A transparent sphere of radius R ahs a cavity of radius `R//2` as shown in figure,. Find the refractive index of the sphere if a parallel beam of light falling on left sureface focuses at point P.

Answer

`(3+sqrt(5))/(2)`

Explanation

Let refractive index of glass be `mu`
Let after first refraction, image distance be v, then
`(mu)/(v)-(1)/(oo)=(mu-1)/(R)rArrV=(muR)/(mu-1)`
Now second refraction will take place.
So distance of first image from O is
`mu_(1)=(muR)/(mu-1)-R=(R)/(mu-1)` and image is formed at R.
`(1)/(R) -(mu(mu-1))/(R)=(2(1-mu))/(R)rArr mu^(2)-3mu+1=0`
So`mu=(3+sqrt(5))/(2)`