A hemispherical portion of the surface of a solid glass sphere (mu = 1.5) of radius r is silvered to make the inner side reflecting. An object is placed on the axis of the hemisphere at a distance 3r from the centre of the sphere. The light from the object is refracted at the unsilvered part, then reflected from the silvered part and again refracted at the unsilvered part. Locate the final image formed.

Explanation

Refraction at 1:
`mu_1=1,mu_2-(3)/(2),u=-2R,R=R,v=?`
`(mu_2)/(v)-(mu_1)/(u)=(mu_2-mu_1)/(R)implies((3)/(2))/(v)-(1)/(-2R)=((3)/(2)-1)/(R)`
`(3)/(2v)=0impliesv=infty`
Image is formed at infinite i.e., after refraction rays become parallel to principal axis.
Refraction at 2: Hemispherical portion (silvered) behave as a concave mirror of focal length `f=(R)//(2)`. For it object is at infinite and hence image `I_2` is formed at focus.
Refraction at 1: After energy refrection direction of incident rays is reversed noe direction of incident rays is right to left and `I_2` is object.
`mu_1=(3)/(2),mu_1=1,u=-(3R)/(2),R=-R,v=?`
`(mu_2)/(v)-(mu_1)/(u)=(mu_2-mu_1)/(R)implies(1)/(v)-((3)/(2))/((-3R)/(2))=(1-(3)/(2))/(-R)`
`(1)/(v)=(1)/(2R)-(1)/(R)=-(1)/(2R)`
`v=-2R` i.e., final image `I_3` is formed at pole of concave mirror.