Primary Rainbow

Consider a ray of sunlight incident at the point B of a raindrop. The ray AB after refraction travels along BC and is refracted along CD and finally comes out along DE. The deviation of the ray BC after refraction is (i – r). The deviation of the ray AB after reflection at C is (180 – 2r) and the deviation of the ray CD after refraction at D is (i – r).

Therefore, total deviation = 2(i – r) + (180 – 2r)

δ = 180 + 2i – 4r                   (i)

For the angle of deviation to be maximum or minimum, the differential coefficient of δ with respect to i must be zero.



Or, sin r = sin i



Equating (ii) and (iii)



4 cos² i = (1 – sin2 r) = – sin² r

But, sin r = sin i

∴ 4 cos² i = – sin² i

3 cos² i = – (sin² i + cos² i) = – 1



Taking the refractive index of water for red light = 1.329,

i = 59.6˚ and δ = 137.2˚

180 – 137.2 = 42.8˚

Taking the refractive index of water for violet light = 1.342,

i = 58.8˚ and δ = 139.2˚

180 – 139.2 = 40.8˚

It is to be remembered that sunlight strikes the raindrops at different angles of incidence and undergoes different deviations. Only those rays produce a rainbow which has the angle of incidence corresponding to minimum angles of deviation. All such rays produce concentrated effect of light in the formation of a rainbow. Also, in the primary rainbow the angle of inclination of red light is more on the eye then the violet. Therefore, the outside of the rainbow appears red and the linear violet. The other spectral colours lie in between violet and red in their order.

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