When r is the radius of curvature of spherical mirror then focal length is fixed according to it. radiusof curvature is distance of pole from centre of curvature. while focal length is distance of pole from focus. we always note that distance between focus and centre of curvature is equal to the distance between pole and focus.
distance between focus and centre of curvature= distance between pole and focus
radius of curvature= distance of focus from pole
radius of curvature =distance between focus and centre of curvature+distance between pole and focus
r=2 *distance between pole and focus
r= 2* f
so focal length= r/2

Relation between Radius of Curvature and Focal Length of a Spherical Mirror

The relation between the radius of curvature (R) and the focal length (f) of a spherical mirror can be understood by considering the shape of the mirror and the behavior of light rays when they interact with the mirror's surface.

Concave and Convex Mirrors:
Spherical mirrors can be of two types: concave and convex. In a concave mirror, the reflecting surface is curved inward, while in a convex mirror, the reflecting surface is curved outward.

Definition of Radius of Curvature:
The radius of curvature (R) of a spherical mirror is defined as the radius of the sphere from which the mirror is obtained. It is the distance between the center of curvature (C) and the mirror's surface.

Definition of Focal Length:
The focal length (f) of a spherical mirror is the distance between the mirror's surface and its focal point (F). The focal point is the point where parallel rays of light, when incident on the mirror, converge or appear to diverge from.

Relation between R and f for Concave Mirror:
For a concave mirror, the focal length is half the radius of curvature, i.e., f = R/2. This means that the focal length is equal to half the distance between the center of curvature and the mirror's surface. The focal length of a concave mirror is always positive.

Relation between R and f for Convex Mirror:
For a convex mirror, the focal length is also half the radius of curvature, but with a negative sign, i.e., f = -R/2. This implies that the focal length of a convex mirror is equal to half the distance between the center of curvature and the mirror's surface, but in the opposite direction. The focal length of a convex mirror is always negative.

Summary:
In summary, the relation between the radius of curvature (R) and the focal length (f) of a spherical mirror is as follows:
- For a concave mirror: f = R/2 (positive focal length)
- For a convex mirror: f = -R/2 (negative focal length)

The relationship between the radius of curvature and the focal length is crucial in understanding the behavior of light rays when they interact with spherical mirrors. It allows us to determine the position and size of the image formed by the mirror, as well as the magnification of the mirror system.