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 same will be the angle of reflection. If MP is the perpendicular from M on the principal axis, then from the geometry. The angles `angle MCP = i and angle MFP = 2i` From right angle trianagles `DeltaMCP and Delta MFP`.
`tan i = (PM)/(PC) and tan2i = (PM)/(PF)`
As the angles are small, `tiapproxi,i=(PM)/(PC)and2i=(PM)/(PF)`
Simplifying further, `2(PM)/(PC) = (PM)/(PF):2PF=PC`
PF is focal lenght 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.