A plano-convex lens (focal length `f_2`, refractive index`mu_(2)`, radius of curvature R) fits exactly into a plano-concave lens (focal length`f_(1)` refractive index `mu_1`, radius of curvature R). Their plane surfaces are parallel to each other. Then, the focal length of the combination will be :

Options

A. `f_(1)-f_(2)`
B. `(R)/(mu_(2)-mu_(1)`
C. `(2f_1f_(2))/(f_(1)-f_(2)`
D. `f_(1)+f_(2)`

Explanation

To find the focal length of the combination of a plano-convex lens and a plano-concave lens, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Lengths of Individual Lenses**:
- For the plano-convex lens, the focal length \( f_2 \) can be calculated using the lens maker's formula:
\[ \frac{1}{f_2} = \left(\mu_2 - 1\right) \frac{1}{R} \]
- For the plano-concave lens, the focal length \( f_1 \) is given by:
\[ \frac{1}{f_1} = \left(\mu_1 - 1\right) \frac{1}{R} \]

2. **Combine the Focal Lengths**:
- The focal length \( F \) of the combination of two lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]

3. **Substitute the Values**:
- Substitute the expressions for \( \frac{1}{f_1} \) and \( \frac{1}{f_2} \) into the equation for \( \frac{1}{F} \):
\[ \frac{1}{F} = \left(\mu_1 - 1\right) \frac{1}{R} + \left(\mu_2 - 1\right) \frac{1}{R} \]

4. **Combine the Terms**:
- Since both terms have a common denominator \( R \), we can combine them:
\[ \frac{1}{F} = \frac{(\mu_1 - 1) + (\mu_2 - 1)}{R} \]
\[ \frac{1}{F} = \frac{\mu_1 + \mu_2 - 2}{R} \]

5. **Solve for Focal Length \( F \)**:
- Taking the reciprocal gives us the focal length of the combination:
\[ F = \frac{R}{\mu_1 + \mu_2 - 2} \]

### Final Result:
The focal length of the combination of the plano-convex lens and plano-concave lens is:
\[ F = \frac{R}{\mu_1 + \mu_2 - 2} \]