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 $
$\frac{1}{2}=4sinr$
$sinr=\frac{1}{2 u}$
$sin\left(60 - r\right)\geq \frac{1}{u}$
$\frac{\sqrt{3}}{2}cosr-\frac{1}{2}sinru\cdot \frac{1}{u}$
$\frac{\sqrt{3}}{2}\times \frac{\sqrt{4 u^{2} - 1}}{2 u}-\frac{1}{2}\times \frac{1}{2 u}\geq \frac{1}{u}$
$\sqrt{3}\left(4 u^{2} - 1\right)^{1 / 2}\geq 5$
$3\left(4 u^{2} - 1\right)\geq 25$
$12u^{2}-3\geq 25$
$u^{2}\geq \frac{28}{12}$
$u\geq \sqrt{\frac{7}{3}}$