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

For a spherical mirror, a paraxial ray parallel to the principal axis strikes the mirror at $M$ close to the pole and reflects through the focus $F$, with $CM$ as the normal at the point of incidence (since it passes through the centre of curvature). Using the exterior-angle property of triangle $MCF$, the angle at $C$ is $i$ and the angle at $F$ is $2i$. In the right triangles $\triangle MCP$ and $\triangle MFP$, $\tan i = PM/PC$ and $\tan 2i = PM/PF$; for paraxial (small-angle) rays these tangents are replaced by the angles themselves, giving $i=PM/PC$ and $2i=PM/PF$, so $PF=PC/2$, i.e. $f=R/2$. This relation holds for both concave and convex mirrors, provided the rays used are paraxial -- rays travelling very close to t …