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A ray of light is refracted through a sp...

A ray of light is refracted through a sphere whose material has a refractive index `mu` in such a way that it passes through the extremities of two radii which make an angle `beta` with each other. Prove that if `alpha` is the deviation of the ray caused by its passage through the sphere, then
`cos((beta-alpha)/2)=mu cos(beta/alpha)`

Text Solution

Verified by Experts

The correct Answer is:
A
`delta_(Total)=delta_P+delta_Q`

`:. alpha=(i-r)+(i-r)`
`or i-r=alpha/2….(i)`
Further, in `angleOPQ, r+r+beta=180^@`
`:. r=90^@-beta/2…..(ii)`
From Eq.(i),
`i=r+alpha/2=90^@+((alpha-beta)/2).....(iii)`
`mu=sini/sinr=(sin[90^@+((alpha-beta)/2)])/(sin(90-beta/2))`
`=(cos((beta-alpha)/2))/(cos(beta/alpha))`
or `cos((beta-alpha)/2)=mu cos beta/2` Henced proved
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