What is relation between R and f?

In optics, we have come across different types of mirrors and have studied their properties in detail like the radius of curvature, focal length, focal point, dimension of the mirror, imaging capacity (erect or inverted), thickness, refractive index and many more. Mirrors are divided into two types:

Plane Mirrors
Spherical Mirrors

In this article, let us understand the relationship between focal length and radius of curvature.

The reflecting surface of a spherical mirror may be curved inwards or outwards. The focal length of a mirror is represented as f and is defined as the distance between the focus and the pole of the mirror. The radius of curvature is represented as R and is defined as the radius of the mirror that forms a complete sphere.

A ray of light AB, which is incident on a spherical mirror at point B and is parallel to the principal axis. CB is normal to the surface at point B. CP = CB = R is the radius of curvature. After reflection from mirror the light will pass through the focus of the concave mirror F or will appear to diverge from the focus of the convex mirror F and obeys the law of reflection i.e. i = r.

From the geometry of the figure,

∠BCP = θ = i (As ∠BCP and ∠ABC are alternate angles)

In ΔCBF, ∠CBF = θ = r

∴BF = FC (because i = r)

If the aperture of the mirror is small, B lies close to P, and therefore BF = PF

Or FC = BF = PF

Or PC = PF + FC = PF + PF

Or R = 2 PF = 2f

Or f = R/2

This kind of relation can be applied for convex mirrors too. In this relation, the aperture of the mirror is assumed to be small.

Radius of curvature is observed to be equal to twice the focal length for spherical mirrors with small apertures. Hence R = 2f .

We can say clearly that the principal focus of a spherical mirror lies at the centre between the centre of curvature and the pole.