Question

A man uses a concave mirror for shaving. He keeps his face at a distance of $25\,cm$ 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 will first calculate the distance of the image by using the magnification formula, $m = \dfrac{{ - v}}{u}$. Now, we will calculate the focal length by using the mirror formula, which is as below
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$

Complete step by step answer:
Let $AB$ be the object which is in the form of the face of the man and $A'B'$ be its image in the mirror.Now, it is given in the question that the man keeps his face at a distance of $25\,cm$ from the mirror. Therefore, the distance of the object from the mirror $ = \,25\,cm$. Also, the image is enlarged $1.4$ times from the object, therefore, the magnification, $m = \,1.4$ . Therefore, using the magnification formula, we get
$m = \dfrac{v}{u}$
$ \Rightarrow \,m = \dfrac{{A'B'}}{{AB}} = \dfrac{{ - v}}{u}$
Here, we have taken a negative sign because the magnification in the case of the concave mirror is negative.

Now, putting the value of $u$ , we get
$\,1.4 = \dfrac{{ - v}}{{ - 25}}$
$1.4 = \dfrac{v}{{25}}$
Here, the distance of the object from the mirror is taken as negative because the distance is taken from the mirror to the object.

Now, we will the value of $v$ from the above equation
$v = 1.4 \times 25$
$ \Rightarrow \,v = 35\,cm$

Now, to calculate the focal length, we will use the mirror formula here
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
Now, putting the values of $v$ and $u$ , we get
$\dfrac{1}{f} = \dfrac{1}{{35}} + \left( {\dfrac{1}{{ - 25}}} \right)$
$ \Rightarrow \,\dfrac{1}{f} = \dfrac{1}{{35}} - \dfrac{1}{{25}}$
$ \Rightarrow \,\dfrac{1}{f} = \dfrac{{5 - 7}}{{175}}$
$ \Rightarrow \,\dfrac{1}{f} = \dfrac{{ - 2}}{{175}}$
$ \Rightarrow \,f = \dfrac{{175}}{{ - 2}}$
$ \therefore \,f = \, - 87.5\,cm$

Therefore, the focal length of the concave mirror is $87.5\,cm$.

Note:A concave mirror is a mirror that bends inside from the middle. A concave mirror is a mirror that reflects the light received from an object such that the reflected light is of the same angle as that of the light from the object. A concave mirror is used for shaving because it provides a magnified and virtual image of the object.