Derive the relation between f and R for a spherical mirror.

Explanation

Let C be the centre of curvature of the mirror. Consider a light ray parallel to the principal axis is incident on the mirror at M and passes through the principal focus F after reflection. The geometry of reflection of the incident ray is shown in figure. The line CM is the normal to the mirror at M. Let i be the angle of incidence and the same will be the angle of reflection.
If MP is the perpendicular from M on the principal axis, then from the geometry, The angles
`angleMCP =i` and `angleMFP =2i`, From right angle triangle `triangleMCP` and `triangleMFP`,
`tan i = (PM)/(PC)` and `tan 2i =(PM)/(PF)` and `tan 2i = (PM)/(PF)`
As the angles are small, `tan i ~~ i = (PM)/(PC)` and `2i = (PM)/(PF)`
Simplifying further, `2(PM)/(PC) = (PM)/(PF), 2PF = PC`
PF is focal length f and PC is the radius of curvature R.

`2f = R (or) f= R/2`
`f=R/2` is the relation between f and R.