One side of an equilateral prism is painted by a transparent material of refractive index n₂. The refractive index of prism is 1.6. The minimum value of n₂ required for total internal reflection from painted face is ________. (a) (3√(3))/(1.6) (b) √(3) (c) (3.2)/(√(3)) (d) (4√(3))/(5)

Answer: (d). The ray enters the left face normally, so it goes straight. That face makes 60° with the base, so the ray meets the base at 60° to its normal. TIR at the painted base needs sin 60° ≥ (n₂)/(1.6). n₂ ≤ 1.6×√(3)/2 = 0.8√(3) = (4√(3))/(5). The limiting value of n₂ is (4√(3))/(5) (TIR needs n₂ below this, so it is really a maximum). So the answer is (d).

One side of an equilateral prism is painted by a transparent material… | JEE Main 2026 (2 Apr, Shift 2)

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Total Internal Reflection

One side of an equilateral prism is painted by a transparent material of refractive index $n_2$ . The refractive index of prism is $1.6$ . The minimum value of $n_2$ required for total internal reflection from painted face is ________.

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Answer: (d)

The ray enters the left face normally, so it goes straight. That face makes $60^\circ$ with the base, so the ray meets the base at $60^\circ$ to its normal.
TIR at the painted base needs $\sin60^\circ \ge \frac{n_2}{1.6}$ .
$n_2 \le 1.6\times\frac{\sqrt{3}}{2} = 0.8\sqrt{3} = \frac{4\sqrt{3}}{5}$ .
The limiting value of $n_2$ is $\frac{4\sqrt{3}}{5}$ (TIR needs $n_2$ below this, so it is really a maximum). So the answer is (d).

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