The rainbow

The y-coordinate of the incident light can be changed when clicking a mouse in the area of the upper half of the drop of water and doing a drag.

The angle between the incident light and the refracting light is maximized in the point that a dot has been put on the left. Even if the position of the incident light changes in the neighborhood of this point, the direction of the refracting light doesn't change too much.
Therefore, the rainbow can be seen in this direction.

Applet and text by Sadahisa Kamikawa © 1997, last update 1997.1.4

Angle of deflection

(Mirrored from here) The angle through which the rain drop turns the light ray is called the angle of deflection for the ray. As the diagram shows, the angle of deflection depends on the angle at which the light ray strikes the drop.

If one graphs angle of deflection vs. angle of incidence (from data collected in a virtual -- or real! -- experiment), one can clearly see that there is a minimum angle of deflection of approximately 138^o (depending on color). This may ring a bell, since 138^o is the supplement of 42^o, the "rainbow angle" (sometimes called the "Descartes angle"). Why does this minimum value correspond to the angle at which you view a rainbow?

• Notice, from the graph, that a fairly-wide range of angles of incidence produce an angle of deflection very near 138^o, so a fairly-large proportion of the incoming sunlight is concentrated in a beam of light leaving the rain drop at about 138^o. For other angles of incidence, the angles of deflection are more spread out.
• Notice, from the applet, that larger angles of incidence (near the angle producing the minimum deflection) cause more dispersion of the red and blue rays than small angles of incidence.
• The angle at which a light ray strikes a boundary determines the proportion of the ray that is reflected and the proportion that is transmitted. Rays that strike the back of the rain drop at a small angle will reflect a smaller proportion of the ray (more will be transmitted) than rays that strike the back of the raindrop at a larger angle (near the angle that produces the minimum deflection.

So, rays that strike the rain drop at an angle of incidence near the angle producing the minimum angle of deflection will tend to form a concentrated, strong beam in which the colors will be widely separated. Rays that strike the rain drop at a small angle of incidence will tend to pass through the drop, and the part of the rays that are reflected inside the drop are spread out relative to one another, while the colors within the rays are not noticeably separated.

Other links of interest

• Physics of Rainbows discussion, includes applets
• Rainbows Physics, derivation of the equation for the deflection angle