Question

(a) What are characteristics of image formed by a plane mirror?
(b) By giving sign conventions made, derive the mirror formula for a concave mirror.

Answer 1

(a) : Plane mirror always forms a virtual and erect image of same size as that of object. Image formed by plane mirror is laterally inverted.

(b) : Derivation of mirror formula for concave mirror.
Ray diagram of image formed by the concave mirror is shown above.
We consider the right direction as the positive direction and left to be the negative direction.
Object $AB$ is kept at a distance $u$ from the pole of mirror $P$ and its image $A^{\prime }{B}^{\prime }$ is formed at a distance $v$ from pole of mirror.
Object distance $PA=-u$ (minus sign represents the negative direction)
Image distance $P{A}^{\prime }=-v$ (minus sign represents the negative direction)
$△BAC$ and $△B^{\prime }{A}^{\prime }C$ are similar triangles.
$\therefore$ $\frac{BA}{B^{\prime }{A}^{\prime }}=\frac{AC}{A^{\prime }C}$ .......(1)
$△BPA$ and $△B^{\prime }P{A}^{\prime }$ are similar triangles.
$\therefore$ $\frac{BA}{B^{\prime }{A}^{\prime }}=\frac{PA}{P{A}^{\prime }}$ .......(2)
From (1) and (2) we get $\frac{AC}{A^{\prime }C}=\frac{PA}{P{A}^{\prime }}$
$\therefore$ $\frac{PA-PC}{PC-P{A}^{\prime }}=\frac{PA}{P{A}^{\prime }}$
Or $\frac{-u-(-R)}{-R-(-v)}=\frac{-u}{-v}$
Or $uv-vR=uR-uv$
Or $uR+vR=2uv$
Or $\frac{u+v}{uv}=\frac{2}{R}$
$⟹\frac{1}{u}+\frac{1}{v}=\frac{2}{R}=\frac{1}{f}$
where focal length of mirror $f=\frac{R}{2}$