Question

Calculate the distance of an object of height $h$ cm from a concave mirror of radius of curvature $20{\text{cm}}$, so as to obtain a real image of magnification 2. Find the location of the image also.

Answer 1

Hint: Magnification of an image is ratio of size of image and size of object. Real images formed by a concave mirror are inverted; the magnification should be negative. Magnification of image is also equal to distance of image by distance of object from mirror.

Complete step by step answer:
Given, radius of curvature of mirror is $20{\text{cm}}$ and given mirror is concave then its focal length is given by $f = \dfrac{{ - R}}{2} = \dfrac{{ - 20}}{2} = - 10cm$.
Image is real and magnification is 2.
We know that if a real image formed by a concave mirror is inverted then magnification is negative.
Then, magnification $m = - 2$.
Let $u$ and $v$ be the distance of the object and image from the mirror.
Magnification of image is equal to distance of image by distance of object from mirror.
Then, $m = \dfrac{{ - v}}{u} = - 2$ or $v = 2u$ -(1)
From mirror formula $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$, we get
$\dfrac{u}{v} = \dfrac{u}{f} - 1$ or $\dfrac{{ - v}}{u} = \dfrac{f}{{f - u}}$ -(2)
From equation (1) and (2) we get
$ - 2 = \dfrac{f}{{f - u}}$
Putting focal length in above equation
$ - 2 = \dfrac{{ - 10}}{{ - 10 - u}}$ or $u = - 15cm$
Putting $u$ in equation (1)
$v = 2 \times u = - 30cm$.
Then, the distance of the object from the mirror is $15cm$ and distance of image from mirror is $30cm$.

Note: When object is at distance of double of focal length from concave mirror then image of that object is also formed at same point with same height but the image is inverted (check this by equation (2)). For an object at infinity size of image is always zero or point size image is formed. That is why a concave mirror is used many places due to its large magnification (positive or negative).