Question

A real image of half the object size is obtained in a concave spherical mirror with a radius of curvature of $40cm$. The distance of the object and that of its image from the mirror will be respectively:

• A. $15cmand30cm$
• B. $30cmand15cm$
• C. $30cmand60cm$
• D. $60cmand30cm$

Answer 1

The correct option is D $60cmand30cm$

Given:
Image formed is real, and hence, is inverted in case of concave mirror.
Image size $h_2$ is half of object size $h_1$ :
$\frac{h_2}{h_1}=\frac{-1}{2}$

Let image distance be $v$ and object distance be $u$.
Magnification for concave mirror is given by:
$m=\frac{-v}{u}=\frac{h_2}{h_1}$
$\frac{-v}{u}=\frac{-1}{2}$
$v=\frac{u}{2}$
Radius of curvature, $R=40cm$
Focal length $\left(f\right)$ is half the radius of curvature $\left(R\right)$
$f=-\frac{R}{2}$
$f=-\frac{40}{2}=-20cm$
Using mirror formula,
$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$
$\frac{1}{u}+\frac{2}{u}=-\frac{1}{20}$
$u=-60cm$
$v=\frac{u}{2}$
$v=\frac{-60}{2}=-30cm$