A glass sphere, refractive index 1.5 and radius 10cm, has a spherical cavity of radius 5cm concentric with it. A narrow beam of parallel light is directed into the sphere. Find the final image and its nature.

Explanation

We will have single surface refractions successively at the four surfaces `S_(1), S_(2),S_(3)` and `S_(4)` . Do not forget to shift origin to vertex of the respective surface.
Refractive at first surface `S_(1)` : Light travel from air to glass.
`(1.5)/(upsilon_(1))-(1)/(oo)=((1.5-1))/((+1))`
`upsilon_(1)=30cm`.
First image is the object for the rafraction at second surface.
For refractive at surface `S_(2):` Light travels from glass to air.
`(1)/(upsilon_(2))-(1.5)/((+25))=(1-1.5)/((+5))`
`upsilon _(2)=-25 cm`
For refraction at surface `S_(3):` Light travels from air to glass.
`(1.5)/(upsilon_(3))-(1)/((-35))=((1.5-1))/((-5))`
`upsilon _(3)=-35//3 cm`
For refraction at surface `S_(4):` Light travels from glass to air.
`(1)/(upsilon_(4))-(1.5)/(-(35//3+5))=(1-1.5)/(-10)`
`upsilon_(4)=-25 cm`
The final image is virtual, formed at 25 cm to the left of the vertex of surface `S_(4)`.