Question

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index n up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is (are) correct?

(A) For n = 1.42, h = 50 cm 

(B) For n = 1.35, h = 36 cm 

(C) For n = 1.45, h = 65 cm 

(D) For n = 1.48, h = 85 cm

Answer 1

Correct option is A,B

Since STU is a plane mirror, we can take mirror image of the whole situation about it and final image can be assumed to be at a distance h below the base.

Since object and image are at same distance from equivalent lens, hence \(h= 2F _ {eq}\)

\(\frac {1}{F_{eq}} = \left(\frac {1.6 -1}{1}\right) \left(\frac {2}{9}\right) + \frac {(n-1)}{1}\left (\frac {-2}{9}\right)\)

\(\frac {1}{\frac {h}{2}} = \frac {1.2}{9} + \frac {2(1-n)}{9}\)

\(\frac {2}{h} = \frac {3.2 - 2 n}{9}\)

\(h = \frac {9}{1.6 - n}cm\)

For n=1.42, h = 50 cm (A is correct)

For n=1.35, h = 36 cm (B is correct)

(For n=1.45, h = 60 cm (C is incorrect)

For n=1.48, h = 75 cm (D is incorrect)