Question

An object is placed in front of a convex mirror at a distance of $50\, cm$ a plane mirror is introduced in front of the convex mirror covering the lower half of it. If the distance between the object and the plane mirror is $30\, cm$ and it is found that there is no parallax between the images formed by the two mirrors for the same object. What is the radius of curvature of the convex mirror (in $cm$ )?

Answer 1

Let $O$ be the object placed in front of a convex mirror $M_{1}$ at a distance of $50\, cm$ as shown in figure. The distance of the plane mirror $M_{2}$ from the object is $30\, cm$. We know that in a plane mirror the image is formed behind the mirror at the same distance as the object in front of it. It is also given that there is no parallax between the images formed by the two mirrors, thus the image produced by the convex mirror is formed at a distance of $10 \, cm$ behind the convex mirror which will coincide with the image produced by plane mirror at a distance $30 \, cm$ behind at position $Q$ as shown in figure below.

For reflection at convex mirror using coordinate sign convention for mirror formula we have
$u=50 \,cm , v=-10\, cm$
Using mirror formula, we have $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$
$\frac{1}{f}=\frac{1}{50}-\frac{1}{10}=-\frac{4}{50}$
$f=-\frac{50}{4}$
So, radius of curvature of the mirror is
$R=2 f=-\frac{50 \times 2}{4}=-25\, cm$
The radius of curvature of convex mirror is $25\, cm$.