The plane face of a planoconvex lens is silvered. If `mu` be the refractive index and R, the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature

Options

A. `muR`
B. `(R)/((mu-1))`
C. `(R^(2))/(mu)`
D. `[(mu+1)/((mu-1))]R`

Explanation

To solve the problem, we need to analyze the behavior of a planoconvex lens when its plane face is silvered. The silvering effectively turns the plane face into a mirror, and we want to find out how this system behaves like a concave mirror in terms of its radius of curvature.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a planoconvex lens with one flat surface and one convex surface.
- The flat surface is silvered, making it act like a mirror.

2. **Identifying the Focal Length of the Lens**:
- The focal length (f) of a planoconvex lens can be calculated using the lens maker's formula:
\[ \frac{1}{f} = (μ - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- For a planoconvex lens, \( R_1 = R \) (the radius of curvature of the convex side) and \( R_2 = \infty \) (the flat side).
- Thus, the formula simplifies to:
\[ \frac{1}{f} = (μ - 1) \left( \frac{1}{R} - 0 \right) = \frac{μ - 1}{R} \]
- Therefore, the focal length of the lens is:
\[ f = \frac{R}{μ - 1} \]

3. **Considering the Silvered Surface**:
- When the plane surface is silvered, the system behaves like a combination of a lens and a mirror.
- The mirror's focal length (f_m) is given by the formula:
\[ f_m = \frac{R_m}{2} \]
- Since the mirror is effectively a plane mirror, its radius of curvature \( R_m \) is considered to be infinite, leading to \( f_m = \infty \).

4. **Finding the Equivalent Focal Length**:
- The equivalent focal length (F_eq) of the system can be expressed as:
\[ \frac{1}{F_{eq}} = \frac{1}{f} + \frac{1}{f_m} \]
- Substituting \( f_m = \infty \):
\[ \frac{1}{F_{eq}} = \frac{1}{f} + 0 = \frac{1}{f} \]
- Therefore, \( F_{eq} = f \).

5. **Relating the Focal Length to the Radius of Curvature**:
- The focal length of the equivalent system is related to the radius of curvature (R_eq) of the mirror by:
\[ F_{eq} = \frac{R_{eq}}{2} \]
- Setting the two expressions for \( F_{eq} \) equal gives:
\[ \frac{R}{μ - 1} = \frac{R_{eq}}{2} \]
- Rearranging gives:
\[ R_{eq} = \frac{2R}{μ - 1} \]

6. **Conclusion**:
- The system behaves like a concave mirror with a radius of curvature:
\[ R_{eq} = \frac{2R}{μ - 1} \]