A lens made of material of Refractive index `mu_(2)` is surrounded by a medium of Refractive Index `mu_(1)`. The focal length f is related as

Options

A. `f prop (mu_(2) - mu_(1))`
B. `f propmu_ (1)/((mu_(2)-mu_(1)))`
C. `f prop (mu_(2) + mu_(1))`
D. `f prop (1)/((mu_(2)+mu_(1)))`

Answer

To solve the problem, we need to derive the relationship between the focal length \( f \) of a lens made of material with refractive index \( \mu_2 \) and surrounded by a medium with refractive index \( \mu_1 \).

### Step-by-Step Solution:

1. **Start with the Lens Maker's Formula**:
The formula for the focal length \( f \) of a lens in a medium is given by:
\[ \frac{1}{f} = \left( \frac{\mu_2}{\mu_1} - 1 \right) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
where \( r_1 \) and \( r_2 \) are the radii of curvature of the two surfaces of the lens.

2. **Rearranging the Formula**:
We can rearrange the formula to isolate \( \frac{1}{f} \):
\[ \frac{1}{f} = \frac{\mu_2 - \mu_1}{\mu_1} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

3. **Expressing \( f \)**:
To express \( f \) in terms of \( \mu_1 \) and \( \mu_2 \), we take the reciprocal:
\[ f = \frac{\mu_1}{\mu_2 - \mu_1} \cdot \frac{1}{\left( \frac{1}{r_1} - \frac{1}{r_2} \right)} \]

4. **Identifying the Proportionality**:
From the above expression, we can see that \( f \) is directly proportional to \( \frac{\mu_1}{\mu_2 - \mu_1} \):
\[ f \propto \frac{\mu_1}{\mu_2 - \mu_1} \]

5. **Final Result**:
Thus, the relationship between the focal length \( f \) and the refractive indices is:
\[ f \propto \frac{\mu_1}{\mu_2 - \mu_1} \]

### Conclusion:
The correct option is that the focal length \( f \) is directly proportional to \( \frac{\mu_1}{\mu_2 - \mu_1} \).

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