The radius of curvature of curved surface of a thin plano-convex lens is `10 cm` and the refractive index is `1.5`. If the plano surface is silvered, then the focal length will be.

Options

A. 15 cm
B. 20 cm
C. 5 cm
D. 10 cm

Explanation

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

### Step 1: Identify the parameters
- Radius of curvature (R) of the curved surface = 10 cm
- Refractive index (n) of the lens = 1.5

### Step 2: Use the lens maker's formula
The lens maker's formula for a thin lens is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Where:
- \( f \) = focal length of the lens
- \( n \) = refractive index of the lens
- \( R_1 \) = radius of curvature of the first surface (convex surface)
- \( R_2 \) = radius of curvature of the second surface (plane surface)

For a plano-convex lens:
- \( R_1 = 10 \) cm (convex surface)
- \( R_2 = \infty \) (plane surface)

### Step 3: Substitute the values into the formula
Substituting the values into the lens maker's formula:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{\infty} \right) \]
\[ \frac{1}{f} = 0.5 \left( \frac{1}{10} - 0 \right) \]
\[ \frac{1}{f} = 0.5 \times \frac{1}{10} = \frac{0.5}{10} = \frac{1}{20} \]

### Step 4: Calculate the focal length
Taking the reciprocal to find \( f \):
\[ f = 20 \text{ cm} \]

### Step 5: Consider the silvering of the plano surface
When the plano surface is silvered, the effective focal length of the system changes. The new effective focal length \( f' \) after silvering is given by:
\[ f' = \frac{f}{2} \]
Substituting the previously calculated focal length:
\[ f' = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \]

### Final Answer
The focal length of the lens after silvering the plano surface is **10 cm**.

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