## Proof that the Focal Length of a Spherical Mirror is Half the Radius of Curvature

The focal length of a spherical mirror is the distance of the point on the principal axis from the mirror that a ray of light initially passing parallel to the axis (and towards the mirror) would pass through on reflection. This is the distancein the diagram below.

The radius of curvature of a spherical mirror is the centre of the sphere of which the mirror is a part. Any ray of light passing through the centre of curvature will strike the mirror at right angles, and be reflected right back, passing back through the centre of curvature. The radius of curvature in the diagram below is

We can prove the radius of curvature CF is twice the focal length FP using the diagram above.

The line CM above is at right angles to the mirror, so bisects the angle (this is the law of reflection –).

(alternate angles)

Hence

Thensinceis isosceles.

Since the mirror is very smallhenceso