Question

What is the relationship between focal length (f) and radius of curvature (R).

Answer 1

Hint: Mirror is defined as the smooth surface which has a capability to reflect most of the light that falls on the mirror. There are three types of mirror that are known: plane mirror, concave mirror and convex mirror in which the focal length of plane mirror is infinite.

Complete step by step solution:
It is asked in the problem about the relationship between the focal length (f) and radius of curvature (R). Let us take a concave mirror and deduce the relationship between the focal length (f) and radius of curvature (R).
According to the law of reflection,
The normal from the radius of curvature gets reflected from the mirror and traces its path again.
$\Rightarrow \mathrm{\angle }i=\mathrm{\angle }r$.
Since AB is parallel to CO the alternate angles,
$\Rightarrow \mathrm{\angle }\alpha =\mathrm{\angle }i$
Also,
$\Rightarrow \mathrm{\angle }\alpha =\mathrm{\angle }r$ (Since the angle of incidence is equal to angle of reflection).
Here the triangle BFC is an isosceles triangle and therefore$CF=FB$.
If the aperture of the mirror is very small then point B will lies near point O we get,
$\Rightarrow FB=FO$.
As if point B lies near O then the FB becomes FO and also,
$\Rightarrow FO=CF$.
Since,
$\Rightarrow CO=CF+FO$
And $FO=CF$ so we get,
$\Rightarrow FO={\displaystyle \frac{CO}{2}}$.
$\Rightarrow f={\displaystyle \frac{R}{2}}$.

The FO is focal length (f) and CO is radius of curvature (R). So the focal length is half of the radius of curvature.

Note: The students are advised to remember the relationship between the focal length and radius of curvature. It is very helpful in predicting the image formation of the image when objects are placed at different places in front of the mirror or lens.