If `f_(1)` and `f_(2)` represent the first and second focal lengths of a single spherical refracting surface, then

Options

A. `f_(2) = - f_(1)`
B. `f_(2) = -mu f_(1)`
C. `f_(1) = - muf_(2)`
D. `f_(1) f_(2) =- 1`

Answer

To solve the problem regarding the first and second focal lengths \( f_1 \) and \( f_2 \) of a single spherical refracting surface, we can use the lens maker's formula and the properties of spherical surfaces.

### Step-by-Step Solution:

1. **Understanding Focal Lengths**:
- The first focal length \( f_1 \) is defined as the distance from the surface to the point where parallel rays of light converge after refraction.
- The second focal length \( f_2 \) is defined as the distance from the surface to the point where light rays originating from a point at the focal point converge after refraction.

2. **Using the Lens Maker's Formula**:
- For a single spherical refracting surface, the relationship between the focal lengths and the radius of curvature \( R \) can be expressed as:
\[ \frac{1}{f} = \frac{n - 1}{R} \]
where \( n \) is the refractive index of the material of the lens.

3. **Relating \( f_1 \) and \( f_2 \)**:
- For a single spherical surface, the first and second focal lengths are related by the following equation:
\[ \frac{1}{f_1} + \frac{1}{f_2} = \frac{2}{R} \]

4. **Deriving the Relationship**:
- Rearranging the above equation gives:
\[ \frac{1}{f_2} = \frac{2}{R} - \frac{1}{f_1} \]
- This shows how \( f_1 \) and \( f_2 \) are interconnected based on the radius of curvature \( R \).

5. **Conclusion**:
- Thus, the relationship between the first and second focal lengths of a single spherical refracting surface can be summarized as:
\[ \frac{1}{f_1} + \frac{1}{f_2} = \frac{2}{R} \]