For a plane convex lens `(mu = 1.5)` has radius of curvature 10 cm. It is slivered on its plane surface. Find focal length after silvering:

Options

A. 10 cm
B. 20 cm
C. 15 cm
D. 25 cm

Explanation

To find the focal length of a plane convex lens after silvering, we can follow these steps:

### Step 1: Understand the lens properties
A plane convex lens has one flat surface and one convex surface. The refractive index (μ) of the lens is given as 1.5, and the radius of curvature (R) of the convex surface is 10 cm.

### Step 2: Use the lens maker's formula
For a lens, the focal length (f) can be calculated using the lens maker's formula:

\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( \mu \) is the refractive index of the lens.
- \( R_1 \) is the radius of curvature of the first surface (convex surface, positive).
- \( R_2 \) is the radius of curvature of the second surface (plane surface, negative).

### Step 3: Assign values
For our lens:
- \( R_1 = +10 \, \text{cm} \) (convex surface)
- \( R_2 = 0 \, \text{cm} \) (plane surface, treated as infinite radius)

### Step 4: Substitute values into the formula
Since \( R_2 \) is infinite, \( \frac{1}{R_2} = 0 \). Thus, the formula simplifies to:

\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} \right) \]

Substituting the values:

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} \right) \]

\[ \frac{1}{f} = 0.5 \times \frac{1}{10} = \frac{0.5}{10} = \frac{1}{20} \]

### Step 5: Calculate the focal length
Taking the reciprocal gives us the focal length:

\[ f = 20 \, \text{cm} \]

### Step 6: Consider the effect of silvering
When the lens is silvered on its plane surface, it behaves like a concave mirror on that surface. The effective focal length of a combination of a lens and a mirror can be calculated using the formula:

\[ \frac{1}{f_{effective}} = \frac{1}{f_{lens}} + \frac{1}{f_{mirror}} \]

For a concave mirror, the focal length is negative. Therefore, the focal length of the mirror (which is the silvered surface) is:

\[ f_{mirror} = -\infty \text{ (for plane surface)} \]

Thus, the effective focal length after silvering becomes:

\[ \frac{1}{f_{effective}} = \frac{1}{20} + 0 = \frac{1}{20} \]

So, the effective focal length remains:

\[ f_{effective} = 20 \, \text{cm} \]

### Final Answer
The focal length of the silvered plane convex lens is **20 cm**.

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