How is the focal length of a spherical mirror related to its radius of curvature?

Answer

To find the relationship between the focal length (F) of a spherical mirror and its radius of curvature (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Components**:
- Identify the spherical mirror (let's consider a concave mirror for this explanation).
- Recognize the key points: the Pole (P), the Center of Curvature (C), and the Focus (F).

2. **Define Radius of Curvature (R)**:
- The radius of curvature is defined as the distance from the pole of the mirror to the center of curvature. This distance is represented as R.

3. **Define Focal Length (F)**:
- The focal length is the distance from the pole of the mirror to the focus. This distance is represented as F.

4. **Locate the Focus**:
- The focus (F) is located at a point between the center of curvature (C) and the pole (P). It is important to note that the focus is halfway between the pole and the center of curvature.

5. **Establish the Relationship**:
- Since the focus is the midpoint between the center of curvature and the pole, we can express this relationship mathematically:
- The distance from C to P is R, and since F is the midpoint, we can say:
\[ F = \frac{R}{2} \]
- Rearranging this gives us:
\[ R = 2F \]

6. **Final Relation**:
- Therefore, the relationship between the focal length (F) and the radius of curvature (R) of a spherical mirror can be summarized as:
\[ R = 2F \quad \text{or} \quad F = \frac{R}{2} \]

### Summary:
The focal length of a spherical mirror is half of its radius of curvature. This can be expressed as:
- \( R = 2F \)
- \( F = \frac{R}{2} \)