Question

A converging mirror $M_1$, a point source S and a diverging mirror $M_2$ are arranged as shown in figure. The source is placed at a distance of 30 cm from $M_1$. The focal length of each of the mirrors is 20 cm. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself.

(a) Find the distance between the two mirrors.

(b) Find the location of the image formed by the single reflection from $M_2$.

Answer 1

For 1st reflection in $M_1$,

$\mu =-30\text{cm}$

f = -20 cm

$\Rightarrow \frac{1}{v}+\frac{1}{-30}=-\frac{1}{20}$

$\Rightarrow \frac{1}{v}+\frac{1}{60}\Rightarrow v=-60\text{cm}$

So, for 2nd reflection in M,

u = 60 - (30 + x) = 30 - x

v = x, f = 20 cm

$\Rightarrow \frac{1}{30-x}-\frac{1}{x}=\frac{1}{20}$

$\Rightarrow \frac{x-30+x}{x(30-x)}=\frac{1}{20}$

$\Rightarrow 40x-600=30x-x^2$

$\Rightarrow x^2+10x-600=0$

$\Rightarrow x=\frac{10+50}{2}=\frac{40}{2}$

= 20 cm or - 30 cm

$\therefore$ Total distance between the two lines is 20 + 30 = 50 cm