Question

A luminous object is placed $20cm$ from the surface of a convex mirror is set so that the virtual image formed in the two mirror coincide. If the plane mirror is at a distance $12cm$ from the object, find the focal length of the convex mirror.

Answer 1

Given: A luminous object is placed 20cm from the surface of a convex mirror is set so that the virtual image formed in the two mirror coincide. If the plane mirror is at a distance 12cm from the object

To find the focal length of the convex mirror.

Solution:

As plane mirror is place at a distance of 12 cm from the object, the image formed by plane mirror will be at a distance of 12 cm from the plane mirror as shown in above figure.

AS the virtual image formed by two image coincides, the distance between object and image in both cases is $12+12=24cm$ as shown in figure.

Given object distance in convex mirror is 20 cm from the convex mirror, so the image will be 4cm ($24-20=4$) away from the convex mirror, as shown in the figure.

Hence considering convex mirror,

Object distance, $u=-20cm$

Image distance, $v=+4cm$

Applying the mirror formula,

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}⟹\frac{1}{f}=\frac{1}{4}+\frac{1}{(-20)}⟹\frac{1}{f}=\frac{1}{4}-\frac{1}{20}⟹\frac{1}{f}=\frac{5-1}{20}⟹f=5cm$

is the required focal length of the convex mirror.