Question

An object is kept in front of a concave mirror of focal length 20cm. The image is three times the size of the object. Calculate two possible distances of the object from the mirror.

Answer 1

Hint : Use the formula for the magnification due to a spherical mirror and the mirror formula. When a real image is formed, the magnification (m) is negative and when a virtual image is formed, the magnification is positive.
$m = - \dfrac{v}{u}$
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$

Complete Step By Step Answer:
According to the sign convection, the focal length of a concave mirror is always negative. Therefore, the focal length of the given concave mirror is $f = - 20cm$ .
It is said that the image is three times the size of the object. However, there are two cases that this can happen. One when the image formed is real and the other when the image formed is virtual.
When a real image is formed, the magnification (m) is negative and when a virtual image is formed, the magnification is positive.
Therefore, the two possible magnifications of the image are $+ 3$ or $- 3$ .
This means that $m = \pm 3$ .
The magnification for a spherical lens is given as $m = - \dfrac{v}{u}$ , where v and u are the position of the image and the object according to the sign convection.
$\Rightarrow \pm 3 = - \dfrac{v}{u}$
$\Rightarrow v = \pm 3u$
Also, from the mirror formula we know that $\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$ .
Substitute the value of v and f.
$\Rightarrow \dfrac{1}{{ - 20}} = \dfrac{1}{{ \pm 3u}} + \dfrac{1}{u}$
$\Rightarrow \dfrac{1}{{ - 20}} = \dfrac{{ \pm 1 + 3}}{{3u}}$
$\Rightarrow u = \dfrac{{( \pm 1 + 3)( - 20)}}{3} = \dfrac{{( \pm 20 - 60)}}{3}$ .
$\Rightarrow u = \dfrac{{( + 20 - 60)}}{3} = \dfrac{{ - 40}}{3}cm$ or $u = \dfrac{{( - 20 - 60)}}{3} = \dfrac{{ - 80}}{3}cm$
This means that the two possible distances of the object for the required case are $- \dfrac{{40}}{3}cm$ and $- \dfrac{{80}}{3}cm$ or we can also say that the object is placed at distance of either $\dfrac{{40}}{3}cm$ or $\dfrac{{80}}{3}cm$ in front of the mirror.

Note :
A virtual image is the image that is formed on the side of the mirror. We cannot obtain a virtual image on a screen rather we can see it in the mirror.
A real image is the image that is formed in front of the mirror. Therefore, we can obtain a real image on a screen.