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Thermodynamics

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Secondary Rainbow

In the case of secondary rainbow, there are two reflections and two refractions. Suppose a ray AB from the sun strikes the drop at B and after refraction goes along BC. It is reflected at C and D and after refraction at E, finally emerges along EG.

The total deviation, δ = 2(i – r) + 2(180 – 2r) = 360 + 2i – 6 (i)

The rays are concentrated around the direction of minimum deviation.

Differentiating δ with respect to i

dδ/di = 2 – 6 dr/di

But, dδ/di = 0

2 – 6 dr/di = 0

Or, dr/di = 1/3 (ii)

Or, sin r = sin i

Differentiating,

cos r dr/di = cos i

Equating (ii) and (iii)

But, sin r = sin i

Taking of water for red light = 1.329, the angle of deviation = 360 – 129.2 = 230.8. The acute angle = 230.8 – 180 = 50.8˚. Taking of water for violet light = 1.342, the angle of deviation = 360 – 125.48 = 234.52˚. The acute angle = 234.52˚ – 180 = 54.52˚. The angle of inclination for violet rays is more than for red rays.