Question

A man uses a concave mirror for shaving. He keeps his face at a distance of 25cm from the mirror and gets an image which is $1.4$ times enlarged. Find the focal length of the mirror.

Answer 1

Hint
To find the focal length of the concave mirror, we need to find the image distance and apply the mirror formula to find the focal length. We have to apply the formula to find the magnification of the image.
$\Rightarrow m = - \dfrac{v}{u}$
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
Where, m is the magnification of the mirror, u is the object distance, v is the image distance, f is the focal length of the mirror.

Complete step by step answer
Let us consider the object distance be u, and the image distance be v. the focal length of the mirror be f and its magnification be m.
We know that, the magnification of the lens is given by,
$\Rightarrow m = - \dfrac{v}{u}$
The magnification of the mirror is $1.4$ , the object distance is 25cm. The object distance is always negative for a concave mirror. Then,
$\Rightarrow 1.4 = - \dfrac{v}{{\left( { - 25} \right)}}$
$\Rightarrow v = 35cm$
The image distance is 35cm. Now, to find the focal length of the concave mirror,
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{{\left( { - 25} \right)}} + \dfrac{1}{{35}}$
When we simplify the above equation, we get,
$\Rightarrow \dfrac{1}{f} = \dfrac{{ - 25 + 35}}{{875}}$
$\Rightarrow \dfrac{1}{f} = \dfrac{{ - 10}}{{875}}$
Then when we reciprocate, we get,
$\Rightarrow f = - 87.5$
Thus, we get the focal length of the concave mirror as $ - 87.5$cm.

Note
Concave mirror is used as a shaving mirror. It is because it produces a virtual, erect and enlarged image of the face in the mirror, when held at a distance less than the focal length of the mirror. It gives us a clear and enlarged image of our face, for a better view. It is also used in torches, as a parallel beam of light can travel a greater distance in a concave mirror.