Question

You are asked to design a shaving mirror assuming that a person keeps it $10cm$ from his face and views the face's magnified image at the closest comfortable distance of $25cm$. Then what would be the radius of curvature of the mirror?

Answer 1

Hint: First, we will learn about the science behind the spherical mirror, especially the concave mirror for this problem. After that, we can come up with the formula of the spherical mirrors, and with that, we will proceed to determine the radius of curvature of the given concave mirror.

Formula used:
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$

Complete step by step solution:
A spherical mirror is exceptional from a plane mirror because, unlike a plane mirror, the spherical mirrors can form images with different sizes other than the objects' actual sizes. Concave mirrors especially form a larger image than the size of the object. When an object is placed between the pole and the focus of the mirror, it produces a virtual image. The concave mirrors are used as a shaving mirror because it makes the image larger. For the spherical mirrors, we have a formula written as follows-
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ ……….$(1)$
Here, $v$ is the distance of the formed image
$u$ is the distance of the object
$f$ is the focal length of the spherical mirror
A point of caution, that we will measure the distance behind the mirror as positive numbers and the distance in front of the mirror as negative numbers. Like here the equation $(1)$, we will put-
$u = - 10cm$, because it is in front of the mirror
$v = \left( {25 - 10} \right)cm = 15cm$, because the image is formed behind the mirror
We put these values in the equation $(1)$-
$\dfrac{1}{{15}} - \dfrac{1}{{10}} = \dfrac{1}{f}$
Solving this, we get-
$f = - 30cm$
The negative sign indicates that the focal point is in front of the mirror
We know the radius of curvature of a spherical mirror is-
$R = 2f$
$ \Rightarrow R = \left( {2 \times 30} \right)cm$
Hence $R = 60cm$

Note: A concave mirror can only produce a real image when the object is placed at a point far from the mirror's focal length. Concave mirrors are used in various places like head mirrors, ophthalmoscopes, astronomical telescopes, headlights, and solar furnaces.