Question

A virtual image three times the size of the object is obtained with a concave mirror of radius of curvature 36 cm. Find the distance of the object from the mirror.

Answer 1

We have been given the magnification of the image which can be used to find the relation between object distance and image distance. This relation can then be used in the mirror's formula to obtain the object distance.

Complete step by step solution:
We have been given that the size of the image is three times the size of the object which means that the image is magnified with a magnification of 3. We know the relation between magnification an image and object distances, which is,
$m = - \dfrac{v}{u}$ = 3 …equation (1)
Where, v is the image distance, u is the object distance and m is the magnification. We now find out the relation between object distance and image distance by simplifying equation (1).
$- \dfrac{v}{u} = 3 \Rightarrow v = - 3u$ …equation (2)
We are given the radius of curvature of the mirror, which is R = 36 cm. But since the mirror is concave, the radius of curvature will have a negative sign. Therefore, R = - 36 cm. The focal length of the mirror, which is represented as f, is half of the radius of curvature.
$f = \dfrac{R}{2} = - \dfrac{{36}}{2} = - 18$ cm
We now use the mirror’s formula to find out the object distance.
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
On substituting the values of the focal length and image distance in terms of object distance, we obtain, $\dfrac{1}{{\left( { - 3u} \right)}} + \dfrac{1}{u} = \dfrac{1}{{\left( { - 18} \right)}}$
Upon simplifying, we obtain,
$\dfrac{{1 - 3}}{{3u}} = \dfrac{1}{{18}} \Rightarrow u = - \dfrac{{2 \times 18}}{3} = - 12$ cm
Therefore, the object distance from the mirror is 12 cm to the left of the mirror.

Note: The object distance from the mirror is negative since we measure the distance in the direction opposite to the direction of light ray.