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 will first calculate the distance of the image by using the magnification formula, $m={\displaystyle \frac{-v}{u}}$. Now, we will calculate the focal length by using the mirror formula, which is as below
${\displaystyle \frac{1}{f}}={\displaystyle \frac{1}{u}}+{\displaystyle \frac{1}{v}}$

Complete step by step answer:
Let $AB$ be the object which is in the form of the face of the man and $A^{\prime }{B}^{\prime }$ be its image in the mirror.Now, it is given in the question that the man keeps his face at a distance of $25cm$ from the mirror. Therefore, the distance of the object from the mirror $=25cm$. Also, the image is enlarged $1.4$ times from the object, therefore, the magnification, $m=1.4$ . Therefore, using the magnification formula, we get
$m={\displaystyle \frac{v}{u}}$
$\Rightarrow m={\displaystyle \frac{A^{\prime }{B}^{\prime }}{AB}}={\displaystyle \frac{-v}{u}}$
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
$1.4={\displaystyle \frac{-v}{-25}}$
$1.4={\displaystyle \frac{v}{25}}$
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
$v=1.4\times 25$
$\Rightarrow v=35cm$

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

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

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.