Question

Two mirrors are inclined at an angle $\theta$, as shown in the figure. A light ray is incident parallel to the mirror which is horizontal. It will start retracing its path after the third reflection if,

• A. $\theta ={45}^{\circ }$
• B. $\theta ={30}^{\circ }$
• C. $\theta ={60}^{\circ }$
• D. All three

Answer 1

The correct option is B $\theta ={30}^{\circ }$



As the ray of light retraces its path after third reflection, the incident ray for third reflection must be incident normally as shown in the diagram.

By the geometry of the figure, for first reflection,
$i_1=r_1=2\theta$

For second reflection,
$i_2=r_2=\theta$

As the ray finally retraces the original path but in opposite direction, $\Rightarrow {\delta }_{net}={180}^{\circ }$
$\Rightarrow {\delta }_1+{\delta }_2={180}^{\circ }-\left(1\right)$

Now, the deviation produces during a reflection is given by,
$\delta ={180}^{\circ }-2i$
$\Rightarrow {\delta }_1={180}^{\circ }-4\theta (\because i_1=2\theta )$
And, ${\delta }_2={180}^{\circ }-2\theta (\because i_2=\theta )$

Putting the above values in equation $\left(1\right)$, we get,

${180}^{\circ }-4\theta +{180}^{\circ }-2\theta ={180}^{\circ }$

$\Rightarrow 6\theta ={180}^{\circ }$

$\therefore \theta ={30}^{\circ }$

Hence, $\left(B\right)$ is the correct answer.