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

Options

A. (a)plane mirror
B. (b)convex mirror of focal length 2F
C. (c)concave mirror of focal length `(F)/(2)`
D. (d)none of these

Answer

To solve the problem, we need to analyze the behavior of a plano-convex lens when its plane surface is silvered. Let's break down the solution step by step.

### Step 1: Understand the Configuration
We have a plano-convex lens with a focal length \( f \). The plane surface of the lens is silvered, which means it will act like a mirror on that surface.

### Step 2: Identify 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, the power is positive.

### Step 3: Consider the Silvered Surface
When the plane surface is silvered, the lens will behave like a combination of a lens and a mirror. The silvered surface acts as a concave mirror. The power of a concave mirror is given by:
\[ P_{\text{mirror}} = -\frac{1}{f_{\text{mirror}}} \]
where \( f_{\text{mirror}} \) is the focal length of the mirror.

### Step 4: Calculate the Net Power
The net power of the system (lens + mirror) can be expressed as:
\[ P_{\text{net}} = P_{\text{lens}} + P_{\text{mirror}} \]
Substituting the values, we have:
\[ P_{\text{net}} = \frac{1}{f} + \left(-\frac{1}{\infty}\right) \]
Since the power of the mirror (which is silvered) is effectively zero, we can simplify this to:
\[ P_{\text{net}} = \frac{1}{f} + 0 = \frac{1}{f} \]

### Step 5: Determine the New Focal Length
Since the silvered surface acts as a concave mirror, we need to consider the effective focal length of the system. The effective focal length \( F \) of the combined system can be found using the lens maker's formula for a plano-convex lens with a silvered surface:
\[ \frac{1}{F} = 2 \cdot \frac{1}{f} \]
This gives us:
\[ F = \frac{f}{2} \]

### Conclusion
The system behaves as a concave mirror with a focal length of \( \frac{f}{2} \).

### Final Answer
The plano-convex lens with its plane surface silvered behaves as a concave mirror with a focal length of \( \frac{f}{2} \).

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