Question

A concave mirror has a radius of curvature $30cm$. How far must it be placed from a small object in order to give a five times magnified virtual image?

Answer 1

Hint Here using the magnification we can find the image distance in terms of the object distance. Then using those values and the radius of curvature as the focal length we can find the object distance from the mirror formula.

Formula Used In this solution we will be using the following formula,
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
where $v$ is the image distance
$u$ is the object distance
$f$ is the focal length.

Complete step by step answer:
 It is said that the concave mirror magnifies the image five times its original size. The formula for the magnification for the concave mirrors is given by,
\[m = - \dfrac{v}{u}\]
Now here \[u\] is the object distance and \[v\] is the image distance. So by substituting the value of the magnification as, \[m = 5\], we get
\[\Rightarrow 5 = - \dfrac{v}{u}\]
So we can write by arranging the terms a little,
\[\Rightarrow v = - 5u\]
In the question we are provided the radius of curvature of the concave mirror $R = - 30cm$. The value of the focal length will be half of the radius of curvature. Hence we can write the focal length of the mirror as,
$f = - 15cm$
So now we can use the mirror formula which is given by,
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
So on substituting the values $f = - 15cm$ and \[v = - 5u\], we get
$\Rightarrow - \dfrac{1}{{5u}} + \dfrac{1}{u} = - \dfrac{1}{{15}}$
On taking the LCM in the denominator of the LHS we get
$\Rightarrow \dfrac{{ - 1 + 5}}{{5u}} = - \dfrac{1}{{15}}$
Therefore on taking reciprocal on both the sides we get,
$\Rightarrow \dfrac{5}{4}u = - 15$
Now by keeping only the object distance in the LHS and taking all other terms to the RHS we get,
$\Rightarrow u = - 15 \times \dfrac{4}{5}$
On doing the calculation we get, $u = - 12cm$
So the concave mirror has to be placed at a distance of $12cm$ from the object so as to form an image that is five times magnified.

Note A concave mirror is also called a converging mirror. It is made by cutting up a hollow sphere into parts and then by painting the outer surface so that the inner surface becomes reflective. When a concave mirror is placed very close to an object a highly magnified virtual image is formed as the light rays converge at a point after being reflected by this mirror.