Two thin lenses of focal lengths `f_(1)` and `f_(2)` are in contact. The focal length of this combination is

Options

A. `f_(1)+f_(2)`
B. `(1)/(f_(1)+f_(2))`
C. `(f_(1)f_(2))/(f_(1)+f_(2))`
D. `(f_(1)+f_(2))/(f_(1)f_(2))`

Explanation

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

### Step 1: Understand the Formula for Combined Focal Length
The formula for the effective focal length \( F \) of two thin lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
where \( f_1 \) and \( f_2 \) are the focal lengths of the individual lenses.

### Step 2: Rearranging the Formula
To find the effective focal length \( F \), we need to rearrange the formula:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
This can be expressed as:
\[ \frac{1}{F} = \frac{f_2 + f_1}{f_1 f_2} \]

### Step 3: Taking the Reciprocal
Taking the reciprocal of both sides to solve for \( F \):
\[ F = \frac{f_1 f_2}{f_1 + f_2} \]

### Step 4: Conclusion
Thus, the focal length of the combination of the two thin lenses in contact is:
\[ F = \frac{f_1 f_2}{f_1 + f_2} \]

### Final Answer
The effective focal length of the combination of the two thin lenses is \( \frac{f_1 f_2}{f_1 + f_2} \).

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