Question

Give the relation between focal length and radius of curvature.

Answer 1

Hint: You could simply consider a spherical mirror of your choice. Then, you could make a diagram for that mirror with all the necessary markings and then you could recall that the incident angle is equal to the reflected angle by laws of reflection. After that you could use simple geometry and express the radius of curvature in terms of focal length and thus get the answer.

Formula used:
Law of reflection, I.e., angle of incidence is equal to the angle of reflection.
$i=r$

Complete answer:
In the question, we are asked to give the relation between the focal length and the radius of curvature.
Focal length by definition is the point at which all the rays parallel to the principle axis converge (concave mirror) after reflection or appear to diverge from (convex mirror). We define the radius of curvature as the radius of that sphere of which the reflecting surface forms a part. We could derive the relation between the two by using simple geometry and laws of reflection.
Let us consider a concave mirror.

In the figure, PF is the focal length and PC is the radius of curvature. The ray is incident at point O at an incident angle $i$ and it is reflected at an angle $r$. If OC is the normal to the point of incidence, we have by laws of reflection that the incident angle is equal to the reflected angle, that is,
$i=r$ ………………………………. (1)
From the figure, we see that $\langle OCF$ and $i$ are alternate interior angle and hence will be equal and hence from (1) we have,
$\langle OCF=r$
We know that sides opposite to equal angles are equal, so,
$OF=CF$
For a very small aperture, we could say that,
$OF\approx PF$
$\Rightarrow OF=PF=CF$ ………………………………………. (2)
From the figure,
$R=CF+PF$
From (2),
$R=PF+PF=2PF$
Clearly, PF is the focal length,
$\therefore R=2f$

Therefore, we found the relation between focal length and radius of curvature to be,
$R=2f$

Note:
Though we have used concave mirror to derive the relation, this relation is also true for convex mirrors. Also, by convention, all the measurements are taken from the pole of the mirror, be it image distance, object distance, focal length or radius. So, for a concave mirror both are negative and for a convex mirror both are positive.