH 1.4]

**is exemplified:** by the coefficient of *x*^{3} in *x*^{4} − 5*x*^{3} − *x*^{2} + 4*x* + 2, which is −5.

coefficient of friction

See coefficient of sliding friction, coefficient of static friction.

coefficient of mutual inductance

**of:** a pair of coils (or circuits), or of a transformer,

**is:** the quantity *M* that relates the magnitude of the induced voltage in one coil to the rate of change of current, *dI*_{2}/*dt*, in the other coil, through the equation

$V_1 = M\left\lvert\,\dfrac{dI_2}{dt}\,\right\rvert$. [PHYS 4.4]

**has as its SI unit:** the henry (H), where 1 H = 1 V s A^{−1}.

**often is abbreviated:** to mutual inductance. [PHYS 4.4]

See also ‘mutual induction’. [PHYS 4.4]

coefficient of self inductance

**of:** a coil (or circuit)

**is:** the quantity *L* that relates the magnitude of the self induced voltage *V*_{ind} in the coil to the rate of change of current *dI*_{ }/*dt* in the coil, through the equation

$V_{\rm ind} = L\left\lvert\,\dfrac{dI}{dt}\,\right\rvert$. [PHYS 4.4, PHYS 4.5]

**has as its SI unit:** the henry (H), where 1 H = 1 V s A^{−1}. [PHYS 4.4, PHYS 4.5]

**often is abbreviated:** to self inductance or inductance. [PHYS 4.4, PHYS 4.5]

See also self–induction and inductance.

coefficient of sliding friction

**for:** an object sliding over a solid surface

**is denoted:** by *μ*_{slide}. [PHYS 2.3]

**is:** the ratio of the magnitude of the sliding frictional force on the object to the magnitude *R* of the reaction force on the object. [PHYS 2.3]

**depends:** on the surfaces involved and their state of lubrication. [PHYS 2.3]

**is largely independent:** of other factors, including the area of contact and the speed of the object. [PHYS 2.3]

**usually is:** smaller than the coefficient of static friction *μ*_{static}. [PHYS 2.3]

coefficient of static friction

**for:** an object on a solid surface, being prevented by friction from moving

**is denoted:** by *μ*_{static}. [PHYS 2.3, PHYS 2.6]

**is:** the ratio of the magnitude of the maximum frictional force on the object before it moves, to the magnitude *R* of the reaction force on the object. [PHYS 2.3, PHYS 2.6, PHYS 7.6]

**depends:** on the surfaces involved and their state of lubrication. [PHYS 2.3, PHYS 2.6]

**is largely independent:** of other factors, including the area of contact. [PHYS 2.3, PHYS 2.6]

**usually is:** larger than the coefficient of sliding friction *μ*_{slide}. [PHYS 2.3, PHYS 2.6]

coefficient of thermal conductivity

**of:** a substance (under strictly specified conditions of temperature and pressure)

**is:** the quantity *κ* that describes the relative ease with which heat is transferred through the material between points at different temperatures. [PHYS 11.4]

**is defined:** as *κ* in the relation (a special case of Fourier’s law)

$\dfrac{dQ}{dt} = -\kappa A\dfrac{T_2-T_1}{l}$

where *dQ*_{ }/*dt* is the rate of flow of heat along a well insulated bar of length *l* and uniform cross–sectional area *A* from an end at temperature *T*_{1} to an end at temperature *T*_{2}. [PHYS 11.4]

**has as its SI unit:** W m^{−1} K^{−1}.

See conduction (of heat) and Fourier’s law.

coefficient of viscosity

**is:** the quantity *η* that describes the relative difficulty with which a fluid may flow. [PHYS 7.6]

**is defined:** by the relation (a special case of Newton’s law of viscosity)

$\sigma_x = -\eta\dfrac{d\upsilon_x}{dy}$

where *σ*_{x} is the shear stress applied in a given direction, *dυ*_{x}/*dy* is the velocity gradient in a perpendicular direction, and the minus sign indicates that the velocity decreases with distance from the plane over which the shear stress is applied. [PHYS 7.6]

**is sometimes called:** the viscosity of the fluid. [PHYS 7.6]

**has as its SI unit:** kg m^{−1} s^{−1}, or equivalently N s m^{−2} or Pa s.

coherence

**between:** waves

**is:** the property that enables phase differences known at one position or time to determine phase differences at other positions and times. [PHYS 5.3, PHYS 6.1]

coherent

**describes:** two waves related in such a way that knowing the phase of one at some particular time and position enables the phase of the other to be predicted at some position (if spatially coherent) or time (if temporally coherent) [PHYS 6.1]

**may also be applied:** in its temporal sense, to two oscillations. [PHYS 5.3]

coherent fibre bundle

**is:** an organized or stacked array of optical fibres, such that the relative position of each optical fibrefibre in the bundle is the same at either end of the bundle. [PHYS 6.2]

**can be used:** to transfer image information. [PHYS 6.2]

coil

**is:** a structure consisting of several loops (called turns) of wire wound in a similar sense to form a simple geometric shape, most typically a circular prism (cylinder) or a helix (solenoid), but possibly some other shape such as a rectangle.

coincident roots

**of:** an equation

**are:** repeated roots. (As, for example, the roots of the equation (*x* − 1)^{2} = 0 are repeated_rootrepeated and are therefore coincident at *x* = 0.) [MATH 1.4]

colinear

**means:** acting along the same line. [PHYS 5.1]

collimator

**is:** a device used to produce a parallel beam of radiation from a lamp or other source. An optical collimator usually consists of a converging lens with an illuminated slit or circular aperture placed at its focus. [PHYS 6.4]

**forms:** the first stage of a spectrometer. [PHYS 8.2]

collision

**is:** a brief but strong interaction between two particles or bodies which come into close proximity. [PHYS 2.4, PHYS 2.5]

coma

**in:** an image

**is:** the aberration which appears as a comet–like flaring at the edge of an extended image. It is the result of the focal length for non–axial rays varying with their point of incidence on a lens. [PHYS 6.4]

common denominator

**of:** two or more fractions

**is:** any number that is exactly divisible (without remainder) by the denominator of each of the fractions. [MATH 1.1]

**can be obtained:** by multiplying together the denominators of each of the fractions (though the result will not necessarily be the lowest common denominator).

common difference

See arithmetic progression.

common factor

**of:** two of more numbers or algebraic expressions

**is:** any number or algebraic expression which is a factor of each. [MATH 1.1]

common logarithm

**is:** a synonym for the logarithm to the base 10, i.e. log_{10}. [MATH 1.5]

common ratio

See geometric progression.

common tangent

**is:** a straight line that is a tangent_to_a_curvetangent to two (or more) given curves. [MATH 2.1]

commutator

**is:** a device used to periodically reverse the current in a rotating coil, in order to maintain the direction of a magnetic torque on the coil. [PHYS 4.3]

complementary angles

**are:** two angles whose sum is 90°. [MATH 2.1]

complementary function

**forms:** part of the general solution to a second–order differential equationsecond–order linear_differential_equationlinear inhomogeneous differential equation with constant coefficients, of the form

$a\dfrac{d^2y}{dt^2}+b\dfrac{dy}{dt}+cy=f(x)$. [MATH 6.3]

**is:** the general solution to the corresponding linear homogeneous differential equation

$a\dfrac{d^2y}{dt^2}+b\dfrac{dy}{dt}+cy=0$. [MATH 6.3,PHYS 5.5]

completed square form

**is:** the form *y* = *a*_{ }(*x*−*p*)^{2} + *q* of a quadratic function, *y* = *ax*^{2} + *bx* + *c* that makes clear the location of the vertex at (*p*, *q*) = (−*b*_{ }/(2*a*), [*c* − *b*^{2} /(4*a*)]). [MATH 1.3]

completely inelastic collision

**is:** a collision in which the maximum amount of kinetic energy is converted into other forms of energy, consistent with the principle of conservation of momentum. [PHYS 2.5]

completing the square

**is:** the procedure by which a quadratic function is expressed in completed square form. [MATH 1.3, MATH 1.4]

complex

**means:** pertaining to complex numbers.

complex capacitive reactance

**of:** a capacitor with capacitance *C* when passing alternating current of angular frequency *ω*

**is given:** by *Z*_{C} = −*i*_{ }/*ωC*. [PHYS 5.5]

See complex impedance, capacitive reactance.

complex conjugate

**of:** a complex number, *z* = *x* + *i*_{ }*y*, (where *x* and *y* are real numbers)

**is:** *x* − *i*_{ }*y*. [MATH 3.1, PHYS 5.5]

**is denoted:** by *z**. [MATH 3.1, PHYS 5.5]

complex impedance

**of:** an electrical component or a network of such components subject to an alternating voltage of angular frequency *ω*

**is:** a complex quantity $\mathcal{Z}$ that determines the complex current $\mathscr{I}$ that flows in response to the complex voltage $\mathscr{V}$ through the relation $\mathscr{V} = \mathcal{Z}\mathscr{I}$. (It therefore determines the peak value and the phase lag of the sinusoidally varying current that flows in response to a sinusoidally varying voltage. [PHYS 5.5]

**is given:** for *n* (complex) impedances connected in series, by

$\mathcal{Z} = \mathcal{Z}_1 + \mathcal{Z}_2 + \dots + \mathcal{Z}_n$ [PHYS 5.5]

**is given:** for *n* (complex) impedances connected in parallel, by

$\dfrac{1}{\mathcal{Z}} = \dfrac{1}{\mathcal{Z}_1} + \dfrac{1}{\mathcal{Z}_2} + \dots + \dfrac{1}{\mathcal{Z}_n}$ [PHYS 5.5]

**is given:** for a single resistance by $\mathcal{Z} = R$; for a single inductance by $\mathcal{Z}_L = i\omega L$ and for a single capacitance by $\mathcal{Z}_C = -i/\omega C$. [PHYS 5.5]

See complex capacitive reactance and complex inductive reactance

complex inductive reactance

**of:** an inductor with inductance *L* when passing alternating current of angular frequency *ω*

**is given:** by $\mathcal{Z}_L = i\omega L$. [PHYS 5.5]

See complex impedance, inductive reactance.

complex number

**is:** an expression that may be written in the form *x* + *i*_{ }*y*, where *x* and *y* are real numbers and *i* is a symbol satisfying the algebraic rule *i*^{2} = −1, i.e. $i = \sqrt{-1\os}$. [MATH 1.4, MATH 3.1, PHYS 5.5, PHYS 10.3, PHYS 11.1]

complex plane

**is:** the set of all complex numbers or the representation of them on an Argand diagram. [MATH 3.1]

complex variable

**is:** a variable that may take on complex values.

component vectors

**of:** a vector

**are:** a number of vectors (usually orthogonal) whose vector sum is the original vector. [MATH 2.4, MATH 2.5]

**should not be confused with:** (scalar) components of a vector.

components of a vector

**are:** *n* scalar quantities (*υ*_{x}, *υ*_{y}, *υ*_{z}) that can be used to specify an *n*–dimensional vector in Cartesian form.

**are sometimes referred to:** as the scalar components, in order to emphasize their distinction from component vectors. [MATH 2.4, PHYS 2.1, PHYS 2.2]

**should not be confused with:** component vectors.

See projection.

composite function

**is:** a function obtained through the combination of two or more functions. Given two functions *f*_{ }(*x*) and *g*_{ }(*x*), the composite function *f*_{ }(*g*_{ }(*x*)) is obtained by replacing each occurrence of *x* in *f*_{ }(*x*) by *g*_{ }(*x*). [MATH 1.3, MATH 4.3]

**is also called:** a function of a function. [MATH 1.3, MATH 4.3]

compound

See chemical compound.

compound microscope

**is:** a microscope which consists of an objective lens and an eyepiece lens, although each of these may consist of several component lenses. [PHYS 6.4]

compression

**is:** the process of making something smaller in size.

**is also:** the force within the body of a compressed elastic spring, acting along the axis of the spring in order to restore the spring’s natural length. [PHYS 2.3]

**also can mean:** the externally applied force acting to compress such a spring. [PHYS 2.3]

**also can mean:** the difference in length between the uncompressed and the compressed spring. [PHYS 2.3]

**also can mean:** a region where pressure and hence density are higher than average. [PHYS 5.7]

Contrast with expansion and rarefaction.

Compton effect

**is:** the phenomenon involving the scattering of photons by an electron, which shows that each quantum of electromagnetic radiation has both energy and momentum. [PHYS 10.1]

Compton wavelength

**for:** a particle of mass *m*

**is defined:** as *h*_{ }/*mc*, where *h* is Planck’s constant and *c* is the speed of light. [PHYS 10.1]

**appears:** in the theory of the Compton effect. [PHYS 10.1]

**is of the same order of magnitude:** as the change in wavelength of the scattered photons. [PHYS 10.1]

concave downwards

**describes:** a function whose second derivative is less than zero throughout an interval. [MATH 4.4]

concave lens

**is:** a lens, shaped so that at least one of its surfaces curves inwards into the material. The centre is thinner than the edges. Usually the surfaces are spherical. [PHYS 6.3]

**is also called:** a diverging lens or a negative lens. [PHYS 6.3]

concave meniscus lens

**is:** a lens having two concave surfaces of different radii when viewed from one side and with the centre of the lens thinner than the edges. [PHYS 6.3]

concave mirror

**is:** a mirror shaped so that its reflecting surface curves inwards, away from the incoming light rays. [PHYS 6.3]

concave surface

**is:** a surface which bulges away from the object position, when viewed from that position. [PHYS 6.3]

concave upwards

**describes:** a function whose second derivative is greater than zero throughout an interval, i.e. a function where slope increases continually throughout the interval. [MATH 4.4]

concentric

**describes:** any two objects which have the same centre, used especially of circles and spheres. [MATH 2.1]

condensation

**is:** the process whereby a gas or vapour is converted into a liquid.

Contrast with evaporation.

conductance

**of:** a body of (electrical) resistance *R*

**is:** the reciprocal of the resistance, i.e. 1/*R*.

conduction (of electricity)

**is:** the process whereby electric charge flows from one part of a material to another

**takes place:** at the atomic level, mainly through the movement of electrons from atom to atom.

**therefore is:** a transport process.

conduction (of heat)

**is:** one of three processes (the other two being convection and radiation) in which heat can be transferred. [PHYS 7.3]

**is operative:** only in materials (gases, liquids and solids), i.e. not in a vacuum. [PHYS 7.3]

**takes place:** at the atomic level, through energy being passed from atom to atom by vibration and/or collision. [PHYS 7.3]

**is driven:** at the macroscopic level, by a temperature gradient, with heat being transferred from high temperature to low temperature. [PHYS 7.3, PHYS 7.5]

**therefore is:** a transport process. [PHYS 7.5]

**sometimes is quantified:** by Fourier’s law. [PHYS 7.3]

conduction band

**in:** the band theory of solids

**is:** the lowest energy band that would be completely unoccupied at absolute zero. [PHYS 11.4]

conduction electrons

**in:** the band theory of solids

**are:** those electrons that are relatively free to move through the solid and may therefore give rise to electrical conduction. [PHYS 11.4]

conductivity

**of:** a material

**is:** the reciprocal of the resistivity *ρ* of that material. [PHYS 4.1, PHYS 7.3]

**has as its SI unit:** (Ω m)^{−1}, though S m^{−1} (i.e. siemens per metre) are also used. [PHYS 4.1, PHYS 7.3]

conductor (electrical)

See electrical conductor.

conductor (thermal)

**is:** a substance with a moderate to high coefficient of thermal conductivity, typically a metal, and usually also an electrical conductor.

cone

**is:** the shape formed by rotating a triangle about one of its sides. [MATH 2.3]

cones (of the eye)

**are:** one of two types of light sensor present in the eye, the other type being rods. PHYS 6.4]

**provide:** colour vision, being mainly sensitive to either red, green or blue light but being ineffective at low light levels. [PHYS 6.4]

confinement

See plasma confinement.

congruent

**describes:** two geometric figures which are identical in shape and size. [MATH 2.1]

conic section

**is:** the intersection of a cone with a plane. [MATH 2.3, PHYS 3.2]

**can be defined:** as the locus of all points P, such that the ratio of the distance from P to a fixed point (the focus), to the distance from P to a fixed line (the directrix), is constant. The value of this constant is known as the eccentricity *e*. The conic section is:

an ellipse if *e* < 1,

a parabola if *e* = 1,

an hyperbola if *e* > 1. [MATH 2.3]

**also can be defined:** as the shape described by any second degree equation of the form:

*ax*^{2} + 2*hxy*+*by*^{2} + 2*gx* + 2*fy* + *c* = 0

The conic section is:

an ellipse if *h*^{2} < *ab*,

a parabola if *h*^{2} = *ab*,

an hyperbola if *h*^{2} > *ab*. [MATH 2.3]

See conic sections in the Maths For Science handbook for further information.

conical pendulum

**is:** a mechanical system consisting of a mass, suspended from a point by a thread, undergoing uniform circular motion in a horizontal plane. [PHYS 2.3]

conics

See conic section.

conjugate equation

**is:** an equation which links object and image points for an optical element. [PHYS 6.3]

See conjugate equation for a single spherical surface and conjugate equation for a thin lens.

conjugate equation for a single spherical surface

**is:** an equation which links together the object distance and image distance and the radius of curvature of the spherical surface at which refraction occurs. [PHYS 6.3]

conjugate equation for a thin lens

**is:** an equation which links together the object distance and image distance and the radii of curvature of the lens surfaces at which refraction occurs. [PHYS 6.3]

conjugate planes

**are:** planes perpendicular to the optical axis, containing conjugate points. [PHYS 6.3]

conjugate points

**are:** object and image points linked by a conjugate equation. [PHYS 6.3]

conservation of angular momentum

**is a principle which states:** that when no unbalanced external torque acts on a body or a system of bodies, the total angular momentum of that body or system stays constant. [PHYS 2.8]

conservation of (electric) charge

**is a principle which states:** that the total net charge in the Universe is constant. Charges can be created and destroyed but only if the amounts of positive and negative charge involved are identical so that the net change is zero. [PHYS 3.3]

conservation of energy

**for:** an isolated system (which is therefore not subjected to unbalanced external forces)

**is a principle which states:** that the total amount of energy in the system is always constant (i.e. energy cannot be created or destroyed), although some or all of the energy may be converted from one form into another. [PHYS 2.4, PHYS 2.5]

See conservation of relativistic energy.

conservation of mass

**for:** a system that does not exchange any matter with its environment

**is a principle which states:** that the mass of the system is constant and is unaffected by position, velocity, temperature or any other factor. [PHYS 2.3]

**is approximately true:** when the velocity of the system is much less than the velocity of light. [PHYS 2.3]

See conservation of relativistic energy.

conservation of mechanical energy

**for:** an isolated system in which only conservative forces act

**is a principle which states:** that the total mechanical energy (i.e. the sum of the kinetic and potential energies) stays constant. [PHYS 2.4]

conservation of momentum

**for:** an isolated system (which is therefore not subject to unbalanced external forces)

**is a principle which states:** that the total momentum of the system is constant. [PHYS 2.5]

**implies:** that the total momentum of the system of objects does not change due to mutual interactions between the objects within the system. [PHYS 2.5]

conservation of nucleon number

**is a principle which states:** during radioactive decay, nuclear fusion and nuclear fission, the number of nucleons (sum of protons and neutrons) is constant.

conservation of relativistic energy

**is:** simply the conservation of energy, but named in this way to emphasize that quantities such as kinetic energy should be specified in the form required by Einstein’s special theory of relativity, and that contributions arising from mass energy should be included. [PHYS 9.1]

conservation principle (or law)

**is:** a law or principle which states that, at least under certain conditions, the value of a physical quantity remains fixed and does not vary in time. [PHYS 2.4, PHYS 9.1]

**is exemplified:** by conservation of mass, conservation of charge, conservation of energy, conservation of momentum and conservation of angular momentum. [PHYS 2.4, PHYS 9.1]

conservative force

**is:** a force which may be associated with a unique value of potential energy at each point in space and for which the work done between any two points is independent of the path chosen. As a result, the work done by the force around any closed path is zero. [PHYS 2.4, PHYS 11.2]

**is exemplified by:** gravitational forces, and electrostatic forces. [PHYS 2.4, PHYS 11.2]

conserved quantity

**describes:** any quantity that has the same value at the beginning and end of a wide class of processes, so that it might be made the subject of a suitably formulated conservation law. [PHYS 2.5]

constant

**means:** independent of time.

**is also:** a quantity whose value does not change in the course of a calculation. [MATH 1.1, MATH 1.3]

**may be:** a physical constant, e.g. Planck’s constant or the speed of light in a vacuum.

**may be:** a mathematical constant, e.g. *π* or e.

Contrast with variable.

constant acceleration

See uniform acceleration.

constant acceleration equations

See uniform acceleration equations.

constant addition rule (for summation)

**for:** any constant *a* and any positive integer *N*

**is:** $\displaystyle \sum_{i=1}^N (x_i+a) = \sum_{i=1}^N x_i + Na$ [MATH 1.7]

constant field

**is:** a field that does not change with time. [MATH 2.6, PHYS 3.3]

constant multiple rule (for integration)

**for:** any constant *a*

**is:** $\int af(x)\,dx = a\int f(x)\,dx$. [MATH 5.2]

constant multiple rule (for summation)

**for:** any constant *a* and any positive integer *N*

**is:** $\displaystyle \sum_{i=1}^N ax_i = a\sum_{i=1}^N x_i$ [MATH 1.7]

constant multiple rule (for differentiation)

**for:** any constant *a*

**is:** $\dfrac{d}{dx}\left(af(x)\right) = a\dfrac{d}{dx}\left(f(x)\right)$ [MATH 4.2]

constant of integration

**is:** the arbitrary constant introduced by the process of indefinite integration. [MATH 5.1, MATH 5.2]

**is exemplified:** by the constant *C* in the equation $\int x\,dx = \dfrac{x^2}{2} + C$

constant of proportionality

**between:** two variables *x* and *y* which are proportional (i.e. *x* ∝ *y* )

**is:** the constant *k* such that *x* = *ky*. [MATH 1.1, PHYS 1.3]

**does not depend:** on the values of *x* and *y* though it may depend on the values of other variables that are independent of *x* and *y*. [MATH 1.1, PHYS 1.3]

constant speed

See uniform speed.

constant velocity

See uniform velocity.

constant–volume gas thermometer

**is:** a thermally sensitive device in which the pressure of a gas, constrained to a constant volume, is used as a thermometric property. [PHYS 7.2]

**is:** not particularly convenient to use, but occupies a central role in defining precise scales for the measurement of temperature. [PHYS 7.2]

**defines:** a gas scale which is intimately related to the thermodynamic Kelvin scale, which is the most fundamental of all temperature scales because it is totally independent of the material (gas, liquid, or solid) and the thermometric property chosen. [PHYS 7.2]

construction line

**is:** an imaginary line added to a diagram to help in explanation, proof or problem solving. [PHYS 2.7]

constructive interference

**is:** the condition in which the superposition of two oscillations or waves produces a resultant with larger amplitude than either of the original oscillations or waves. When the two oscillations or waves are in phase, the amplitude of their resultant is the sum of their amplitudes. [PHYS 5.1, PHYS 5.6, PHYS 5.7, PHYS 6.1]

constructive superposition

See constructive interference.

continuous spectrum (emission or absorption)

**of:** electromagnetic radiation (usually from a specified source)

**is:** a spectrum that is (relatively) smooth and unbroken over a wide continuous range of wavelengths. [PHYS 8.2]

**is typical:** of the emission spectrum from a solid or liquid heated to a high temperature. [PHYS 8.2]

**is exemplified:** by white light, which can be dispersed by a diffraction grating or a prism into all the colours of the rainbow. [PHYS 8.2]

**is also exemplified:** by the black–body spectrum. [PHYS 8.2]

continuous flow method

**is:** a standard calorimetry procedure that can be used to measure specific heats of fluids. [PHYS 7.4]

**involves:** a fluid flowing at a constant known rate past a heater delivering a known power which produces a rise in temperature between the inlet and outlet. [PHYS 7.4]

continuous function

**is:** a function whose graph has no breaks. [MATH 4.4]

continuous refraction

**is:** a phenomenon that can occur in a region of a medium where the refractive index varies smoothly with position. [PHYS 6.2]

**can cause:** an appropriately directed ray to change its direction continuously. [PHYS 6.2]

continuous variable

**is:** a variable that changes only in a smooth fashion (with no sudden jumps in its value). [MATH 1.3]

continuous X–ray spectrum

**is:** the spectrum of X–rays that results when electrons are accelerated through a potential difference of several thousand volts and then strike a target. [PHYS 8.3]

**is created:** as the electrons come to rest. Because the energy of the electrons in the target ranges from zero to a maximum value, the energy of the X–rays emitted will also vary continuously from zero up to a maximum. [PHYS 8.3]

**also known as:** bremsstrahlung Contrast with characteristic X–ray spectrum.

continuum

**is:** the continuous range of available energies for an electron moving under the influence of an atom or ion to which it is not bound. The electron is sometimes said to be in an unbound state of the atom or ion. [PHYS 8.2]

**can be contrasted:** with the discrete energy levels of the bound states of the atom which the electron might otherwise occupy. [PHYS 8.2]

**is reached:** by a bound electron which is given sufficient energy to exceed the ionization level of the atom or ion and therefore to escape from it. [PHYS 8.2]

continuum level

See ionization level.

contraction

**is:** the process of making something smaller is size.

control rod

**is:** a rod of a material that readily absorbs thermal neutrons. [PHYS 9.3]

**is lowered:** into a nuclear fission reactor to control or stop the nuclear chain reaction. [PHYS 9.3]

convection

**is:** one of three processes (the other two being conduction and radiation) in which heat can be transferred. [PHYS 7.3]

**operates:** only in fluids (i.e. gases and liquids), where the relative movement of parts of the fluid at different temperatures is the means by which heat is carried from hot regions to cold regions. [PHYS 7.3]

**is classified:** in two broad categories: ‘forced convection’, in which the fluid is being moved by external means (a breeze blowing across your face, or a coolant being pumped past a hot object), and ‘free convection’, in which the flow is induced by buoyancy caused by thermal expansion of hotter regions of the fluid relative to cooler regions (around fins designed to cool the electronics in your hi–fi amplifier, in central heating by electric convectors or, despite their common name, by water–filled radiators). [PHYS 7.3]

**involves:** a generally very complicated relationship between heat flow and temperature difference, depending on temperature difference in a non-linear way and on other factors including many thermal properties of the fluid and the geometry and orientation of the object exchanging heat with the fluid. There are many empirical formulae employed by engineers for situations commonly encountered, but when in doubt, or when information is lacking, the best one can do is to use Newton’s law of cooling *dQ* = *hA* Δ*T* where *dQ*_{ }/*dt* is the rate of heat *dt* flow between two surfaces of area *A* that differ in temperature by an amount Δ*T*, and *h* is an appropriately chosen convective heat transfer coefficient. [PHYS 7.3]

convective heat transfer coefficient

See convection.

converge

See convergent series and convergent sequence.

convergent integral

**is:** an improper integral with a finite value. [MATH 5.2]

convergent sequence

**is:** a sequence, *S*_{1}, *S*_{2}, *S*_{3}, *S*_{4}, ... all of whose members, beyond some particular member, are as close as we please to some particular number. This number is called the limit of the sequence. [MATH 1.7]

convergent series

**is:** a series whose partial sums form a convergent sequence. The limit of a sequence of partial sums is known as the sum of the series. [MATH 1.7]

converging lens

**is:** a lens which increases the convergence or reduces the divergence of light rays passing through it. [PHYS 6.3]

**is also called:** a convex lens or a positive lens. [PHYS 6.3]

conversion factor

**is:** a dimensionless factor, such as (10^{3} m/km), which is actually equal to one, but which is expressed as a ratio of two quantities which have different units. [PHYS 1.1]

**can be used:** to convert a quantity expressed in terms of certain units into an equivalent quantity expressed in terms of other units. [PHYS 1.1]

convex lens

**is:** a lens, shaped so that at least one of its surfaces curves outwards, away from the centre of the material. The centre is thicker than the edges. Usually the surfaces are spherical. [PHYS 6.3]

**is also called:** a converging lens or a positive lens. [PHYS 6.3]

convex meniscus lens

**is:** a lens having two convex surfaces of different radius of curvature when viewed from one side, and with the centre of the lens thicker than the edges. [PHYS 6.3]

convex mirror

**is:** a mirror shaped so that its reflecting surface curves outwards, towards the incoming light rays. [PHYS 6.3]

convex surface

**is:** a surface which bulges towards the object position, when seen from that side. [PHYS 6.3]

coolant

**in:** a nuclear fission reactornuclear reactor

**is:** a fluid (liquid or gas) that keeps a reactor cool and transfers the energy released in the reactor so that it may be used to produce steam and hence drive electricity generators. [PHYS 9.3]

coordinate axes

See Cartesian coordinates.

coordinate system

**is:** a system for associating a set of values, called coordinates, with points in space so that each point may be uniquely identified and distinguished from every other point.

See Cartesian coordinates, polar coordinates and spherical polar coordinates.

coordinates

**of:** a point

**are:** the (unique) set of values associated with that point by a coordinate system that distinguish that point from other points.

**are exemplified by:** the *x*- and *y*–coordinates of a point on a graph. [PHYS 1.3]

Copenhagen interpretation

**is:** the most commonly accepted view of quantum physics. [PHYS 10.2]

**holds that:** the Universe operates according to probabilistic laws which tell us as much as can be known, even in principle, about future events. [PHYS 10.2]

**was formulated:** by a group of scientists (including Heisenberg) who worked in Copenhagen in the 1920s. [PHYS 10.2]

**contrasts:** with the many universe interpretation.

coplanar

**means:** in the same plane. [PHYS 2.7]

correspondence principle

**states:** that in the classical limit the predictions of quantum mechanics are in agreement with those of (non-relativistic) classical physics. [PHYS 11.2, PHYS 11.3]

corkscrew rule

**is:** a rule for working out the direction of a vector product such as **a** **×** **b**.

**states that:** if the handle of a (right–handed) corkscrew is aligned with the vector **a** and oriented in such a way that its handle may be twisted into alignment with **b** by turning it through an angle less than 180°, then the direction of **a** **×** **b** is the direction in which the corkscrew would advance.

**more briefly states:** the direction of **a** **×** **b** is the direction of advance of a corkscrew as its handle is rotated from **a** to **b**. [MATH 2.7, PHYS 4.3]

Compare with the right–hand screw rule and the (preferred) right–hand rule.

cornea

**is:** the transparent protective outer covering to the eye. [PHYS 6.4]

**is:** the first surface at which refraction takes place for light entering the eye. [PHYS 6.4]

corresponding angles

See transversal.

cosecant, cosec

See trigonometric function.

cosech

See hyperbolic function.

cosh

See hyperbolic function.

cosine rule

**states:** that given a triangle with angles *A*, *B* and *C* opposite to sides *a*, *b* and *c* then *c*^{2} = *a*^{2} + *b*^{2} − 2*ab*_{ }cos_{ }*C*. Likewise *a*^{2} = *b*^{2} + *c*^{2} − 2*bc*_{ }cos_{ }*A* and *b*^{2} = *a*^{2} + *c*^{2} − 2*ac*_{ }cos_{ }*B*. [MATH 1.6]

**reduces:** to Pythagoras’s theorem when the chosen angle is 90°.

See trigonometric functions in the Maths For Science handbook for further details.

cosine, cos

See trigonometric function.

cosmic rays

**are:** high energy particles (mainly protons) which enter the Earth’s upper atmosphere from space. They may collide with nuclei in the atmosphere, producing radioactive isotopes. [PHYS 9.3]

cotangent, cot

See trigonometric function.

coth

See hyperbolic function.

Coulomb force

See electrostatic force and Coulomb’s law. [PHYS 3.3]

Coulomb’s law

**is:** the law, first formulated by Charles Augustin de Coulomb (1736–1806), which describes the electrostatic force between charged particles. [PHYS 3.1]

**states that:** for two particles of charge *q*_{1} and *q*_{2} separated by a distance *r*, the force on particle 2 due to particle 1 is

$F_{\rm el} = \dfrac{q_1q_2}{4\pi\varepsilon_0r^2}\hat{\boldsymbol r}$

where *ε*_{0} is the permittivity of free space, *q*_{1} and *q*_{2} are signed quantities, and *r*^{^} is a unit vector pointing from *q*_{1} to *q*_{2}. The direction of the force is therefore along the line joining the charges, and like charges repel while unlike charges attract. [PHYS 3.3]

coulomb, C

**is:** the SI unit of charge.

**is defined:** as the amount of charge transferred when a current of 1 ampere flows for 1 second, so 1 C = 1 As. [PHYS 3.3]

couple

**is:** a pair of forces of equal magnitude acting in opposite directions along different lines of action. [PHYS 2.7, PHYS 4.3]

**may be characterized:** by a non–zero torque about any point, the magnitude of which is equal to the magnitude of either one of the forces multiplied by the perpendicular distance between their lines of action. [PHYS 2.7]

**causes:** rotation but not translation, when applied to rigid body that is entirely free to move [PHYS 2.7]

coupled oscillators

**are:** two oscillators connected in such a way that the displacement of one oscillator affects the restoring force acting on the other. [PHYS 5.1, PHYS 5.3]

**exhibit:** normal modes. [PHYS 5.1, PHYS 5.3]

**may be generalized:** to a system of many oscillators.

covalent bond

**is:** a bond in which one or more electrons is shared between two (or more) atoms. [PHYS 11.4]

covalent bonding

**is:** a type of chemical bonding in which the chemical bonds are created by electron pairs shared between atoms. [PHYS 8.4]

**has typical energy:** of 1 to 5 eV. [PHYS 7.1]

**is characterized:** by an increased electron density between the nuclei of the atoms. [PHYS 11.4]

creep

**is:** the condition in which the strain in a material exhibits a slow time- dependence under constant stress in the region of plasticity. [PHYS 7.6]

critical

**describes:** the condition inside a nuclear reactor (or similar device) in which a nuclear chain reaction is just able to self–sustain at a steady rate, i.e. where, on average, exactly one neutron released in the fission of one nucleus goes on to produce fission in one further nucleus. [PHYS 9.3]

Contrast with subcritical, supercritical.

critical angle

**for:** light rays passing from a medium of given refractive index into a medium of lesser refractive index

**is:** the minimum angle of incidence that corresponds (via Snell’s law) to an angle of refraction of 90°. A ray meeting the interface at a greater angle of incidence will suffer total internal reflection unless special steps are taken to frustrate the process. [PHYS 5.7, PHYS 6.2]

critical damping

**is:** the condition in which a damped oscillator just fails to oscillate and comes to rest in the shortest possible time following release from a given position. It is the intermediate condition between light damping (i.e. underdamping) and heavy damping (i.e. overdamping). [PHYS 5.2]

**is accompanied by:** no more than one overshoot of the equilibrium value before coming to rest.

**is exemplified electrically:** by a series a.c. circuit containing a capacitor of capacitance *C*, an inductor of inductance *L*, and a resistor of resistance *R*, wherein the damped oscillations of stored charge (or current) are critically damped when $R = 2\sqrt{L/C}$. [PHYS 5.4]

**is exemplified mechanically:** by a damped mechanical oscillator containing an oscillating body of mass *m*, a spring of spring constant *k*, and a linear damping force with damping coefficient *b*, wherein oscillations are critically damped when $b = 2\sqrt{km}$. [PHYS 5.4, PHYS 5.5]

**is described by:** *x*_{ }(*t*) = (*H* + *Jt*)_{ }e^{−γt /2} where *γ* = *b*_{ }/*m* for a mechanical oscillator, and *γ* = *R*_{ }/*L* for an electrical oscillator. *H* and *J* are constants determined by the initial conditions.

critical mass

**is:** the mass of a fissile material that is just capable of maintaining a nuclear chain reaction. [PHYS 9.3]

**therefore is:** the mass of a fissile material that is just capable of keeping a nuclear chain reaction at the critical condition. [PHYS 9.3]

critical opalescence

**is:** a phenomenon displayed by normally transparent fluids under the conditions that define the critical point. Illuminated by a beam of light, the substance takes on an intensely white, diffuse cloudy appearance. [PHYS 7.4]

critical point

**of** a substance

**is:** the unique point on a *PVT*–surface (or some similar surface), or on one of its projections, representing the state in which the liquid and vapour phases of a substance become indistinguishable. [PHYS 7.4]

See also critical opalescence.

critical pressure

**is:** the pressure of a substance at its critical point. [PHYS 7.4]

critical temperature

**is:** the temperature of a substance at its critical point.

**is also:** the maximum temperature at which a gas can be liquefied by an isothermal process. [PHYS 7.4]

critical volume

**is:** the volume of a substance at its critical point. [PHYS 7.4]

critically damped

See critical damping.

cross product

See vector product.

cross–sectional area

**generally is:** the area of intersection of a geometrical solid and a plane. Usually the plane is normal to an axis of symmetry, but could be some other specified direction. [MATH 2.1]

See also prism.

crown glass

**is:** a glass of relatively low refractive index and thus low dispersive power. [PHYS 6.4]

**is used as a component:** in an achromatic doublet. [PHYS 6.4]

crystal

**is:** any material with a crystalline structure.

crystalline structure

**is:** a regular array of atoms in three–dimensional space that can be described by associating the same arrangement of one or more atoms with every point of a given three–dimensional lattice. [PHYS 11.4]

cubic equation

**is:** a polynomial equation of degree 3. [MATH 1.4]

cubic function

**is:** a polynomial function of degree 3. [MATH 1.3]

cuboid

**is:** any rectangular block. [MATH 2.1]

current

See electric current.

current balance

**is:** a device designed to measure the force between two current–carrying coils or wires. [PHYS 4.3]

**can be used:** to measure currents accurately and hence to determine the current of magnitude one ampere. [PHYS 4.3]

current divider equations

**are:** a pair of equations which describe the way in which an electric current is divided between two resistors in parallel. [PHYS 4.1]

curve

**is:** a continuous set of points, often (though not necessarily) in a plane.

cut–off wavelength

**of:** a continuous X–ray spectrum.

**is:** the sharply defined wavelength, below which there is no continuous spectrum. [PHYS 8.3]

**corresponds to:** the situation in which the maximum kinetic energy of an incident electron is entirely radiated away as a single X–ray photon. [PHYS 8.3]

cycle

**of:** a periodic motion (or a more general oscillation)

**is:** the motion or behaviour which occupies exactly one period. [PHYS 5.1]

cyclotron

**is:** a device which can accelerate charged particles by applying a periodic electric field to the particle as it moves, constrained in a circular or spiral path, by an applied magnetic field. [PHYS 4.3]

cyclotron frequency

**is:** the frequency of the circular or helical motion of a charged particle in a uniform magnetic field. [PHYS 4.3]

**is dependent:** only on the particleparticle’s charge–to–mass ratio *q*_{ }/*m* and on the magnetic field strength *B*:

$f_{\rm cyclotron} = \dfrac{\left\lvert\,q\,\right\rvert B}{2\pi m}$. [PHYS 4.3]

cyclotron motion

**of:** a charged particle

**in:** a magnetic field

**is:** the periodic motion of the particle in the plane perpendicular to the magnetic field. [PHYS 4.3]

cyclotron period

**is:** the time to complete one period of cyclotron motion and the reciprocal of the cyclotron frequency. [PHYS 4.3]

d’Alembert’s ratio test

**is:** one of several tests for the convergence or divergence of a series. If *a*_{n} is the *n*^{th} term in the series, the test consists of calculating:

$\displaystyle R = \lim_{n\to\infty}\left(\dfrac{a_{n+1}}{a_n}\right)$

There are three possible outcomes:

*R* < 1 implying convergence,

*R* > 1 implying divergence,

*R* = 1 implying that the test is incapable of providing a definite answer.

[MATH 1.7]

DC circuit, d.c. circuit

**is:** an electrical circuit in which a direct current flows, or may be presumed to flow. [PHYS 4.1]

DC isolation, d.c. isolation

**of:** two circuits

**is:** their separation such that they may have independent d.c. potentials but yet may be mutually influenced by each other’s a.c. currents. [PHYS 4.4]

**can be achieved:** via the mutual inductance between the circuits, through a transformer or via a capacitor. [PHYS 4.1, PHYS 4.4, PHYS 5.4]

damped (electrical) oscillator

**is:** an electrical system in which a quantity such as charge or current exhibits oscillatory behaviour while energy is dissipated to the environment.

**is exemplified:** by a circuit in which an inductance *L*, capacitance *C*, and resistance *R* are connected in series, so that the charge *q* stored in the capacitor at time *t* obeys the differential equation:

$L\dfrac{d^2q}{dt^2} = -\dfrac1Cq - R\dfrac{dq}{dt}$

and is consequently described, in the case of light damping (*R*^{2} < 4*L*_{ }/*C*), by an oscillation with an exponentially decaying amplitude:

*q*_{ }(*t*) = *q*_{0}_{ }e^{−γt /2}cos(*ωt* + *ϕ*)

where γ = R/L, $\omega = \sqrt{\dfrac{1}{LC}-\dfrac{R^2}{4L^2}}$ and *q*_{0} and *ϕ* are arbitrary constants. [PHYS 5.4, PHYS 5.5]

See critical damping. [PHYS 5.4, PHYS 5.5]

damped oscillator

See damped (electrical) oscillator, damped (mechanical) oscillator.

damped (mechanical) oscillator

**is:** a mechanical system in which a quantity such as displacement exhibits oscillatory behaviour while energy is dissipated to the environment.

**is exemplified:** by a particle of mass *m* on a spring of spring constant *k*, moving subject to a damping force with damping coefficient *b* so that its displacement from equilibrium, *x*, at time *t* satisfies the equation of motion:

$m\dfrac{d^2x}{dt^2} = -kx -b\dfrac{dx}{dt}$

and is consequently described in the case of light damping (*b*^{2} < 4*mk*) by an oscillation with an exponentially decaying amplitude:

*x*_{ }(*t*) = *x*_{0}_{ }e^{−γt /2}cos(*ωt* + *ϕ*)

where *γ* = *b*_{ }/*m*, $\omega = \sqrt{\dfrac km - \dfrac{b^2}{4m^2}}$ and *x*_{0} and *ϕ* are arbitrary constants. [PHYS 5.2, PHYS 5.5]

See critical damping and heavy damping.

damped oscillation

See damped (electrical) oscillator, damped (mechanical) oscillator.

damping

**is:** any phenomenon involving dissipation (such as friction, viscosity or electrical resistance) that causes a system (particularly an oscillating system) to lose energy. [MATH 6.3, PHYS 5.2, PHYS 5.4]

See damping force, damping constant.

damping coefficient

is: the constant *b* that appears in the equation for a linearly damped harmonic oscillator: $m\dfrac{d^2x}{dt^2}+b\dfrac{dx}{dt}+kx=0$

damping constant

**for:** an oscillating particle of mass *m* subject to a dissipative force of magnitude *bυ*, where *υ*, is the speed of the particle

**is given:** by *γ* = *b*_{ }/*m*. [PHYS 5.2]

**is equal:** to twice the decay constant *α* for the amplitude of the oscillation. [PHYS 5.2]

See damping force, damped mechanical oscillator.

damping force

**in:** a mechanical oscillator.

**is:** a dissipative force which opposes the motion and which therefore causes damping. [MATH 6.3, PHYS 5.2, PHYS 5.4]

See damping constant.

data

**is:** recorded information, particularly numerical or statistical information that can be used in an analysis or calculation.

daughter isotope

See daughter nucleus.

daughter nucleus

**is:** an isotope produced in the radioactive decay of a parent nucleus. [PHYS 9.2]

de Broglie hypothesis

**states:** that the propagation of all matter is determined by an associated de Broglie wave, from which the diffraction and interference behaviours may be predicted. [PHYS 10.2]

de Broglie wave

**is:** a wave associated with the propagation of matter. [PHYS 10.2]

**can be used:** to predict the diffraction and interference behaviours of matter. [PHYS 10.2]

See de Broglie wavelength.

de Broglie wavelength

**of:** a particle or, more generally, of a free quantum

**is given:** by *λ*_{dB} = *h*_{ }/*p*, where *p* is the magnitude of the momentum of the particle and *h* is Planck’s constant. [PHYS 10.2, PHYS 11.1, PHYS 11.2]

**determines:** the diffraction when the quantum meets an obstacle. [PHYS 11.1, PHYS 11.2]

Debye model

**is:** a model of the specific heats of solids

**postulates:** that the solid behaves like an elastic body capable of exhibiting quantized oscillations characterized by a specific distribution of classical frequencies. [PHYS 11.4]

**predicts:** that near absolute zero the specific heat is proportional to *T*^{3}, where *T* is the absolute temperature. [PHYS 11.4]

decay

**is:** a general term describing the tendency to decrease with time.

See decay constant.

decay channels

**are:** the different ways in which a particular radioactive nucleus may decay. [PHYS 9.3]

decay constant

**is:** the constant of proportionality, *α* that relates the rate of radioactive decay, *R* to the number, *N* of unstable nuclei present: *R* = *αN*. [PHYS 9.2]

**is:** a property of radionuclides, unaffected by the physical or chemical environment. [PHYS 9.2]

**more generally is:** the reciprocal of the time constant *τ* in any exponential decay process: *A*_{ }(*t*) = *A*_{0}_{ }e^{−αt} = *A*_{0}_{ }e^{−t /τ}. [PHYS 5.2]

deceleration

**is:** the slowing down of an object, and an associated reduction in speed. [MATH 4.1, PHYS2.1]

**is commonly misconstrued:** as negative acceleration. This may be, but is not necessarily, the case, since acceleration is a vector quantity and has an associated sign. [PHYS 2.1]

decibel, dB

**is:** a unit of (acoustic) intensity level. [PHYS 5.7]

**permits representation:** of intensity level in terms of a reference intensity level: given a sound of intensity *I* (measured in W m^{−2}), its intensity level is given by

$\beta = 10 \times \log_{10}\left(\dfrac{I}{I_0}\right)$ decibel

where *I*_{0} = 1 × 10^{−12} W m^{−2}. Audible, non–painful sounds usually have intensity levels in the range 0 to 120 dB. [PHYS 5.7]

decimal number

**is:** a number expressed in base ten notation, so that 345.6 means 3 × 10^{2} + 4 × 10^{1} + 5 × 10^{0} + 6 × 10^{−1}. [MATH 1.2]

decimal notation

**is:** a way of representing integer and real numbers. [MATH 3.2]

**consists:** of a string of one or more base ten digits in the case of an integer number.

**consists:** of a string of one or more base ten digits, a decimal point and then another string of base ten digits in the case of an real number. The total number of digits after the decimal point is the number of significant figures, and is often used to denote the precison of a numerical value [MATH 1.2, MATH 1.5, PHYS 1.1]

**may be:** preceeded by a minus (-) or a plus (+) symbol to indicate whether the number is less than or greater then zero, respectively.

**compare with:** scientific notation.

decimal places

**describes:** the number of digits which a decimal number has after the decimal point. [MATH 1.2]

decreasing function

**is:** a function *f*_{ }(*x*) for which *f*_{ }(*a*) > *f*_{ }(*b*) for all *a* < *b*. [MATH 4.4]

**exists:** on an interval if *f*_{ }′(*x*) is negative at all points of the interval. [MATH 4.4]

definite integral

**of:** a function *f*_{ }(*x*) defined on an interval from *x* = *a* to *x* = *b*

**is denoted:** $\displaystyle \int_a^b f(x)\,dx$

where the values *a* and *b* are known as the lower and upper limits of integration, *f*_{ }(*x*) is called the integrand, and the symbol *dx* is the element of integration which shows that *x* is the integration variable with respect to which the integration is to be performed. [MATH 5.1, MATH 5.2, PHYS 2.4]

**is defined:** by the limit of a sum:

$\displaystyle \int_a^b f(x)\,dx = \lim_{\Delta x\to0}\left(\sum_{i=1}^n f(x_i)\,\Delta x_i\right)$ with Δ*x*_{i} = *x*_{i+1} − *x*_{i}

where the sequence of values *x*_{1}, *x*_{2}, ... *x*_{n+1} is such that *a* = *x*_{1} < *x*_{2} < ... < *x*_{n+1} = *b*, and Δ*x* is the largest of the Δ*x*_{i}. [MATH 5.1, MATH 5.2,PHYS 2.4]

**may be interpreted:** for a given function between given limits, as the area under a graph of that function between the given limits, provided that due regard is paid to signs (areas of regions below the horizontal axis must be treated as negative quantities). [MATH 5.1, MATH 5.2, PHYS 2.4]

**can be evaluated:** according to the fundamental theorem of calculus using

$\displaystyle \int_a^b f(x)\,dx = \left[F(x)\right]_a^b = F(b) - F(a)$

where *F*_{ }(*x*) is any indefinite integral of *f*_{ }(*x*) (i.e. any function *F*_{ }(*x*) that satisfies *dF/dx* = *f*_{ }(*x*)). [MATH 5.1, MATH 5.2, PHYS 2.4]

**also can be evaluated:** by means of numerical integration. [MATH 5.1, MATH 5.2, PHYS 2.4]

degeneracy

**is:** the phenomenon in which different quantum states of a system (e.g. the states of electrons in an atom) have the same characteristic energy and therefore belong to the same energy level of the system. [PHYS 8.3]

**therefore is also:** the existence of more than one independent wavefunction, characterized by different sets of quantum numbers, corresponding to the same energy level. [PHYS 10.3]

degenerate

**describes:** an energy level or a wavefunction, when degeneracy is present. [PHYS 10.3]

degree, °

**is:** the unit of plane angle corresponding to 1/360th of a circle, written as 1°. In other words, a rotation through 360° is a full rotation.

**is equal:** to 0.01745 radian, (to five decimal places). [MATH 1.6] [MATH 2.1]

degree Celsius, °C

**is:** a non-SI unit of temperature and temperature difference.

**is defined:** to be equal in size to the SI unit of absolute temperature, the kelvin (K), but the zeros of the thermodynamic Kelvin temperature scale and the Celsius temperature scale are different (0 °C = 273.15 K).

degree (of a differential equation)

**is:** the highest power_mathematicalpower to which the highest order of derivative in the differential equation is raised. [MATH 6.1]

**for:** a linear differential equation is equal to 1. [MATH 6.1]

degree (of a polynomial)

**is:** the integer *n* that appears in a polynomial expression of the form *a*_{0} + *a*_{1}*x* + *a*_{2}*x*_{2} + ... + *a*_{n}^{−1}*x*_{n}^{−1} + *a*_{n}*x*_{n} = 0, that is, the highest power_mathematicalpower of the variable in the polynomial expression. [MATH 1.3, MATH 1.4]

degrees of freedom

**of:** a system

**are:** the characteristics of a system’s configuration that can be varied independently. [PHYS 5.1]

**are exemplified:** by the three position coordinates that determine the location of a particle in three–dimensional space.

**correspond:** to the independent variables required to describe the motion of the system fully. [PHYS 7.5]

**are reduced:** by constraints in the system which limit the possible motions. For instance, a system consisting of two independent particles has six degrees of freedom, but a ‘dumb–bell’ in which two particles are separated by a fixed distance has only five degrees of freedom (these can be thought of as three translational and two rotational degrees of freedom). [PHYS 7.5]

Demoivre’s theorem

**states:** that for any real number, *n*

[cos(*θ*) + *i*_{ }sin(*θ*)]^{n} = cos(*nθ*) + *i*_{ }sin(*nθ*) [MATH 3.3, PHYS 5.5]

denominator

**is:** the number or expression at the bottom of a fraction. [MATH 1.1]

density

**of:** a uniform body of mass *M* and volume *V*

**is:** the mass per unit volume of the body, *M*_{ }/*V*

**is defined more generally:** at a point in a (possibly non-uniform) body by

$\displaystyle \rho = \lim_{\Delta V\to 0}\left(\dfrac{\Delta m}{\Delta V}\right)$

where Δ*m* is the mass of a small element of the body, of volume Δ*V* centred on the specified point.

dependent error

**in:** a measurement

**when:** the errors arising in the measurement are being analysed

**is:** any error whose size is determined, wholly or partly, by the size of another. [PHYS 1.2]

See uncertainty.

dependent variable

**in:** an experiment (or a calculation)

**is:** the quantity whose value is monitored by the experimenter (or by the person doing the calculation). [PHYS 1.3]

**is controlled by:** the value of the independent variable to which it is connected by a set of experimental observations (or by a mathematical function). [MATH 1.3]

**on graphs is plotted:** conventionally along the vertical axis. [PHYS 1.3]

depth of field

**is:** the range of distances of an object from a lens, for which the image will appear to be sharp for a particular lens position. [PHYS 6.4]

**increases:** as the lens aperture is reduced in size. [PHYS 6.4]

Contrast with depth of focus.

depth of focus

**is:** the range of lens positions for which the image of an object will appear to be sharp for a particular distance of the object from the lens. [PHYS 6.4]

**increases:** as the lens aperture is reduced in size. [PHYS 6.4]

Contrast with depth of field.

derivative

**of:** a function *y* = *f*_{ }(*x*)

**is:** its rate of change with respect to *x* at any particular value of *x*

**is given by:**

$\displaystyle f'(x) = \dfrac{dy}{dx} = \lim_{\Delta x\to0}\left(\dfrac{\Delta y}{\Delta x}\right) = \lim_{\Delta x\to0}\left[\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\right]$

where *f*_{ }′(*x*) is known as the first derivative or derived function.

**is defined:** over any domain in which a unique limit exists for all values of *x*. [MATH 4.1, MATH 4.2, PHYS 2.1]

derived function

See derivative.

derived units

**are:** SI units created by specified combinations of the base units. [PHYS 1.1]

See Table 2 in Section 0 of the Maths For Science handbook for a detailed listing.

destructive interference

**is:** the condition in which the superposition of two oscillations or waves results in an oscillation or wave with smaller amplitude than either of the original oscillations or waves. When the two oscillations or waves are in anti–phase, the amplitude of their resultant is the difference of their amplitudes. [PHYS 5.1, PHYS 5.6, PHYS 5.7, PHYS 6.1]

**also known as:** destructive superposition.

destructive superposition

See destructive interference.

determinism

**is:** a belief that the Universe operates according to laws whose nature is such that the state of the Universe at one time completely determines its state at any later time. [PHYS 10.2]

deterministic system

**is:** a system for which a complete knowledge of the laws governing it and of its initial state allows its subsequent evolution in time to be predicted exactly. [MATH 6.1]

deuterium

**is:** the isotope of hydrogen that has mass number *A* = 2. [PHYS 9.3]

**is also called:** heavy hydrogen. [PHYS 9.3]

deuteron

**is:** a deuterium nucleus, ^{2}_{1}H. [PHYS 9.3]

**is also represented:** as D or sometimes d. [PHYS 9.3]

deviation

**is:** the difference between a particular measurement *x*_{i} (from a set of measurements) and the mean $\langle x\rangle$ of that set. The deviation of the *i*^{th} measurement is therefore $d_i = x_i - \langle x\rangle$. [PHYS 1.2]

See also standard deviation.

diameter

**of:** a circle, sphere or ellipse.

**is:** a line segment passing through the centre of the circle, sphere or ellipse. [MATH 2.1]

**touches:** the boundary at two ‘diametrically opposite’ points. [MATH 2.1]

**is also:** the length of such a line segment, which will be twice the radius in the case of a circle or sphere, but will depend on orientation in the case of an ellipse. [MATH 2.1]

diatomic ideal gas

**is:** an ideal gas in which the internal energy is a function of temperature *T* that (classically) rises from 3*nRT*_{ }/2 at low temperature, to 5*nRT*_{ }/2 at moderate temperature (due to the excitation of the rotational degrees of freedom), to 7*nRT*_{ }/2 at high temperature (due to the excitation of vibrations). [PHYS 7.4]

**can be used:** to model the behaviour of a real gas with two atoms per molecule at low density. [PHYS 7.4]

diatomic molecules

**are:** molecules made of two atoms chemically bonded together.
The atoms can be of the same element (homonuclear molecules), or of different elements (heteronuclear molecules).

**are exemplified by:** the five gaseous state diatomic elements: Cl_{2}, F_{2}, H_{2}, N_{2}, O_{2} and at room temperatures Br_{2} (liquid) and I_{2} (solid).

dielectric

**is:** a term used to describe an insulator in situations where its dielectric constant is (or may be) of significance (e.g. between the plates of a capacitor). [PHYS 4.5]

dielectric constant

**of:** a medium

**is:** the ratio of the permittivity of the medium to the permittivity of free space, *ε*_{0}. [PHYS 4.5]

**is synonymous:** with relative permittivity, *ε*_{r}.

difference

See operation.

differential equation

**is:** an equation which involves the first derivative and/or higher derivatives of a quantity. [PHYS 5.3, PHYS 5.4, MATH 6.1]

**has as its order:** the order of the highest derivative appearing in the equation. [MATH 6.1, PHYS 5.3, PHYS 5.4]

**has as its degree:** the highest power_mathematicalpower of the derivative of highest order appearing in the equation. [MATH 6.1, PHYS 5.3, PHYS 5.4]

**has a general solution:** which involves one or more arbitrary constants with values that have to be determined by boundary conditions which are characteristic of the problem being considered. [MATH 6.1, PHYS 5.3, PHYS 5.4]

See differential equations in the Maths For Science handbook for further details.

differential operator

**is:** an operator (i.e. a symbolic instruction to carry out a mathematical operation) that involves the process of differentiation. [MATH 4.3]

**usually acts:** on whatever is immediately to its right. [MATH 4.3]

**is exemplified:** by $\hat{\rm p}_x = -i\hbar\dfrac{d}{dx}$ which, in quantum mechanics, corresponds to the *x*–component of momentum. [PHYS 10.4]

**is also exemplified:** by $\hat{\rm E}_{\rm kin} = -\dfrac{\hbar^2}{2m\vphantom{^0}}\dfrac{d^2}{dx^2}$ which, in quantum mechanics, corresponds to the kinetic energy of a particle moving in one dimension. [PHYS 10.4]

See also eigenfunction, eigenvalue and eigenvalue equation.

differentiation

**is:** the process of finding the derived function, or derivative, of a function. [MATH 4.1, MATH 4.2]

diffraction

**is:** the ability of waves to bend around obstacles or to be spread by apertures. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

**depends for its amount:** on the relationship between the wavelength of the wave and the size of the obstacle or aperture. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

**is negligible:** when the wavelength is much less than the size of the obstacle or aperture. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

**is greatest:** when the wavelength is about the same size as the obstacle or aperture. [PHYS 5.7, PHYS 6.1, PHYS 6.2]

diffraction grating

**is:** an optical device consisting of a flat plate with a series of equally spaced, parallel slits on its surface. The distance between the slits is usually a few wavelengths of the radiation involved, and is called the grating spacing. The plate may be transparent (a transmission grating) or reflecting (a reflection grating) and the slits may have been produced by ruling them with an appropriate machine (ruled grating), or by taking a cast of an existing ruled grating (replica grating). [PHYS 6.1]

**produces:** when illuminated by normally incident monochromatic light of wavelength *λ* an interference pattern which has primary intensity maxima at angles *θ*_{n} from the straight–through position given by

$\sin\theta_n = \dfrac{n\lambda}{d}$

where *n* is the order of diffraction and *d* is the grating spacing. [PHYS 5.5, PHYS 6.1]

diffraction pattern

**is:** an interference pattern from an identifiable obstruction, for example a circular aperture or slit, or a pair of slits (as in Young’s experiment), or an array of slits (as in a diffraction grating). [PHYS 5.7, PHYS 6.1, PHYS 6.2]

See diffraction.

diffuse reflection

**is:** reflection from a rough surface, so that rays incident from the same direction are reflected in different directions by different parts of the surface. [PHYS 5.7]

diffusion

**is:** the process by which molecules spread from regions of high to low concentration. [PHYS 7.5]

**therefore is:** a transport process. [PHYS 7.5]

digit

**is:** a symbol used in the specification of a number 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are the ten digits used to specify decimal numbers. [MATH 1.2, PHYS 1.1]

dimension

**of:** a coordinate system (e.g. a system of Cartesian coordinates)

**is:** a ‘direction’ in which measurements may be made (usually) independently of measurements in other dimensions. In the case of Cartesian coordinates the directions of the *x*–axis, *y*–axis and *z*–axis each represent one of three independent dimensions. The number of dimensions (the dimensionality of the system) is therefore the minimum number of coordinates needed to uniquely identify any point in the region covered by the system of coordinates.

See also dimensional analysis and dimensions for a different meaning.

dimensional analysis

**is:** the process of assigning appropriate combinations of dimensions to physical quantities and using such assignments to test the plausibility of proposed relationships between physical quantities. [MATH 1.3, PHYS 1.1]

dimensionless

**refers:** to a quantity with no overall dimensions, such as a pure number or a ratio of two quantities which have the same dimensions. [MATH 1.2, PHYS 1.1]

dimensionless ratio

**is:** a ratio of two quantities which have the same dimensions. [MATH 1.2]

dimensions

**are:** basic measurable quantities such as mass (M), length (L) and time (T). [MATH 1.2]

**can be used:** singly or in appropriate combinations to characterize physical quantities. Speed, for example, can be measured in the same units as the ratio of a length to a time and is therefore said to have the same dimensions as length/time, a relationship shown by writing [speed] = [length/time] = L T^{−1}. Quantities with units that differ only by a dimensionless conversion factor are said to have the same dimensions. [MATH 1.2, PHYS 1.1]

diminished

**means:** made smaller – as for an image formed by a lens or a mirror, when the image is smaller than the object. [PHYS 6.3]

dioptre

**is:** the unit of optical power of a lens, being the reciprocal of the focal length of the lens and expressed in m^{−1}. [PHYS 6.3]

dipole

See electric dipole, magnetic dipole.

dipole moment

See electric dipole moment, magnetic dipole moment.

direct current, d.c.

**is:** an electric current whose direction does not vary with time. [PHYS 4.1]

**more generally refers:** to other associated electrical quantities whose direction or polarity does not vary with time, e.g. d.c. voltage. [PHYS 4.1]

**is abbreviated:** DC at the beginning of a sentence, and d.c. elsewhere. [PHYS 4.1]

direct integration

**is:** a method of solution which can be applied to differential equations of the form $\dfrac{dy}{dx} = f(x)$. [MATH 6.1, MATH 6.2]

See inverse differentiation.

directed line segment

**is:** a line of finite length with an arrow head drawn on it. The length and orientation of such a line can be used to represent the magnitude_of_a_vector_or_vector_quantitymagnitude and direction of a vector or a vector quantity in a diagram or illustration. [MATH 2.4]

direction (of a vector)

**is:** a characteristic property of a vector which determines its orientation with respect to a coordinate systemsystem of coordinates. [PHYS 2.2]

**usually is specified:** in two dimensions relative to a two–dimensional Cartesian coordinate system, by quoting the angle (measured in the anticlockwise sense) from the positive *x*–axis to the vector. [PHYS 2.2]

**may be more generally specified:** by expressing the vector in terms of its components of a vectorcomponents relative to a given Cartesian coordinate system.

See scalars and vectors in the Maths For Science handbook.

direction (of propagation)

**is:** the direction of motion of a wave. [PHYS 5.6, PHYS 6.1]

See transverse wave and longitudinal wave.

direction cosines

**of:** a straight line relative to a three–dimensional system of Cartesian coordinates.

**are:** three numbers that represent the cosines of the angles between the line and the coordinate axes. [MATH 2.2]

**are:** in the same ratio as the direction ratios of the line.

direction ratios

**for:** a straight line

**in:** three dimensions

**are:** the constants *l*, *m*, *n* in the equation for the straight line:

$\dfrac{x-a}{l} = \dfrac{y-b}{m} = \dfrac{z-c}{n}$

where (*a*, *b*, *c*) is a point on the line. [MATH 2.2]

directly proportional

**describes:** two variables *x* and *y*, if their ratio *x*_{ }/*y* remains constant as *x* and *y* are varied. [MATH 1.1]

**is symbolized:** by *x* ∝ *y.* [MATH 1.1]

**generally is abbreviated:** to ‘proportional’. [MATH 1.1]

See constant of proportionality.

Contrast with inversely proportional.

directrix

See conic section.

disc

**is:** a circle together with the points enclosed by its circumference. [MATH 2.1]

discharge tube

**is:** a device used to investigate the conduction of electricity through a gas. [PHYS 8.1]

**consists:** in its simplest form, of a gas–filled glass tube containing an anode and a cathode, in which the pressure can be reduced by means of a pump. [PHYS 8.1]

discrete (variable)

**is:** a variable that only takes certain separated values and is therefore not a continuous variable. [MATH 1.3]

discriminant

**for:** a quadratic equation *ax*^{2} + *bx* + *c* = 0

**is:** the quantity *b*^{2} − 4*ac.* [MATH 1.3, MATH 1.4]

**determines:** the number of times that the graph of the quadratic function will intersect the *x*–axis, i.e., the number of roots that the equation has. [MATH 1.3, MATH 1.4]

dispersion

**is:** the phenomenon in which a wave travels through a material with a phase speed that depends on its frequency. [PHYS 5.6, PHYS 6.1, PHYS 6.2, PHYS 6.3, PHYS 10.3]

**arises from:** variation of the refractive index of the material with the frequency of the wave, for an electromagnetic wave. [PHYS 5.6, PHYS 6.1, PHYS 6.2, PHYS 6.3, PHYS 10.3]

**therefore causes:** light of different frequencies to be refracted by different angles on entering the material, and hence enables light of different frequencies to be refracted in different directions. [PHYS 5.6, PHYS 6.1, PHYS 6.2, PHYS 6.3, PHYS 8.2, PHYS 10.3]

dispersion relation

**of:** a given type of wave in a specified medium

**is:** an expression which describes the variation of the wave’s wavelength (or some related quantity such as wavenumber) with the frequency of the wave. [PHYS 10.3]

**is exemplified:** for an electromagnetic wave of wavelength *λ* travelling through a medium with a frequency–dependent refractive index *μ*_{ }(*x*), by *λ* = *c*_{ }/*f*_{ }*μ*_{ }(_{ }*f*_{ }) where *c* is the speed of light in a vacuum.

**is also exemplified:** by the dispersion relation for the de Broglie wave of a free particle $\omega = \dfrac{\hbar k^2}{2m}$, where *ω* is the angular frequency and *k* is the corresponding angular wavenumber.

dispersive power

**is:** the ability of an optical medium to produce dispersion for a given optical power or focal length. High or low dispersive power corresponds to high or low refractive index, respectively. [PHYS 6.4]

displacement

**from:** one point in space to another

**is:** the change in position from the first point to the second. [PHYS 2.1]

**is represented:** by a vector. The displacement **s** from a point with position vector **r**_{1} to a point with position vector **r**_{2} is given by **s** = **r**_{2} − **r**_{1}. [PHYS 2.2]

**has magnitude:** equal to the distance between the two points. [PHYS 2.2]

**has direction:** along the line from the first point to the second. [PHYS 2.2]

**may be measured:** from any selected reference point, unlike a position vector. [PHYS 2.2]

**has as its SI unit:** the metre (m). [MATH 2.4]

**in one dimension can be represented:** by a single scalar component *s*_{x}. If the selected reference point is at the initial position of the particle, then the displacement of the particle at time *t* is *s*_{x} = *x*_{ }(*t*) − *x*_{ }(0). [MATH 4.1, PHYS 2.1]

**in linear motion is given:** for displacement of an object from its position at time *t*_{1} to its position at time *t*_{2} by the area under a grapharea under the corresponding velocity–time graph between *t*_{1} and *t*_{2}. [MATH 5.1]

displacement–time graph

**for:** a particle moving in one dimension

**is:** a graph of the displacement (from an agreed reference point) of the particle against time. The convention is to plot the displacement vertically and the time horizontally. The gradient of the displacement–time graph is the velocity in that dimension. [PHYS 2.1]

dissipation

**is:** the irreversible loss of energy by a system to its environment as a result of the action of dissipative forces.

dissipative forces

**are:** forces arising from friction, viscosity or similar effects that cause a reduction in relative motion, and are usually accompanied by the production of heat. [PHYS 5.2]

dissociation

**is:** the process of breaking a molecule (or part of a molecule) into its constituent atoms. [PHYS 8.2]

distance

**from:** one point to another

**is:** the magnitude_of_a_vector_or_vector_quantitymagnitude of the displacement from the first point to the second. [MATH 4.1]

**therefore is:** a positive quantity. [PHYS 2.1, PHYS 2.2]

**has as its SI unit:** the metre. [PHYS 2.2]

See basic coordinate geometry in the Maths For Science handbook.

See also path length.

distance–time graph

**is:** a graph used in the analysis of one–dimensional linear motion, where the distance of an object from a reference point is plotted against the time. [PHYS 2.1]

distant–action force

**is:** a force that always exists between two particles without their being in contact and regardless of any intervening matter.

**is exemplified:** by the gravitational force. [PHYS 3.1]

distribution

**of:** values of a given physical quantity *x*

**over:** a number of particles or entities.

**is:** a function *f*_{ }(*x*) which specifies the fraction of the total number of particles which have values of *x* lying within the small interval between *x* and *x* + Δ*x*. [MATH 5.4]

**is defined:** so that this fraction is equal to *f*_{ }(*x*)Δ*x*. [MATH 5.4]

divergent (integral)

**is:** an improper integral with no finite value. [MATH 5.2]

divergent sequence

**is:** a sequence that does not converge. [MATH 1.7]

divergent series

**is:** a series that does not converge. [MATH 1.7]

diverging lens

**is:** a lens which increases the divergence or reduces the convergence of light rays passing through it. [PHYS 6.3]

**is also called:** a concave lens or a negative lens. [PHYS 6.3]

domain (of a function)

**of:** a function

**is:** the range of values of the independent variable over which the function is defined. [MATH 1.3]

Doppler effect

**is:** the effect in which the observed frequency of a wave (such as an acoustic wave or an electromagnetic wave) is changed when the source of the wave and the observer are moving with respect to each other. [PHYS 5.7]

**causes:** an increase in the observed frequency of the wave if the source and observer are moving closer together, and a decrease in the observed frequency of the wave if the source and observer are moving apart. [PHYS 5.7]

dose equivalent

**is:** a quantity that quantifies the biological hazard of ionizing radiation [PHYS 9.3]

**is defined:** as the product of the absorbed dose and the appropriate radiation weighting factor. [PHYS 9.3]

**has as its SI unit:** the sievert, Sv. [PHYS 9.3]

dot product

See scalar product.

double–angle formulae

**are:** a class of trigonometric identities. [MATH 1.6]

See trigonometric functions in the Maths For Science handbook for details.

double bond

**is:** a chemical bond between two atoms, which is equivalent to two single bonds. [PHYS 8.4]

**arises:** in electronic theories of bonding, from the sharing of two pairs of electrons. [PHYS 8.4]

double cone

**is:** the surface produced by extending to infinity (in both directions) every straight line on the surface of a cone. [MATH 2.3]

double–argument identities

**are:** members of a class of hyperbolic function identities. [MATH 4.6]

See hyperbolic functions in the Maths For Science handbook.

doublet

**in:** a line spectrum

**consists:** of two spectral lines whose wavelengths are almost equal. [PHYS 8.2]

**arises:** when two transitions have almost the same energy difference.

**appears:** if at all, in each order of diffraction from a diffraction grating (except in the zeroth order). [PHYS 8.2]

driven oscillations

**describes:** the behaviour exhibited by a driven oscillator. [PHYS 5.4, PHYS 5.5]

driven oscillator

**is:** an oscillating system that is supplied with energy (continuously or periodically) by an externally applied driving force.

**is exemplified:** by a mechanical oscillator consisting of a particle of mass *m* moving in one dimension along the *x*–axis subject to a restoring force −*kx* a damping force −*bυ*_{x} and a driving force *F*_{0}_{ }sin(*Ωt*), so that its displacement from equilibrium, *x* at time *t* satisfies the equation of motion:

$m\dfrac{d^2x}{dt^2} = -kx - b\dfrac{dx}{dt} + F_0\sin({\it\Omega}t)$

and consequently will eventually exhibit forced oscillations described by

$x(t) = A_0\sin({\it\Omega}t+\phi)$

where $A_0 = \dfrac{F_0/m}{\sqrt{(\omega_0^2-{\it\Omega}^2)^2+(\gamma{\it\Omega})^2}}$ and $\phi = \arctan\left(\dfrac{-\gamma{\it\Omega}}{\omega_0^2-{\it\Omega}^2}\right)$ with $\omega = \sqrt{k/m}$ and *γ* = *b*_{ }/*m*. [PHYS 5.2, PHYS 5.3]

**is also exemplified:** by an electrical oscillator consisting of an inductance *L* in series with a capacitance *C* and a resistance *R* driven by an applied voltage *V*_{0}_{ }sin(*Ωt*). In such a system the charge *q* stored on the capacitor at time *t* is described by the same equations as the driven mechanical oscillator, subject to the replacement of *m*, *k*, *b* and *F*_{0} by *L*, (1/*C*), *R* and *V*_{0}, respectively. [PHYS 5.4]

**has angular frequency:** *Ω* which is completely independent of the natural frequency *ω* of the oscillating system in the absence of driving or damping forces. [PHYS 5.4]

**displays amplitude:** *A*_{0}, which is generally dependent on the angular frequency (Ω) of the driver and which may exhibit resonance at a particular driving frequency. [PHYS 5.4]

driving force

**is:** one of a trio of forces that determine the behaviour of a driven oscillator: restoring force, damping force and driving force. [PHYS 5.2, PHYS 5.3, PHYS 5.5]

ductile region

**is:** the part of the loading curve (the graph of stress against strain) of a material over which it exhibits plasticity.

**is also called:** the plastic region. [PHYS 7.6]

dummy variable

**is:** the variable of integration which is used in a definite integral. [MATH 5.2]

**is named:** ‘dummy’ since it does not appear in the final answer, so its identity is unimportant. [MATH 5.2]

**more generally is:** in a particular calculation, any variable that does not appear in the final result of that calculation.

dynamic equilibrium

**is:** a state of a multi–member system in which there is no time–dependence in the average properties of the system as a whole, but in which there are changes and fluctuations in the states of the individual members of the system. [PHYS 7.6]

dynamic friction

See sliding friction.

dynamics

**is:** the study of how forces give rise to changes in motion. [PHYS 2.3]

Compare with kinematics.

dynamo

**is:** a device that generates an induced voltage by rotating a coil within a magnetic field. [PHYS 4.4]

**produces:** depending on the arrangement of the connections to the external circuit, an output which may be either a.c. or d.c. An a.c. dynamo is also known as an alternator. [PHYS 4.4]

e

**is:** a numerical constant, whose value to eight decimal places is 2.718 281 83 [MATH 1.5]

**can be defined:** by $\displaystyle {\rm e} = \lim_{n\to\infty}(1+1/n)^n$. [MATH 1.5]

**equivalently can be defined:** by $\displaystyle {\rm e} = \lim_{m\to0}(1+m)^{1/m}$. [MATH 1.5]

**is the basis:** of the exponential function e^{x}. [MATH 1.5]

**is used:** as the base of natural logarithms. [MATH 1.5]

**is:** an irrational number. [MATH 1.5]

Contrast with the (italic) *e* used to represent the charge on the proton.

*e*

**is:** the symbol used to represent the electric charge on a proton, one of the fundamental physical constants.

**has the value:** 1.602 × 10^{−19} C, to three decimal places.

**is equal in magnitude:** to the negative charge carried by the electron. [PHYS 3.3]

See quantization of charge.

Contrast with the (non–italic) e used to represent the base of natural logarithms.

Earth satellite

**is:** any object in orbit around the Earth, whether natural (the Moon) or artificial (e.g. communication or meteorological satellites). [PHYS 2.6]

**must have:** an orbit that is circular or elliptical (to a first approximation). [PHYS 2.6]

earth potential

**is usually defined:** to be at zero potential and is used as a reference potential in conventional circuit measurements. [PHYS 4.1]

earthed

**describes:** a conduction_of_electricityconducting body, or a point on a body, that is connected to the Earth by an electrical_conductorelectrically conducting pathway. [PHYS 4.1]

**implies:** that the conduction_of_electricityconducting body or point is at earth potential. (The Earth may be regarded as an enormous reservoir of mobile charge at a fixed potential (earth potential), so any conduction_of_electricityconducting body (or point) that is earthed will quickly acquire earth potential.) [PHYS 4.1]

earthing

**is:** the process of connecting a body to the Earth by a conduction_of_electricityconducting pathway so that it is earthed. [PHYS 4.1]

**allows:** charge on a charged electrical_conductorconductor to flow to the Earth until the electric potential of the electrical_conductorconductor is equal to that of the Earth, i.e. is at earth potential. [PHYS 4.1]

**is a special case:** of charge sharing. [PHYS 3.3]

eccentricity

**of:** a given conic section

**is:** the ratio of the distance PF from any point P on the conic section to a focus F of the conic section, to the perpendicular distance PD from the point P to the directrix (i.e. *e* = PF/PD). [MATH 2.3]

**is exemplified:** by the eccentricity *e* of an ellipse for which 0 ≤ *e* < 1, and the lengths of the semi–major axis *a* and the semi–minor axis *b* are related by $b = a\sqrt{1-e^2}$.

eddy current

**is:** an induced current which circulates entirely within the body of a electrical_conductorconductor. [PHYS 4.4]

effective area

**of:** a (current–carrying) coil of *N* turns, all in the same plane and each of geometrical area *A*

**is equal:** to *NA* [PHYS 4.3]

See magnetic dipole moment.

efficiency

**of:** a piece of equipment

**generally is:** the dimensionless ratio of the amount of a physical quantity extracted from the equipment to the amount of the same physical quantity supplied to the equipment.

efficiency (of a heat engine)

**is:** the ratio of the useful work delivered from the heat engine, to the heat supplied to the heat engine, *η* = Δ*W*_{ }/(*Q*_{1} − *Q*_{2}). [PHYS 7.4]

efficiency of a reversible heat engine

**operating:** between two fixed temperatures *T*_{hot} and *T*_{cold}

**is:** *η* = 1 − *T*_{cold}/*T*_{hot}. [PHYS 7.4]

eigenfunction

**of:** a mathematical operator as used in quantum mechanics.

**is:** a function *ψ*_{ }(*x*) which, when operated on by the operator, produces a real number multiplied by *ψ*_{ }(*x*). The real number is the eigenvalue of the operator. [PHYS 10.4, PHYS 11.1, PHYS 11.2, PHYS 11.3]

See eigenvalue, eigenvalue equation and spatial wavefunction.

eigenvalue

**of:** a mathematical operator as used in quantum mechanics.

**is:** the real number which appears when the operator acts on one of its eigenfunctions to produce the eigenfunction multiplied by a real number. [PHYS 10.4, PHYS 11.1, PHYS 11.2, PHYS 11.3]

See eigenfunction, eigenvalue equation and energy level.

eigenvalue equation

**is:** an equation in which an operator acts on an eigenfunction to produce the eigenfunction, multiplied by an eigenvalue. That is, for an operator $\hat{\rm O}$,

$\hat{\rm O}f = \lambda f$

where *f* is an eigenfunction of $\hat{\rm O}$, and *λ* is the eigenvalue of $\hat{\rm O}$ belonging to the particular eigenfunction. [PHYS 10.4, PHYS 11.1, PHYS 11.2, PHYS 11.3]

**permits:** more than one (and possibly an infinite number) of eigenvalues and eigenfunctions for a given operator. In physical problems $\hat{\rm O}$ is most commonly a differential operator, but it can take other forms. In quantum physics, *f* is commonly a spatial wavefunction (i.e. an eigenfunction of the energy operator, the Hamiltonian). [PHYS 10.4, PHYS 11.1, PHYS 11.2, PHYS 11.3]

Einstein model

**is:** a model of the specific heat of a solid.

**postulates:** that a solid behaves as though composed of independent quantum harmonic oscillators characterized by a common classical frequency. [PHYS 11.4]

**predicts:** that the specific heat will be small near absolute zero. [PHYS 11.4]

See Debye model.

Einstein’s mass–energy equation

**is:** the equation, *E* = *mc*^{2}, which gives the mass *m* associated with an amount of energy *E*, where *c* is the speed of light in a vacuum. [PHYS 2.4, PHYS 9.1]

**is one of the consequences:** of Einstein’s special theory of relativity. [PHYS 2.4, PHYS 9.1]

Einstein’s photoelectric equation

**is:** an equation that relates the maximum kinetic energy of electrons released in the photoelectric effect to the frequency *f* of the incident light, the work function *ϕ* of the surface and Planck’s constant *h*:

$hf-\phi = \frac12m_{\rm e}\upsilon_{\rm max}^2$. [PHYS 10.1]

Einstein’s special theory of relativity

**is based:** on two postulates:

Postulate 1: The laws of physics can be written in the same form in all inertial frames of reference.

Postulate 2: The speed of light (in a vacuum) has the same constant value, *c* in any inertial frame of reference.

**has deep consequences:** among which are these:

1. If two spatially separated events are measured as simultaneous in one inertial frame, they will not generally be measured as simultaneous in another inertial frame which is moving relative to the first frame.

2. If a clock is measured as moving in an inertial frame, it will also be measured as running slow (losing time) in that inertial frame.

3. If an object is measured as moving in an inertial frame, it will also be measured as contracted in the direction of its motion in that inertial frame. Moreover, its mass will be measured as greater than if it were at rest.

**has been confirmed:** repeatedly by experiment.

elastic

**describes:** the ability of a body to recover fully from a distortion and to store energy (as strain potential energy) while distorted, so long as it is not strained beyond its elastic limit. [PHYS 2.4, PHYS 5.2, PHYS 5.7]

elastic body

**is:** a deformable body that returns to its original shape when the cause of any deformation is removed, unless the amount of deformation exceeds the elastic limit of the body. [PHYS 2.4]

elastic collision

**is:** a collision during which the total kinetic energy of the system of interacting bodies is conserved. [PHYS 2.4, PHYS 2.5]

elastic limit

**of:** an elastic body.

**is:** the maximum change in length under which the body still obeys Hooke’s law. [PHYS 2.3]

**is also:** the maximum stress that a solid can sustain without undergoing permanent deformation. [PHYS 7.6]

**equivalently is:** the point on the loading curve which marks the end of the elastic region and the start of the plastic region. [PHYS 7.6]

**is also called:** the yield point. [PHYS 7.6]

elastic material

**is:** a material which fully recovers its previous physical and mechanical state, with zero strain, when the stress is removed. [PHYS 7.6]

elastic modulus

See modulus of elasticity.

elastic region

**is:** the part of the loading curve of a material, over which the material behaves as an elastic material. [PHYS 7.6]

**extends:** from zero stress to the elastic limit or yield point. [PHYS 7.6]

electric cell

**is:** a device essentially consisting of two dissimilar electrodes dipping into an electrolyte solution. Chemical reactions between the electrodes and electrolyte produce ions. When the cell is connected to an external circuit, there is a flow of charge within the electrolyte and around the external circuit, i.e. the cell is a source of direct current. [PHYS 4.5]

electrical charge

**is:** a fundamental property of matter which determines whether or not particles or bodies experience electrical interactions. [PHYS 3.3]

**is classified:** into two types: positive and negative. Charges of the same type repel each other, charges of opposite types attract each other. [PHYS 3.3]

**is carried:** by some fundamental particles, e.g. the electron carries a charge of −*e*, the proton a charge of +*e*. Some others carry none, e.g. the neutron is uncharged. [PHYS 3.3]

**has as its SI unit:** the coulomb (C), where 1 C = 1 A s (i.e. 1 ampere second).

See quantization of charge.

electric current

**through:** a surface

**is:** the rate *dq*_{ }/*dt* at which (net) charge *q* is transferred across that surface. [PHYS 4.1, PHYS 5.5]

**is due:** in metallic conductors, to the movement of electrons. In other media, it can be due to the movement of other charged particles (e.g. ions in solution_chemicalsolution). [PHYS 4.1]

**has direction:** which is defined conventionally as the direction in which positive charge would move, though in many cases the current is actually a flow of negatively-charged particles in the opposite direction. [PHYS 4.1]

**has as its SI unit:** the ampere (A). [PHYS 4.1]

electric dipole

**consists:** of equal and opposite electric charges +*q* and −*q* separated by a distance *d* [PHYS 3.3]

**can be found:** in molecules containing a variety of atoms, where the electrons forming the bonds between atoms of different chemical elements are not shared equally between the two atoms involved. The result is equivalent to a dipole, in which one atom has a slight positive charge and the other a slight negative charge. [PHYS 3.3]

See electric dipole moment. [PHYS 3.3]

electric dipole moment

**is:** the product of the charge magnitude and charge separation in an electric dipole. For a dipole consisting of charges +*q* and −*q* separated by a distance *d*, the dipole moment is *qd*. [PHYS 3.3]

**is strictly:** a vector quantity whose magnitude_of_a_vector_or_vector_quantitymagnitude is as defined above, and whose direction is the same as for the displacement from the negative to the positive charge.

electric field

**throughout:** a region of space

**is:** a vector field that gives rise to an electrical force on a test charge placed at any point in the region. [PHYS 3.1]

**is defined:** at any point specified by a position vector **r**, as the electrostatic force per unit positive charge that would act on a test charge placed at that point. So, generally,

${\boldsymbol E}({\boldsymbol r}) = \dfrac{{\boldsymbol F}_{\rm el}(\text{on }q\text{ at }{\boldsymbol r})}{q}$

whether the test charge *q* is positive or negative. [PHYS 3.1, PHYS 3.2]

**is related:** to the electric potential by the requirement that it points in the direction of most rapid decrease of the potential, and has a magnitude given at every point by the magnitude of the rate of change of the potential in that direction (e.g. in the radial direction from an isolated point charge, so that *E*_{r} = −*dV*_{el}/*dr*). It therefore always points in a direction at right angles to lines or surfaces of equipotential surfaceequal potential, and from high potential towards low potential. [PHYS 3.1, PHYS 3.3]

**has as its SI unit:** the newton per coulomb (N C^{−1}) or, equivalently, the volt per metre (V m^{−1}). [PHYS 3.1, PHYS 3.3]

electric field lines

**are:** a means of representing an electric field. [PHYS 3.3]

**are drawn:** so that at any point the tangent_to_a_curvetangent to the line represents the direction of the field at that point. [PHYS 3.3]

**therefore are directed:** away from a positive charge and towards a negative charge. [PHYS 3.3]

**have spacing:** which is related to the electric field strength, i.e. where the lines are close together the field is strong and where they are further apart the field is weaker. [PHYS 3.3]

**always cut:** equipotential surfaces at right angles. Where these are closest together, their rate of change is greatest, and so there the electric field is strongest. [PHYS 3.1, PHYS 3.3]

electric field strength

**at:** any point

**is:** the magnitude of the electric field at that point. [PHYS 3.1]

electric force

See: electrostatic force.

electric potential

**at:** a given point in space

**is:** the electric potential energy per unit positive charge at that point. [PHYS 3.1, PHYS 3.3]

**is also:** the electric potential difference (i.e. voltage difference) between the given point and a point at which the electric potential energy is defined to be zero. In an electrical circuit the earth, or the negative terminal of a power supply, is usually taken to be at zero potential. [PHYS 3.1, PHYS 3.3]

**has as its SI unit:** the volt, (V). [PHYS 4.1]

electric potential difference

**between:** point A and point B in an electric field

**is:** the difference *V*_{B} − *V*_{A}, in electric potential energy per unit positive charge between the two points (i.e. Δ*V*_{el} = Δ*E*_{el}/*q*). [PHYS 2.6, PHYS 4.1]

**is therefore:** the negative of the work done per unit charge by an electric field when a unit charge is moved from A to B. [MATH 2.6]

**is also called:** voltage difference. [PHYS 4.1]

**has as its SI unit:** the volt, (V), where 1 V = 1 J C^{−1} (i.e. 1 joule per coulomb). [PHYS 4.1]

electric potential energy

**is:** the energy a charged particle has by virtue of its position in an electric field. [PHYS 3.1, PHYS 3.3, PHYS 4.1]

**requires for its full definition:** a position of zero electric potential energy to be arbitrarily chosen, since only differences in electric potential energy are physically meaningful. [PHYS 3.1, PHYS 3.3]

**changes:** in going from point A to point B, by an amount equal to the negative of the work done by the electric field when the charged particle is moved from A to B. [PHYS 3.1]

**is exemplified:** by the electric potential energy of a particle of charge *q*_{2} in the electric field of a particle of charge *q*_{1}, when the distance between the two particles is *d*. Subject to the conventional choice that *E*_{el} = 0 when *r* → ∞, this is given by

$E_{\rm el} = \dfrac{q_1q_2}{4\pi\varepsilon r}$

where *ε* is the permittivity of the medium between the charges. [PHYS 3.1, PHYS 3.3]

**is related:** to the electric potential *V*_{el} in a region by *E*_{el} = *qV*_{el}, so when a charge *q* moves through a voltage difference (i.e. an electric potential difference) Δ*V*_{el}, the change in electric potential energy Δ*E*_{el}, is given by Δ*E*_{el} = *q*Δ*V*_{el}. [PHYS 4.1]

**often is referred:** to as ‘electrical energy’ or electrostatic potential energy. [PHYS 3.1, PHYS 3.3]

**has as its SI unit:** the joule (J).

electrical

**means:** pertaining to electricity.

See also electrostatics and electromagnetism.

electrical breakdown

**in:** an electrical insulator which is subjected to an electric field above a certain threshold

**occurs:** when some of the electrons become detached from their parent atoms and flow through the material − which thus becomes, temporarily, an electrical conductor. [PHYS 3.3]

electrical components

**is:** a general term for electrical devices, particularly those that are used in circuits.

electrical conductor

**is:** a material containing an abundance of mobile charged particles that are free to move throughout the whole of the material. [PHYS 3.3, PHYS 4.1]

**has:** a low resistivity. [PHYS 4.1]

**has typically:** in terms of the band theory of solids, a partly filled valence band at absolute zero. [PHYS 11.4]

**is exemplified:** by any metal. [PHYS 3.3, PHYS 4.1, PHYS 11.4]

**is the opposite:** of an electrical insulator. [PHYS 3.3, PHYS 11.4]

electrical energy

**is:** energy supplied by an electrical power supply.

See also electric potential energy.

electrical insulator

**is:** a material containing a negligible number of mobile charged particles. [PHYS 3.3, PHYS 4.1]

**has:** an extremely high resistivity. [PHYS 4.1]

**has typically:** in terms of the band theory of solids, an empty conduction band separated by a substantial gap (e.g. 5 eV) from a full valence band at absolute zero. [PHYS 11.4]

**is the opposite:** of an electrical conductor. [PHYS 3.3]

**can be used:** to prevent the flow of current between points at different potential. [PHYS 4.1]

See also electrical breakdown.

electrical interaction

See electromagnetic interaction.

electrical oscillator

**is essentially:** an inductor connected across a capacitor to form a simple circuit in which charge stored on the capacitor may oscillate, possibly also containing a resistor (to provide damping) and possibly subject to an externally supplied voltage to make it a driven oscillator.

See simple harmonic oscillator, damped electrical oscillator, driven oscillator, as appropriate.

electricity

**is:** a general term for electric charge, whether static or moving, as in an electric current.

electrochemical series

**is:** a listing of chemical elements in order of their electrode potential. The further apart two element_chemicalelements are in the series, the greater is the open circuit voltage (e.m.f.) produced when they form the two electrodes in a simple electric cell. The element_chemicalelement with the greater (more positive) electrode potential forms the positive terminal of the cell. [PHYS 4.5]

electrode

**is:** an electrically conducting structure used to emit or collect charge, often (though not always) a metal plate or grid.

electrode potential (of an element)

**is:** the open circuit voltage (e.m.f.) obtained by using the element to make one terminal of an electric cell, whose other terminal is a hydrogen electrode. The (theoretical) magnitude of the open circuit voltage (e.m.f.) of any simple cell is found by subtracting the two electrode potentials one from the other. [PHYS 4.5]

electrolyte

**is:** a substance, usually in the form of a solution_chemicalsolution, which allows the conduction_of_electricityconduction of electricity by the movement of positive and negative ions. [PHYS 4.5]

electrolytic capacitor

**is:** a capacitor whose plates are made from two different materials separated by an electrolyte.

**must be connected:** the correct way round in a d.c. circuit. [PHYS 4.5]

electromagnet

**is:** a coil or solenoid would around a core of ferromagnetic material and which then exhibits strong magnetic induction when a current flows. [PHYS 4.2]

electromagnetic force

**is:** the total force on a charged particle in an electric field and/or magnetic field, found by adding the separate electrostatic force and magnetic force that would be produced by each field acting independently. [PHYS 4.3]

**is described:** by the Lorentz force law

${\boldsymbol F} = q({\boldsymbol E}+{\boldsymbol\upsilon}~{\boldsymbol\times} ~{\boldsymbol B})$. [PHYS 4.3]

**is also called:** the Lorentz force. [PHYS 4.3]

**arises:** from the electromagnetic interaction, one of the four known fundamental interactions in nature. [PHYS 9.2]

electromagnetic induction

**is:** the phenomenon that results in the production of an induced voltage in a electrical_conductorconductor by changing a magnetic field near the electrical_conductorconductor, or by moving the electrical_conductorconductor within a magnetic field (motional induction). [PHYS 4.4]

See Faraday’s law and Lenz’s law.

electromagnetic interaction

**is:** the fundamental interaction that gives rise to electromagnetic force. [PHYS 9.2]

**comprises:** together with the weak_interactionweak, strong_interactionstrong and gravitational_interactiongravitational interactions, the four known fundamental interactions of nature. [PHYS 9.2]

See gravitational force, strong nuclear force, weak_interactionweak nuclear force.

electromagnetic pick–up

**is:** the induced voltage caused in a circuit by magnetic field fluctuations near the circuit. [PHYS 4.4]

electromagnetic radiation

**is:** radiation consisting of fluctuating electric_fieldelectric and magnetic fields that can propagate through space, or through suitable media, as electromagnetic waves characterized by a wavelength *λ* and a frequency *f*. Many aspects of the interaction of electromagnetic radiation with matter require the use of quantum theory for their accurate description.

**is exemplified:** by familiar phenomena such as visible light, radio waves and X–rays, which are all parts of the electromagnetic spectrum, corresponding to different wavelengths of electromagnetic radiation.

**can transfer:** energy and momentum. [PHYS 7.3]

See radiation pressure.

electromagnetic spectrum

**is:** the complete range of electromagnetic waves. [PHYS 6.1, PHYS 7.3]

**extends:** from long-wavelength radio waves, through microwaves, infrared, visible light, ultraviolet and X–rays to short-wavelength γ–rays. [PHYS 6.1, PHYS 7.3]

electromagnetic wave

**is:** a pattern of mutually perpendicular, oscillating electric_fieldelectric and magnetic fields that can travel through a vacuum at the speed of light, *c*. [PHYS 6.1]

**is a form:** of transverse wave. [PHYS 6.1]

**is characterized:** in the simplest case (a linearly polarized, monochromatic, plane wave), by its direction of propagation, plane of polarization, amplitude, wavelength and frequency. (In a vacuum the wavelength and frequency are related by *c* = *f*_{ }*λ*). [PHYS 6.1]

**has speed:** *c*_{ }/*μ* in materials other than a vacuum, where *μ* is the refractive index of the material. [PHYS 6.1]

electromagnetism

**is:** the branch of physics that encompasses all electrical and magnetic phenomena, including the interactions of charges and magnets with electric and magnetic fields and the production and propagation of electromagnetic waves. [PHYS 4.2]

electromotive force (e.m.f.)

**is:** an alternative term for the open circuit voltage of a voltage generator. [PHYS 4.1]

**is not:** a force in the sense defined by Newton’s second law.

electron

**is:** an elementary particle that is a constituent of every atom. [PHYS 3.3, PHYS 8.1]

**has:** charge −*e* = −1.602 × 10^{−19} C and mass *m* = 9.109 56 × 10^{−31} kg, approximately 1/1836 times the mass of a proton. [PHYS 3.3, PHYS 8.1]

**is liberated:** from atoms when the atoms are ionized in a discharge tube, as deduced by its discoverer J.J. Thomson (1856–1940). [PHYS 8.1]

**has:** no known internal structure at the time of this writing. [PHYS 8.1]

electron antineutrino

**is:** an elementary particle, the antiparticle of the electron neutrino. [PHYS 9.2]

**always accompanies:** the electron emitted in β_decayβ^{−}–decay. [PHYS 9.2]

electron band

See energy band.

electron cloud

**in:** the quantum model of the atom

**is:** the concept that replaces the electron orbits of more primitive models, such as the Bohr model.

**has:** for a given stationary state of the atom, a density at every point that is proportional to the probability density |_{ }*Ψ*_{ }(*r*, *θ*, *ϕ*)_{ }|^{2} of the associated wavefunction. [PHYS 11.3]

electron diffraction

**is:** the diffraction of electrons by a regular array of atoms (as in a crystal). [PHYS 7.1]

**is a consequence:** of the wave–like behaviour of electrons, as described by quantum physics. [PHYS 7.1]

**results in:** a diffraction pattern with sharp local maxima of intensity in directions determined by Bragg’s law. [PHYS 7.1]

See de Broglie wave.

electron microscope

**is:** a microscope that uses the (short wavelength) wave–like behaviour of beams of electrons to produce images with much better resolution than those possible with optical microscopes. [PHYS 7.1]

electron neutrino

**is:** an elementary particle that has zero charge and such a small mass (if any) that it is currently indistinguishable from zero.

**always accompanies:** the positron which is emitted in β_decayβ^{+}–decay. [PHYS 9.2]

electron pair

**is:** two electrons that occupy the same quantum state apart from having opposed spin angular momentumspins.

electron shell

See shell.

electron spin

**is:** the intrinsic angular momentum of an electron. [PHYS 8.3]

**is described:** by an electron spin quantum number *s* = 1/2 and hence permits two possible values for the spin magnetic quantum number, *m*_{s} = 1/2 or *m*_{s} = −1/2, implying that the *z*–component of the spin must be either $+\hbar/2$ or $-\hbar/2$ when measured along an arbitrarily chosen *z*–axis. [PHYS 8.3]

**creates:** electron spin magnetism. [PHYS 4.2]

**helps to account:** for the magnetic properties of the electron and those of atoms that contain unpaired electrons. [PHYS 8.3]

electron spin magnetism

**is:** an intrinsic property of an electron (like electric charge), such that the electron behaves as a magnet with a measurable magnetic dipole moment. [PHYS 4.2]

electron subshell

See subshell.

electron tunnelling

**is:** a special case of quantum tunnelling, in which an electron tunnels through a potential barrier whose height exceeds the total energy of the electron. [PHYS 10.4]

**is important:** in various electronic devices, including the tunnel diode.

electronegativity

**is:** a numerical measure of the ability of an atom to attract electrons to itself during chemical reactions. [PHYS 8.4]

**is highest:** (~4.0) in the region of the periodic table occupied by fluorine and chlorine. [PHYS 8.4]

electronic configuration

**is:** a description of the distribution of electrons within shells and subshells in an atom, using the quantum numbers that describe the quantum states of the electrons. [PHYS 8.3, PHYS 8.4]

**often is presented:** in shorthand form using the s–p–d–f notation for subshells. For example, the ground state configuration of sodium is represented as 1s^{2}2s^{2}2p^{6}3s^{1}, meaning that

- in the
*n* = 1 shell (the lowest-energy shell, closest to the nucleus), there are two electrons in the s subshell (the subshell in which (*l* = 0)). This subshell is full.
- in the
*n* = 2 shell, there are two electrons in the s subshell (*l* = 0)
- in the
*n* = 2 shell, there are six electrons in the p subshell (*l* = 1)
- in the
*n* = 3 shell, there is one electron in the s subshell (*l* = 0) [PHYS 8.3, PHYS 8.4]

electronic structure

**is:** a synonym for electronic configuration. [PHYS 8.3]

electronvolt, eV

**is:** a non-SI unit of energy.

**is defined:** as the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt. [PHYS 8.3, PHYS 9.1]

**is equal:** to 1.602 × 10^{−19} J (to four significant figures). [PHYS 3.3]

**is commonly used:** in large multiples such as MeV (1 MeV = 10^{6} eV) and GeV (1 GeV = 10^{9} eV) in nuclear physics and elementary particle physics. [PHYS 9.1]

electrostatic constant

**is:** the physical constant 1/(4*πε*_{0}) that appears in Coulomb’s law. [PHYS 3.1]

**has the value:** 1/(4*πε*_{0}) = 8.988 × 10^{9} N m^{2} C^{−2} (to four significant figures). [PHYS 3.1]

See Coulomb’s law, permittivity of free space (*ε*_{0}) [PHYS 3.1]

electrostatic force

**is:** the force that acts on a charged body due to its location in a static electric field. For a test charge *q* located at a point with position vector **r**, where the electric field is E(r), the electrostatic force is

**F**_{el}_{ }(on *q* at **r**) = *q***E**_{ }(**r**). [PHYS 3.3]

**is exemplified:** by the force (described by Coulomb’s law) that one charged particle exerts on another by virtue of the electric field that it creates. Two particles with charge of the same sign repel one another, and two particles with charge of the opposite sign attract one another. [PHYS 3.1]

**is given:** for a positive unit charge by the negative derivative of electric potential energy in the direction of maximum change (e.g. in the radial direction from an isolated point charge, *F*_{r} = −*dE*_{el}/*dr*). [PHYS 3.1, PHYS 3.3]

electrostatic induction

**is:** the process by which a region of an initially electric_chargeuncharged object can become charged due to the influence of an electric field (usually due to another charged object) which causes a rearrangement of charge on the original object. [PHYS 3.3]

electrostatic potential energy

See electric potential energy.

electrostatic screening

**is created:** by a perfectly conduction_of_electricityconducting shell containing no free charges. [PHYS 3.3]

**ensures:** that no electric field can exist inside the shell. [PHYS 3.3]

electrostatics

**is:** the study of the electrical interaction between charged particles which are not moving in relation to one another, or in relation to the observer. [PHYS 3.3]

element

**is:** a small part of something, often of a given solid body, or a volume of fluid. For example, a body of mass *M* may be considered to be composed of many separate elements of mass Δ*m*_{i} such that $\displaystyle M = \sum_i m_i$

element (chemical)

See chemical element.

element (of a set)

**is:** an entity that is a member of a set.

element of integration

**is:** an infinitesimal increment in the variable with respect to which an integration is to be performed. [MATH 5.1, MATH 5.2]

**is exemplified:** by the *dx* which appears at the end of the definite integral $\displaystyle \int_a^b f(x)\,dx$. [MATH 5.1,MATH 5.2]

elementary entity

See mole.

elementary functions

**are:** a slightly ill–defined class of functions including the common (and ‘uncomplicated’) functions, such as sine, logarithm and arctangent. [MATH 1.7]

See the Maths For Science handbook, which includes graphs of many of these functions.

elementary particles

**are:** subatomic particles believed, or formerly believed, not to have any constituents. Examples include electrons, protons, neutrons and photons. It is now widely believed that protons and neutrons are in fact composed of constituents called quarks, and a modern listing of ‘truly’ elementary particles would consist of three families: the leptons (including the electron), the quarks (including the charged constituents of many other ‘elementary particles’), and the exchange particles (including the photon and the various other particles that are responsible for the fundamental interactions between quarks and leptons).