The focal length of a planoconvex glass lens is 20 cm `(mu_(g)=1.5)`. The plane face of it is silvered. An illuminating object is placed at a distance of 60 cm from the lens on its axis along the convex side. Then the distance (in cm) of the image is

Options

A. 20
B. 30
C. 40
D. 12

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given data
- Focal length of the plano-convex lens, \( f = 20 \, \text{cm} \)
- Refractive index of glass, \( \mu_g = 1.5 \)
- Object distance, \( u = -60 \, \text{cm} \) (negative because the object is on the same side as the incoming light)

### Step 2: Determine the equivalent focal length of the silvered plano-convex lens
Since the plane face of the lens is silvered, it behaves like a concave mirror. The equivalent focal length \( f' \) of the system can be calculated using the formula for the power of the lens and the mirror.

1. The power of the lens \( P_L \) is given by:
\[ P_L = \frac{1}{f} = \frac{1}{20} \, \text{cm}^{-1} \]

2. The power of the plane mirror \( P_M \) is zero because the focal length of a plane mirror is infinite:
\[ P_M = 0 \]

3. The equivalent power \( P \) of the system (lens + mirror) is:
\[ P = P_L + P_M = \frac{1}{20} + 0 = \frac{1}{20} \, \text{cm}^{-1} \]

4. The equivalent focal length \( f' \) of the silvered lens is:
\[ \frac{1}{f'} = P \Rightarrow f' = 20 \, \text{cm} \]

However, since the plane face is silvered, we need to consider that the effective focal length of a silvered plano-convex lens is given by:
\[ f' = \frac{f}{2} = \frac{20}{2} = 10 \, \text{cm} \]
But since it's a concave mirror, we take it as negative:
\[ f' = -10 \, \text{cm} \]

### Step 3: Apply the mirror formula
The mirror formula is given by:
\[ \frac{1}{f'} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{-10} = \frac{1}{v} + \frac{1}{-60} \]

### Step 4: Solve for \( v \)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{60} \]

Finding a common denominator (which is 60):
\[ \frac{1}{v} = \frac{-6 + 1}{60} = \frac{-5}{60} \]
Thus:
\[ \frac{1}{v} = -\frac{1}{12} \]
So:
\[ v = -12 \, \text{cm} \]

### Step 5: Conclusion
The negative sign indicates that the image is formed on the same side as the object, which is consistent with the behavior of a concave mirror. Therefore, the distance of the image from the lens is \( 12 \, \text{cm} \).

### Final Answer
The distance of the image is \( 12 \, \text{cm} \).

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