Question

A short linear object of length b lies along the axis of a concave mirror of focal length $f$ at a distance $u$ from the pole of the mirror, what is the size of image?

• A. $\left(\frac{f}{u-f}\right)b$
• B. ${\left(\frac{f}{u-f}\right)}^{2}b$
• C. $\left(\frac{f}{u-f}\right){}^2$
• D. $\left(\frac{f}{u-f}\right)$

Answer 1

Answer: (C)

Using the relation for the focal length of concave mirror
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}...(1)$
Differentiating equation (1), we obtain
$0=-\frac{1}{v^2}dv-\frac{1}{u^2}du$
So, $dv=-\frac{v^2}{u^2}\times b...(2)$
(Here: $du=b$ )
From equation (1)
$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{u-f}{fu}$
or $\frac{u}{v}=\frac{u-f}{f}$
$\frac{v}{u}=\frac{f}{u-f}...(3)$
Now, from equations (2) and (3), we get
$dv=-{\left(\frac{f}{u-f}\right)}^{2}b$
Therefore, size or image is $={\left(\frac{f}{u-f}\right)}^{2}b$