When two thin lenses are kept in contanct, the focal length of the combination is

Options

A. the geometricamean of the two focal lenghts.
B. the same as the larger focal length
C. gretter than either focal length
D. smaller than either focal length

Answer

To find the focal length of a combination of two thin lenses kept in contact, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Lengths of the Lenses**:
Let the focal length of the first lens be \( f_1 \) and the focal length of the second lens be \( f_2 \).

2. **Use the Lens Formula for Combination**:
The formula for the focal length \( f \) of two thin lenses in contact is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]

3. **Rearranging the Formula**:
To find the focal length \( f \), we can rearrange the equation:
\[ \frac{1}{f} = \frac{f_1 + f_2}{f_1 f_2} \]
Thus, we can express \( f \) as:
\[ f = \frac{f_1 f_2}{f_1 + f_2} \]

4. **Understanding the Result**:
This formula shows that the focal length of the combination \( f \) is less than either \( f_1 \) or \( f_2 \) when both lenses are converging (positive focal lengths). If one of the lenses is diverging (negative focal length), the effect will depend on the magnitudes of \( f_1 \) and \( f_2 \).

5. **Conclusion**:
Therefore, the focal length of the combination of two thin lenses in contact is given by:
\[ f = \frac{f_1 f_2}{f_1 + f_2} \]