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

To solve this problem, we need to apply the principles of refraction and the lens formula to determine the height \( h \) of the liquid at which the image of the object formed is onto itself.

1. Understanding the Setup:
We are given a plano-convex base of refractive index \( n = 1.60 \), with the radius of curvature of the convex surface \( R = 9 \, \text{cm} \). The planar surface acts as a mirror. The beaker is filled with liquid of refractive index \( n \), and the liquid level is \( QPR \). The task is to find the height \( h \) of the liquid such that the image of the object formed is onto itself.

2. Refraction at the Curved Surface:
The glass beaker has a curved surface, and the liquid is filling the beaker. The point object \( O \) is located at a height \( h \). We apply the refraction at the curved surface of the beaker using the lens-maker’s formula for refraction at a spherical surface:

\[ \frac{n_2 - n_1}{R} = \frac{1}{f} \] where: - \( n_2 = 1.60 \) (refractive index of the glass), - \( n_1 = n \) (refractive index of the liquid), - \( R = 9 \, \text{cm} \) (radius of curvature), - \( f \) is the focal length of the curved surface.

3. Using the Lens Formula:
We apply the lens formula to find the relationship between object distance \( u \), image distance \( v \), and the focal length \( f \):

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Here, the image formed is real and is formed onto the object itself. Therefore, the object and image distances are equal, so \( v = -u \). Now, substituting the appropriate values, we can solve for the height \( h \). After solving, we find the value of \( h \) for different refractive indices of the liquid.

4. Evaluating the Given Options:
From the calculations, we find that the correct height \( h \) for the liquid is given by two options as follows:

For \( n = 1.42 \), the height \( h = 50 \, \text{cm} \), and for \( n = 1.35 \), the height \( h = 36 \, \text{cm} \). These are the correct solutions based on the setup and calculations.

Final Answer:
The correct options are \( \boxed{(A, B)} \).