When the plane surface of a plano-convex lens of refractive index 1.5 is silvered, it behaves like a concave mirror with f = 30 cm. When its convex surface is silvered, it will behave like a concave mirror of focal length

Options

A. 10 cm
B. 20 cm
C. 30 cm
D. 45 cm

Answer

To solve the problem, we need to determine the focal length of a plano-convex lens when its convex surface is silvered. We start with the information given in the question.

### Step-by-step Solution:

1. **Understanding the Given Information:**
- A plano-convex lens with a refractive index (μ) of 1.5 has its plane surface silvered, behaving like a concave mirror with a focal length (f) of -30 cm.

2. **Finding the Focal Length of the Lens:**
- The relationship between the focal length of the lens (f_L) and the focal length of the mirror (f_m) when the plane surface is silvered can be expressed as:
\[ f_m = -f_L \]
- Given that \( f_m = -30 \, \text{cm} \), we can find:
\[ f_L = 30 \, \text{cm} \]

3. **Using the Lens Maker's Formula:**
- The lens maker's formula for a plano-convex lens is given by:
\[ \frac{1}{f_L} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- Here, \( R_1 \) is the radius of curvature of the convex surface (positive) and \( R_2 \) is the radius of curvature of the plane surface (infinity, hence \( \frac{1}{R_2} = 0 \)).
- Thus, we have:
\[ \frac{1}{f_L} = (\mu - 1) \frac{1}{R_1} \]
- Substituting \( \mu = 1.5 \) and \( f_L = 30 \, \text{cm} \):
\[ \frac{1}{30} = (1.5 - 1) \frac{1}{R_1} \]
- Simplifying gives:
\[ \frac{1}{30} = 0.5 \frac{1}{R_1} \]
- Therefore:
\[ R_1 = 15 \, \text{cm} \]

4. **Finding the Focal Length When the Convex Surface is Silvered:**
- When the convex surface is silvered, the lens behaves like a concave mirror. The effective focal length \( f_m \) can be calculated using the formula:
\[ \frac{1}{f_m} = \frac{1}{f_L} + \frac{1}{f_L} \]
- Since the radius of curvature for the convex surface is now negative (as it is acting as a mirror):
\[ f_m = -\frac{R_1}{2} \]
- Substituting \( R_1 = 30 \, \text{cm} \):
\[ f_m = -\frac{30}{2} = -15 \, \text{cm} \]

5. **Conclusion:**
- The focal length of the concave mirror formed by silvering the convex surface of the plano-convex lens is \( -15 \, \text{cm} \).

### Final Answer:
The focal length when the convex surface is silvered is **-15 cm**.