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

Explanation

(i) C be the centre of curvature of the mirror.
(ii) 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.
(iii) 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.
(iv) If MP is the perpendicular from M on the principal axis, then from the geometry,
(v) The angles `angleMCP = i` and `angleMFP = 2i`
(vi) From right angle triangles `DeltaMCP` and `Delta MFP`
` tan i = (PM)/(PC) ` and tan 2i ` = (PM)/(PF)`
(vii) As the angles are small, tan i = i,
` i= (PM)/(PC) " and " 2i =(PM)/(PF)`
(viii) Simplifying
` 2(PM)/(PC) = (PM)/(PF) , 2PF = PC`