# Incident angle for the formation of rainbow

A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. The following figure shows a ray that refracts into a drop at point A, is reflected from the back surface of the drop at point B, and refracts back into air at point C....

For complete problem and figure, please see attached file, scroll down to Bonus question.

1. Show that

2. Show that the angle in radians between the ray before it enters the drop at A and after it exits at C (the total angular deflection of the ray is Delta =

3. Use Snell's law to write delta in terms of ** and n, the refractive index of the water in the drop.

4. A rainbow will form when the angular deflection delta is stationary in the incident angle - in other words, when = 0. If this condition is satisfied then all of the rays with incident angle close to will be sent back in the same direction, producing a bright zone in the sky. Let be the value of for which this occurs. Show that *** .

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#### Solution Preview

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I have picked the symbols above to denote the angles to make the typing process simple. Angles used correspond to the angles provided in the figure as follows.

---------------- Diagram ------------

(a) Show that ib = ra, ic = ra and rc = ia

In triagle AOB,

Triangles AOF and OFB are congruent by the sss rule. Hence

ra = ib (proof done)

In similar fasion, ...

#### Solution Summary

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