Question

Consider an equilateral prism $ABC$ ABC as shown in the figure. A ray of light is incident on the face $AB$ and gets transmitted through the prism. Then total internal reflection takes place at the face $BC$ and the ray comes out of prism through the face $AC$ . The total angle of deviation is $120^{\circ }.$ Minimum value of refractive index $m$ of the material of the prism should be

• A. $\sqrt{\frac{4}{3}}$
• B. $\sqrt{\frac{5}{3}}$
• C. $\sqrt{2}$
• D. $\sqrt{\frac{7}{3}}$

Answer 1

Answer: (D)

$30+x+60+x=180^{\circ }$
$\left(90-r\right)$
$i-r+i-r+60+2r=120$
$2i=60^{\circ }$
$i=30^{\circ }$
$\frac{1}{2}=4sinr$
$sinr=\frac{1}{2u}$
$sin\left(60-r\right)\ge \frac{1}{u}$
$\frac{\sqrt{3}}{2}cosr-\frac{1}{2}sinru\cdot \frac{1}{u}$
$\frac{\sqrt{3}}{2}\times \frac{\sqrt{4u^2-1}}{2u}-\frac{1}{2}\times \frac{1}{2u}\ge \frac{1}{u}$
$\sqrt{3}{\left(4u^2-1\right)}^{1/2}\ge 5$
$3\left(4u^2-1\right)\ge 25$
$12u^2-3\ge 25$
$u^2\ge \frac{28}{12}$
$u\ge \sqrt{\frac{7}{3}}$