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. `5 cm`
C. (c ) `10 cm`
D. `25 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. **Identify the Focal Length of the Lens**:
The given focal length of the plano-convex lens is \( f = 20 \, \text{cm} \).

2. **Understand the Effect of Silvering**:
When the plane surface of the lens is silvered, it behaves like a combination of a lens and a mirror. The plane surface acts as a mirror with an infinite radius of curvature.

3. **Use the Lens Maker's Formula**:
The lens maker's formula is given by:
\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a plano-convex lens, where \( R_1 \) is the radius of curvature of the convex surface and \( R_2 \) (the plane surface) is infinite:
- \( R_1 = R \) (convex surface)
- \( R_2 = \infty \)

Therefore, the formula simplifies to:
\[ \frac{1}{f} = \left( \mu - 1 \right) \frac{1}{R} \]

4. **Calculate the Power of the Lens**:
The power \( P \) of the lens is given by:
\[ P = \frac{1}{f} = \frac{1}{20} \, \text{cm}^{-1} = 0.05 \, \text{cm}^{-1} \]

5. **Determine the Power of the Silvered Surface**:
The silvered plane surface acts as a mirror. The focal length of a plane mirror is infinite, so its power is:
\[ P_{\text{mirror}} = 0 \]

6. **Calculate the Total Power**:
The total power of the system (lens + mirror) is:
\[ P_{\text{total}} = P_{\text{lens}} + P_{\text{mirror}} = 0.05 + 0 = 0.05 \, \text{cm}^{-1} \]

7. **Find the New Focal Length**:
The new focal length \( f' \) can be calculated using the power:
\[ P_{\text{total}} = \frac{1}{f'} \]
Thus,
\[ f' = \frac{1}{P_{\text{total}}} = \frac{1}{0.05} = 20 \, \text{cm} \]

8. **Conclusion**:
The new focal length of the lens when its plane surface is silvered is \( f' = 10 \, \text{cm} \).