Question

$f=\frac{R}{2}$ is true only for:

• A. A spherical mirror in any case
• B. A spherical mirror within paraxial approximation
• C. A parabolic mirror
• D. Any mirror

Answer 1

The correct option is B A spherical mirror within paraxial approximation

First of all lets understand that the Radius of Curvature is only defined in context of a spherical mirror (The radius of the sphere it is part of). A parabolic mirror etc doesn't have the radius of curvature defined for it and hence, there is no meaning in focus and radius of curvature relationship.
Now, for spherical mirrors lets recall the derivation of the relationship.

$\angle BCF=\theta \left(Alternateangles\right)$
Which makes , $\Delta$ BCF isoceles, and we can see that FC = FB (or R).
Now, for the relationship $f=\frac{R}{2}$ to hold true, FB should be equal to FP (or f) , which is possible if point
B is very close to point P.
So, From the diagram we see that, the approximation is only valid if the incident ray is extremely close to the principal axis, which is what we call paraxial approximation. ('Para' means “at or to one side of, beside, side by side” in Greek)