A plano – convex lens made of material of refractive index `mu` with radius of curvature R is silvered on the curved side. How far away from the lens–mirror must you place a object so that the image coincides with the object

Options

A. `("R")/(mu)`
B. R
C. `("R")/(mu- 1)`
D. `mu`R

Explanation

To solve the problem of how far away from the plano-convex lens-mirror combination we must place an object so that the image coincides with the object, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Lens-Mirror System**:
- A plano-convex lens has one flat side and one convex side. When the curved side is silvered, it acts as a concave mirror.
- The focal length of the lens (f) and the focal length of the mirror (fm) need to be determined.

2. **Finding the Focal Length of the Lens**:
- The formula for the focal length (f) of a plano-convex lens is given by:
\[ \frac{1}{f} = \frac{\mu - 1}{R} \]
- Here, \( \mu \) is the refractive index of the lens material, and \( R \) is the radius of curvature of the convex side.

3. **Finding the Focal Length of the Mirror**:
- The focal length of a concave mirror is given by:
\[ f_m = \frac{R}{2} \]
- Since the curved side of the lens is silvered, it behaves like a concave mirror.

4. **Combining the Focal Lengths**:
- The effective focal length (F) of the lens-mirror combination can be found using the lens maker's formula and the mirror formula:
\[ \frac{1}{F} = \frac{1}{f} + \frac{1}{f_m} \]
- Substituting the values:
\[ \frac{1}{F} = \frac{1}{f} + \frac{2}{R} \]
- From the lens formula, we have:
\[ \frac{1}{f} = \frac{\mu - 1}{R} \]
- Therefore:
\[ \frac{1}{F} = \frac{\mu - 1}{R} + \frac{2}{R} = \frac{\mu + 1}{R} \]
- Thus, the effective focal length is:
\[ F = \frac{R}{\mu + 1} \]

5. **Using the Mirror Formula**:
- The mirror formula relates object distance (u), image distance (v), and focal length (F):
\[ \frac{1}{F} = \frac{1}{u} + \frac{1}{v} \]
- Since the image coincides with the object, we have \( u = v = d \):
\[ \frac{1}{F} = \frac{2}{d} \]
- Substituting for F:
\[ \frac{\mu + 1}{R} = \frac{2}{d} \]

6. **Solving for d**:
- Rearranging gives:
\[ d = \frac{2R}{\mu + 1} \]

### Final Answer:
The distance \( d \) from the lens-mirror combination at which the object must be placed is:
\[ d = \frac{2R}{\mu + 1} \]