A planoconvex lens behaves like a concave mirror of 30 cm focal length on silvering the plane surface of the lens, but on polishing by silver on convex surface, it acts as a concave mirror of 10 cm focal length. What will be the refractive index of the material of the lens?

Options

A. 1
B. `1.25`
C. `1.33`
D. `1.5`

Answer

To find the refractive index of the plano-convex lens, we will use the information provided about its behavior when the surfaces are silvered.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A plano-convex lens behaves like a concave mirror of focal length -30 cm when the plane surface is silvered.
- It behaves like a concave mirror of focal length -10 cm when the convex surface is silvered.
- We need to find the refractive index (μ) of the lens material.

2. **Using the Lens-Mirror Combination Formula**:
- When the plane surface is silvered, the lens behaves like a concave mirror. The formula for the equivalent focal length (F) of a lens-mirror combination is:
\[ \frac{1}{F} = \frac{1}{F_m} - \frac{2}{F_l} \]
- Here, \(F_m\) is the focal length of the mirror (which is -30 cm) and \(F_l\) is the focal length of the lens.

3. **Calculate the Focal Length of the Lens**:
- Setting \(F_m = -30\) cm:
\[ \frac{1}{F} = \frac{1}{-30} - \frac{2}{F_l} \]
- Rearranging gives:
\[ \frac{1}{F_l} = \frac{1}{-30} + \frac{2}{F} \]
- Since the equivalent focal length \(F\) is 30 cm (as a concave mirror):
\[ \frac{1}{F_l} = -\frac{1}{30} + \frac{2}{30} = \frac{1}{30} \]
- Thus, \(F_l = 60\) cm.

4. **Using the Second Condition**:
- Now, when the convex surface is silvered, it behaves like a concave mirror with a focal length of -10 cm.
- Using the same formula:
\[ \frac{1}{F} = \frac{1}{F_m} - \frac{2}{F_l} \]
- Setting \(F_m = -10\) cm:
\[ \frac{1}{-10} = \frac{1}{F_m} - \frac{2}{60} \]
- Rearranging gives:
\[ \frac{1}{F_m} = -\frac{1}{10} + \frac{1}{30} \]
- Finding a common denominator (30):
\[ \frac{1}{F_m} = -\frac{3}{30} + \frac{1}{30} = -\frac{2}{30} = -\frac{1}{15} \]
- Thus, \(F_m = 15\) cm.

5. **Finding the Radius of Curvature**:
- The radius of curvature \(R\) of the mirror is given by:
\[ R = 2 \times F_m = 2 \times 15 = 30 \text{ cm} \]

6. **Using the Lens Maker's Formula**:
- The lens maker's formula is:
\[ F_l = \frac{R}{(μ - 1)} \]
- Here, \(R = 30\) cm and \(F_l = 60\) cm:
\[ 60 = \frac{30}{(μ - 1)} \]
- Rearranging gives:
\[ μ - 1 = \frac{30}{60} = \frac{1}{2} \]
- Thus, \(μ = 1 + \frac{1}{2} = 1.5\).

### Final Answer:
The refractive index of the material of the lens is \(μ = 1.5\).