A hollow sphere of glass of inner and outer radii R and 2R respectively has a small mark on its inner surface. This mark is observed from a point outside the sphere such that the centre of the sphere lies in between. Prove that the mark will appear nearer than it really it, by a distance `(mu-1)(R)//(3mu-1)`

Explanation

Refraction at 1:
`mu_1=1,mu=mu,u=-2R,R=-R,v=?`
`(mu)/(v)-(1)/(-2R)=(mu-1)/(-R)`
`(mu)/(v)=(1-mu)/(R)-(1)/(2R)=(2-2mu-1)/(2R)`
`v=(2muR)/(1-2mu)=-(2muR)/(2mu-1)`
Refractiojn at 2:
`mu_1=mu,mu_2=1,u=-(R+(2muR)/(2mu-1))=-((4mu-1)/(2mu-1))R`.
`(1)/(v)-(mu)/(-((4mu-1)/(2mu-1))R)=(1-R)/(-2R)`
`(1)/(v)=(mu-1)/(2R)-(mu(2mu-1))/((4mu-1)R)`
`=((mu-1)(4mu-1)-2mu(2mu-1))/(2(4mu-1)R)`
`=(-3mu+1)/(2(4mu-1)R)`
`v=-(2(4mu-1)R)/((3mu-1))`
`P_2O=3R,P_2I_2=(2(4mu-1)R)/(3mu-1)`
`OI_2=P_2O-P_2I_2=3R-(2(4pi-1)R)/((3mu-1))=((mu01)R)/((3mu-1))`