The radius of curvature of a thin planoconvex lens is 10 cm and the refractive index of its glass is 1.5. If the plane surface is silvered, then it will behave like a

Options

A. concave mirror of focal length 10 cm
B. concave mirror of focal length 20 cm
C. convex mirror of focal length 10 cm
D. convex mirror offocal length 20 cm

Explanation

To determine how a thin plano-convex lens behaves when its plane surface is silvered, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Parameters**:
- Radius of curvature (R) of the plano-convex lens = 10 cm
- Refractive index (n) of the glass = 1.5

2. **Understand the Configuration**:
- A plano-convex lens has one flat surface (plane) and one convex surface. When the plane surface is silvered, it effectively acts like a mirror.

3. **Calculate the Focal Length (F) of the Lens**:
- The formula for the focal length (F) of a thin lens is given by:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- For a plano-convex lens:
- \(R_1 = R = +10 \, \text{cm}\) (convex surface)
- \(R_2 = \infty\) (plane surface)
- Plugging in the values:
\[ \frac{1}{F} = (1.5 - 1) \left( \frac{1}{10} - 0 \right) \]
\[ \frac{1}{F} = 0.5 \times \frac{1}{10} = \frac{0.5}{10} = \frac{1}{20} \]
- Therefore, the focal length (F) is:
\[ F = 20 \, \text{cm} \]

4. **Determine the Behavior of the Silvered Lens**:
- When the plane surface of the lens is silvered, it behaves like a concave mirror. The focal length of a concave mirror is considered negative in sign convention.
- Thus, the effective focal length of the silvered plano-convex lens is:
\[ F = -20 \, \text{cm} \]

5. **Conclusion**:
- The silvered plano-convex lens behaves like a concave mirror with a focal length of 20 cm.

### Final Answer:
The silvered plano-convex lens behaves like a concave mirror of focal length 20 cm.