Question

Light enters at an angle of incidence in a transparent rod of refractive index n. For what value of the refractive index of the material of rod the light once entered into it will not leave it through its lateral face whatsoever be the value of angle of incidence?

• A. $n>\surd 2$
• B. $n=1$
• C. $n=1.1$
• D. $n=1.3$

Answer 1

Answer: (A)

$n>\surd 2$

From the following figure
$r+i={90}^o\Rightarrow i={90}^o-r$
For ray not to emerge from curved surface $i>C$
$\Rightarrow \sin i>\sin C\Rightarrow \sin ({90}^o-r)>\sin C\Rightarrow \cos r>\sin C$
$\Rightarrow \sqrt{1-{\sin }^{2}r}>\frac{1}{n}\{\because \sin C=\frac{1}{n}\}$
$\Rightarrow 1-\frac{{\sin }^2\alpha }{n^2}>\frac{1}{n^2}\Rightarrow 1>\frac{1}{n^2}(1+{\sin }^2\alpha )$
$\Rightarrow n^2>1{\sin }^2\alpha \Rightarrow n>\sqrt{2}\{\sin i\to 1\}$
$\Rightarrow$ Least value $=\sqrt{2}$