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)

$\left( \frac{f}{u-f} \right){^{2}}$

Answer 2

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 $0=-\frac{1}{v^{2}} d v-\frac{1}{u^{2}} d u$ So, $d v=-\frac{v^{2}}{u^{2}} \times b\,\,\,...(2)$ (Here: $d u=b$ ) From equation (1) $\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{u-f}{f u}$ or $\frac{u}{v}=\frac{u-f}{f}$ $\frac{v}{u}=\frac{f}{u-f} \,\,\,\,...(3)$ Now, from equations (2) and (3), we get $d v=-\left(\frac{f}{u-f}\right)^{2} b$ Therefore, size or image is $=\left(\frac{f}{u-f}\right)^{2} b$