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

Answer

To solve the problem, we need to find the focal length of a plano-convex lens when its plano surface is silvered. Let's break down the steps:

### Step 1: Understand the Lens Configuration
A plano-convex lens has one flat surface (plano) and one curved surface (convex). The radius of curvature (R) of the convex surface is given as 10 cm. The plano surface can be considered to have an infinite radius of curvature (R2 = ∞).

### 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:
- \( f \) is the focal length of the lens,
- \( n \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface (convex),
- \( R_2 \) is the radius of curvature of the second surface (plano).

### Step 3: Substitute Values into the Formula
Given:
- \( n = 1.5 \)
- \( R_1 = 10 \, \text{cm} \) (convex surface, positive)
- \( R_2 = \infty \) (plano surface, negative)

Substituting these values into the formula:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{\infty} \right) \]
\[ \frac{1}{f} = 0.5 \left( \frac{1}{10} - 0 \right) \]
\[ \frac{1}{f} = 0.5 \cdot \frac{1}{10} = \frac{0.5}{10} = \frac{1}{20} \]

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

### Step 5: Consider the Effect of Silvering
When the plano surface is silvered, it behaves like a mirror. The effective focal length of a lens with one surface silvered is half of the original focal length:
\[ f_{\text{effective}} = \frac{f}{2} = \frac{20}{2} = 10 \, \text{cm} \]

### Final Answer
The focal length after silvering the plano surface is:
\[ \boxed{10 \, \text{cm}} \]