Question

Consider the refraction of light ray as shown in the figure below. Incident ray is rotating with constant angular velocity $2rad/s$. What is the angular velocity of refracted ray, when the angle of incidence becomes ${30}^{\circ }$? (Given ratio of refractive indices $\frac{n_1}{n_2}=\sqrt{3}$)

• A. $2rad/s$
• B. $0.5rad/s$
• C. $4rad/s$
• D. $6rad/s$

Answer 1

The correct option is D $6rad/s$

Given
${\theta }_1={30}^{\circ }$
${\omega }_1=2rad/s$
$\frac{n_1}{n_2}=\sqrt{3}$
From Snells law
$n_{1}sin{\theta }_1=n_{2}sin{\theta }_2\dots \left(i\right)$

$\Rightarrow sin{\theta }_2=\frac{\sqrt{3}}{2}$
$\Rightarrow {\theta }_2={60}^{\circ }$
differentiating eq $\left(i\right)$ with respect to $\theta$

$\left(n_{1}cos{\theta }_1\right){\omega }_1=\left(n_{2}cos{\theta }_2\right){\omega }_2\dots \left(ii\right)$

$\left(\frac{n_1}{n_2}\frac{cos{\theta }_1}{cos{\theta }_2}\right){\omega }_1={\omega }_2\dots \left(ii\right)$

$(\sqrt{3}.\frac{\sqrt{3}}{1}).2={\omega }_2$

$\Rightarrow {\omega }_2=6rad/s$