Question

How are focal length and radius of curvature connected? Prove the relation between focal length f and radius of curvature R.

Answer 1

The focal length of a spherical mirror, whether it is concave or convex, is equal to half of its radius of curvature.
Focal length = Radius of curvature/2
f = R/2

In the diagram given above, M and N represent the spherical mirror. However, P is the principal axis, C is the centre of curvature, F is the focal point where the two rays meet each other. In addition, the distance between the C and P is called the radius of curvature, and the distance between F and P is called the focal length. The ray of light incident on the mirror is parallel to the principal axis. The ray of light AB generally strikes on the surface. CP is equal to R (radius of curvature ). However, when the two parallel rays A and B strike on the mirror, they reflect and pass through the focus point. As the law of reflection is followed, i = r,

According to the figure of the geometry,

∠BPC = θ = i

Also, θ = r

BF = FC (according to the law of reflection i = r)

When the mirror’s aperture is smaller than, B lies near to P, and therefore, the condition becomes BF = PF.

FC = FP = Pf,

PC = PF + FC = PF + PF

R = 2 PF = 2f

So, f = R/2