A plano convex lens has focal length `f = 20 cm`. If its plane surface is silvered, then new focal length will be

Options

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

Explanation

To find the new focal length of a plano-convex lens when its plane surface is silvered, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Lens Configuration**:
A plano-convex lens has one flat surface and one convex surface. The given focal length (f) of the lens is 20 cm.

2. **Silvering the Plane Surface**:
When the plane surface of the lens is silvered, it acts as a mirror. Therefore, we need to consider both the lens and the mirror effects together.

3. **Use the Lens Maker's Formula**:
The power (P) of the lens is given by:
\[ P = \frac{1}{f} \]
where \( f \) is the focal length of the lens. For our lens:
\[ P_{lens} = \frac{1}{20 \text{ cm}} = 0.05 \text{ cm}^{-1} \]

4. **Power of the Silvered Surface**:
The silvered plane surface acts like a mirror. The focal length of a plane mirror is considered to be infinite, hence its power is:
\[ P_{mirror} = 0 \text{ (since } P = \frac{1}{f} \text{ and } f = \infty\text{)} \]

5. **Combine the Powers**:
When the plane surface is silvered, the effective power of the system (lens + mirror) can be expressed as:
\[ P_{total} = P_{lens} + P_{mirror} \]
Since \( P_{mirror} = 0 \):
\[ P_{total} = P_{lens} + 0 = P_{lens} \]

6. **Calculate the New Focal Length**:
The new focal length \( f' \) can be calculated from the total power:
\[ P_{total} = \frac{1}{f'} \]
Therefore:
\[ \frac{1}{f'} = 2 \cdot P_{lens} \]
Since the effective power of the lens is doubled when the plane surface is silvered:
\[ P_{total} = 2 \cdot 0.05 = 0.1 \text{ cm}^{-1} \]
Thus:
\[ f' = \frac{1}{P_{total}} = \frac{1}{0.1} = 10 \text{ cm} \]

### Conclusion:
The new focal length of the plano-convex lens when its plane surface is silvered is **10 cm**.