Question

Figure shows a silvered transparent cylinder of radius R. A light ray is incident on cylinder and traverse the path as shown in figure. The value of m is found to be $\alpha \sqrt{\frac{2}{\beta }}$, Find the value of $\beta -\alpha$

Answer 1

For Snell's law : $\left(1\right)sin2\theta =\left(\mu \right)sin\theta$

By geometry: $sin2\theta =\frac{\frac{3}{5}R}{R}=\frac{3}{5}$

$\Rightarrow \mu =2cos\theta \Rightarrow {\mu }^2=4co{s}^2\theta =2(1+cos2\theta )$

$\Rightarrow {\mu }^2=2(1+\frac{4}{5})=\frac{18}{5}\Rightarrow \mu \sqrt{\frac{18}{5}}=3\sqrt{\frac{2}{5}}$

$\Rightarrow \alpha =3,\beta =5\Rightarrow (\beta -\alpha )=2$