A thin plano-convex lens acts like a concave mirror of radius of curvature `20 cm` when its plane surface is silvered. The radius of curvature of the curved surface if index of refraction of its matarial is `1.5` will be

Options

A. `40 cm`
B. `30 cm`
C. `10 cm`
D. `20 cm`

Explanation

To solve the problem, we need to determine the radius of curvature of the curved surface of a plano-convex lens that behaves like a concave mirror when its plane surface is silvered. The given radius of curvature of the concave mirror is 20 cm, and the refractive index of the lens material is 1.5.

### Step-by-Step Solution:

1. **Understanding the System**:
- A plano-convex lens has one flat surface and one convex surface. When the flat surface is silvered, it acts like a concave mirror.

2. **Using the Mirror Formula**:
- The focal length \( F \) of a concave mirror is related to its radius of curvature \( R \) by the formula:
\[ F = \frac{R}{2} \]
- Given that the lens behaves like a concave mirror with a radius of curvature of 20 cm, we can find the focal length of the mirror:
\[ F = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

3. **Relating Focal Length to Lens Parameters**:
- The focal length \( F \) of a thin lens is given by the lens maker's formula:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- For a plano-convex lens, \( R_1 \) (the radius of curvature of the convex surface) is \( R \) (unknown), and \( R_2 \) (the radius of curvature of the flat surface) is infinite (since it is flat).

4. **Substituting Values**:
- The refractive index \( n \) is given as 1.5. Therefore:
\[ \frac{1}{F} = (1.5 - 1) \left( \frac{1}{R} - 0 \right) \]
- Simplifying this gives:
\[ \frac{1}{10} = 0.5 \cdot \frac{1}{R} \]

5. **Solving for R**:
- Rearranging the equation:
\[ \frac{1}{R} = \frac{1}{10 \cdot 0.5} = \frac{1}{5} \]
- Therefore, the radius of curvature \( R \) is:
\[ R = 5 \, \text{cm} \]

6. **Conclusion**:
- The radius of curvature of the curved surface of the plano-convex lens is \( 5 \, \text{cm} \).

### Final Answer:
The radius of curvature of the curved surface is \( 5 \, \text{cm} \).

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