Question

Two plane mirrors $M_1$ and $M_2$ are inclined to each other at angle of $\theta = \pi/n$. Find the value of n for which a ray of light incident parallel to $M_2$ on $M_1$ undergoes a total of seven reflection and emerges retracing its path

Answer 1

n = 14

Answer 2

1. In the unfolded diagram, each reflection rotates the ray by an angle of $2\theta$. After 7 reflections the total rotation is:

   $$7\times 2\theta = 14\theta.$$

2. For the ray to retrace its path (i.e. exit in the reverse direction), the overall rotation must be $180^\circ$ (or $\pi$ radians). Therefore, we have:

   $$14\theta = \pi.$$

3. Given that $\theta = \frac{\pi}{n}$, substitute it into the equation:

   $$14\left(\frac{\pi}{n}\right) = \pi.$$

4. Cancel $\pi$ from both sides:

   $$\frac{14}{n} = 1 \quad \Longrightarrow \quad n = 14.$$