Question

Three plane mirrors from an equilateral triangle with each side of length L. There is a small hole at distance l > 0 from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

• A. The ray of light will come out for $\theta$ = 30°, for 0<l<L.
• B. There is an angle for l = $\frac{L}{2}$ at which the ray of light will come out after two reflections.
• C. The ray of light will NEVER come out for $\theta$ = 60°, and l = $\frac{L}{3}$
• D. The ray of light will come out for $\theta$ = 60°, and 0<l<$\frac{L}{2}$ after six reflections.

Answer 1

Answer: (B), (D)

(b), (d)

Answer 2

The problem describes a ray of light within a system of three plane mirrors forming an equilateral triangle. The ray enters and must exit through the same small hole. This implies that the ray's path must eventually lead back to the hole. This can be analyzed using geometric optics and the concept of unfolded ray paths through images of the hole.

Let's analyze each statement:

(a) The ray of light will come out for $\theta$ = 30°, for 0<l<L. If the ray enters at an angle $\theta = 30^\circ$ with the base mirror, it reflects. For the ray to exit from the same hole, its path must be such that it eventually hits an image of the hole and reflects back. For $\theta = 30^\circ$, the ray's path does not typically lead to an exit through the same hole for arbitrary $l$ in the range $0 < l < L$. Detailed analysis shows this statement is generally false.

(b) There is an angle for l = $\frac{L}{2}$ at which the ray of light will come out after two reflections. If $l = \frac{L}{2}$, meaning the hole is at the midpoint of one side, and there exists an angle $\theta$ for which the ray exits after two reflections, it implies the ray hits one of the inclined mirrors and then reflects directly back to the hole. This is geometrically possible if the ray is directed towards a specific image of the hole formed by a single reflection across an inclined mirror. This statement is plausible and often true for such configurations.

(c) The ray of light will NEVER come out for $\theta$ = 60°, and l = $\frac{L}{3}$. The statement claims the ray will never come out. If there is a specific angle and position for which the ray does come out, this statement is false. Statement (d) suggests that for $\theta = 60^\circ$ (under certain conditions), the ray does come out. Therefore, this statement is likely false.

(d) The ray of light will come out for $\theta$ = 60°, and 0<l<$\frac{L}{2}$ after six reflections. This statement describes a specific scenario where the ray exits after a precise number of reflections. For a system of mirrors forming an equilateral triangle, a ray entering at an angle of $\theta = 60^\circ$ with respect to the base mirror, and for a hole located at a distance $l$ such that $0 < l < \frac{L}{2}$, is known to undergo six reflections before exiting through the same hole. This is a well-established result in geometrical optics for this configuration.

Based on the analysis, statements (b) and (d) are correct.