A planocovex lens is made of a material of refractive in `mu=1.5` The radius of curvature of curved surface of the lens is 20.cm. If its plane surface3 is silvered, the focal length of the silvered lens will be

Options

A. `10 cm`
B. `20 cm`
C. `40 cm`
D. `80 cm`

Answer

To find the focal length of the silvered planoconvex lens, we will follow these steps:

### Step 1: Understand the Configuration
A planoconvex lens has one flat surface and one convex surface. When the flat surface is silvered, it acts as a mirror. The lens will have both refraction (through the lens) and reflection (from the silvered surface).

### Step 2: Use the Lensmaker's Formula
The focal length \( f_L \) of a planoconvex lens can be calculated using the Lensmaker's formula:
\[ \frac{1}{f_L} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Where:
- \( \mu \) is the refractive index of the lens material (1.5 in this case).
- \( R_1 \) is the radius of curvature of the convex surface (20 cm).
- \( R_2 \) is the radius of curvature of the plane surface (infinity for a plane surface).

### Step 3: Substitute Values into the Formula
For our planoconvex lens:
- \( R_1 = 20 \, \text{cm} \)
- \( R_2 = \infty \)

Substituting these values into the formula:
\[ \frac{1}{f_L} = (1.5 - 1) \left( \frac{1}{20} - 0 \right) \]
\[ \frac{1}{f_L} = 0.5 \cdot \frac{1}{20} \]
\[ \frac{1}{f_L} = \frac{0.5}{20} = \frac{1}{40} \]

### Step 4: Calculate the Focal Length of the Lens
Taking the reciprocal gives us the focal length of the lens:
\[ f_L = 40 \, \text{cm} \]

### Step 5: Find the Focal Length of the Silvered Lens
When the plane surface is silvered, the focal length \( f \) of the silvered lens is given by:
\[ \frac{1}{f} = \frac{2}{f_L} + \frac{1}{f_M} \]
Where \( f_M \) is the focal length of the plane mirror, which is infinite:
\[ \frac{1}{f_M} = 0 \]
Thus:
\[ \frac{1}{f} = \frac{2}{f_L} \]
Substituting \( f_L = 40 \, \text{cm} \):
\[ \frac{1}{f} = \frac{2}{40} = \frac{1}{20} \]
Taking the reciprocal gives:
\[ f = 20 \, \text{cm} \]

### Conclusion
The focal length of the silvered planoconvex lens is **20 cm**.

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