Question

A thin rod of length $f/3$ is placed along the optical axis of a concave mirror of focal length $f$ such that its image which is real and elongated just touches the rod. Calculate the magnification.

Answer 1

Let $l$ be length of image

then

$m=\frac{l}{\frac{f}{3}}$

$l=\frac{mf}{3}$

Also image of one end coincide with object

$u^{\prime }=2f$

$u^{\prime }=u+\frac{f}{3}$

$u=2f-\frac{f}{3}=\frac{5f}{3}$

$v=-(u+\frac{f}{3}+\frac{mf}{3})$

Putting mirror formula

$\frac{1}{u+\frac{f}{3}+\frac{mf}{3}}+\frac{1}{u}=\frac{1}{f}$

$\frac{3}{5f+f+mf}+\frac{3}{5f}=\frac{1}{f}$

$\frac{1}{m+6}=\frac{2}{15}$

$m=\frac{3}{2}$