Three plane mirrors form an equilateral triangle with each side of length L. There is a small hole at a distance from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle and can only come out through the same

Question

Three plane mirrors form an equilateral triangle with each side of length L. There is a small hole at a 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 $θ$ 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. [2022]

Which of the following statement(s) is(are) correct

Which of the following statement(s) is(are) correct?

Smart Achievers · Accepted Answer

(1, 2)

As we can see, for $θ = 30 °$ the ray will be incident normally and hence will retrace its path $\Rightarrow$ (1) is correct.

For $θ = 60 °, ℓ = \frac{L}{2},$ then, we get the ray diagram shown below. Clearly, the ray of light comes out after two reflections $\Rightarrow$ (2) is correct.

If $θ = 60 °$ and $ℓ = \frac{L}{3},$ then we get the ray diagram as shown below.

Clearly, after 5 reflections, the ray comes out. So (3) and (4) are incorrect.

Solution

1 The ray of light will come out for $θ = 30 °$ , for $0 < l < L$ .

2 There is an angle for $l = \frac{L}{2}$ at which the ray of light will come out after two reflections.

3 The ray of light will NEVER come out for $θ = 60 °$ , and $l = \frac{L}{3} .$

4 The ray of light will come out for $θ = 60 °$ , and $0 < l < \frac{L}{2}$ after six reflections.

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Solution

1 The ray of light will come out for θ=30°, for 0<l<L.

2 There is an angle for l=L2 at which the ray of light will come out after two reflections.

3 The ray of light will NEVER come out for θ=60°, and l=L3.

4 The ray of light will come out for θ=60°, and 0<l<L2 after six reflections.

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1 The ray of light will come out for $θ = 30 °$ , for $0 < l < L$ .

2 There is an angle for $l = \frac{L}{2}$ at which the ray of light will come out after two reflections.

3 The ray of light will NEVER come out for $θ = 60 °$ , and $l = \frac{L}{3} .$

4 The ray of light will come out for $θ = 60 °$ , and $0 < l < \frac{L}{2}$ after six reflections.