Give relation between focal length and radius of curvature for spherical mirror.

Explanation

Let C be the centre of curvature of the mirror.
Consider a ray parallel to the principal axis striking the mirror at M. Then CM will be perpendicular to the mirror at M.
Let `theta` be the angle of incidence and MD be the perpendicular from M on the principal axis. Then `angleMCP = theta" and "angleMFP = 2theta` Because, `DeltaMCP` is an obtuse angle for `angleMCP" and "angleMCF" and "angleCMF` are its internal angles.
`DeltaMDC" and "DeltaMDF` are equiangular. [D `~~` P] `therefore tan theta=(MD)/(CD)" and "tan2theta=2theta` For small `theta`,which is true for paraxial rays, `tan theta ~~ theta" and "tan 2theta ~~ 2theta` `therefore theta=(MD)/(CD)" and "2 theta=(MD)/(FD)` ... (1)
Now, for small 8, the point D is very close to the point P. Therefore, CD = CP = R and FD = FP = f
From equation (1),
`2theta=(MD)/(FD)`
`therefore 2 theta=(MD)/(f)` ... (2)
and `theta=(MD)/(CD)`
`therefore theta=(MD)/(R)`... (3)
`therefore` From equation (2) and (3),
`2xx(MD)/(R)=(MD)/(f)`
`therefore 2f=R`
`therefore` R=2f or f=`R/2`
This relation is true for both concave and convex mirrors.