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

The Correct Option is A, B

Optical System Analysis 

Focal Length of Convex Surface SPU

The convex surface SPU acts as a lens with the following focal length:

\[ \frac{1}{f_1} = (1.6 - 1) \times \left( \frac{1}{9} - \frac{1}{\infty} \right) = \frac{0.6}{9}. \]

Focal Length of Planar Surface STU

For the planar surface STU acting as a mirror:

\[ \frac{1}{f_2} = \frac{-2}{\infty} = 0. \]

Effective Focal Length of the System

The effective focal length \( f_e \) of the system is given by:

\[ \frac{1}{f_e} = \frac{1}{f_1} + \frac{1}{f_2} = \frac{1.6 - n}{9}. \]

Image Formation Distance

The distance \( h \) for the image to form onto itself is given by:

\[ h = 2f = \frac{18}{1.6 - n}. \]

Calculations for Given Options

• For \( n = 1.42 \):
• For \( n = 1.35 \):
• For \( n = 1.45 \):
• For \( n = 1.48 \):

Final Answer:

(A), (B)