The plane surface of a plano-convex lens of focal length f is silvered. It will behave as

Options

A. Plane mirror
B. Convex mirror of focal length 2 f
C. Concave mirror of focal length `f//2`
D. None of the above

Answer

To solve the problem, we need to analyze the behavior of a plano-convex lens when its plane surface is silvered. Here’s a step-by-step solution:

### Step 1: Understanding the Configuration
We have a plano-convex lens with a focal length \( f \). When the plane surface of this lens is silvered, it effectively turns that surface into a mirror.

**Hint:** Recall that a plano-convex lens has one flat surface and one convex surface. Silvering the flat surface means that light will reflect off this surface.

### Step 2: Identifying the Powers
The power \( P \) of a lens is given by the formula:
\[ P = \frac{1}{F} \]
where \( F \) is the focal length of the lens. For a plano-convex lens with focal length \( f \), the power is:
\[ P_{\text{lens}} = \frac{1}{f} \]

The power of a plane mirror is zero because its focal length is infinite:
\[ P_{\text{mirror}} = 0 \]

**Hint:** Remember that the power of a mirror is defined as the reciprocal of its focal length.

### Step 3: Calculating the Net Power
When the plane surface is silvered, the effective power of the system (lens + mirror) can be calculated as:
\[ P_{\text{net}} = P_{\text{lens}} + P_{\text{mirror}} = \frac{1}{f} + 0 = \frac{1}{f} \]

However, since the convex surface of the lens is still functioning, we need to consider the effect of the silvered plane surface. The effective power of the system becomes:
\[ P_{\text{net}} = 2 \times P_{\text{lens}} = 2 \times \frac{1}{f} = \frac{2}{f} \]

**Hint:** The silvered surface acts like a concave mirror, effectively doubling the power of the lens.

### Step 4: Finding the Effective Focal Length
To find the effective focal length \( F_{\text{effective}} \) of the system, we use the relationship between power and focal length:
\[ P_{\text{net}} = \frac{1}{F_{\text{effective}}} \]
Thus,
\[ F_{\text{effective}} = \frac{f}{2} \]

**Hint:** The effective focal length is half of the original focal length of the plano-convex lens.

### Step 5: Conclusion
Since the effective focal length is positive and the system behaves like a converging lens, we conclude that the silvered plano-convex lens behaves like a lens with a focal length of \( \frac{f}{2} \).

**Final Answer:** The silvered plano-convex lens behaves like a converging lens with a focal length of \( \frac{f}{2} \).