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The speed of light in media $M_1$ and $M_2$ is $1.5\times10^8$ m/s and $2.0\times10^8$ m/s respectively. A ray of light enters from medium $M_1$ to $M_2$ at an incidence angle i. If the ray suffers total internal reflection, the value of i is
A
Equal to $\sin^{-1}\left(\frac{2}{3}\right)$
B
Equal to or less than $\sin^{-1}\left(\frac{3}{5}\right)$
C
Equal to or greater than $\sin^{-1}\left(\frac{3}{4}\right)$
D
Less than $\sin^{-1}\left(\frac{2}{3}\right)$
Detailed Solution
Refractive index $\mu = \frac{c}{v}$: $\mu_1 = \frac{3\times10^8}{1.5\times10^8} = 2$ and $\mu_2 = \frac{3\times10^8}{2.0\times10^8} = \frac{3}{2}$
Light goes from the denser medium $M_1$ to the rarer medium $M_2$, so total internal reflection is possible.
At the critical angle C: $\mu_1\sin C = \mu_2\sin90^\circ$
$\sin C = \frac{\mu_2}{\mu_1} = \frac{3/2}{2} = \frac{3}{4}$
For total internal reflection the angle of incidence must not be less than the critical angle: $\sin i \geq \frac{3}{4}$
$i \geq \sin^{-1}\left(\frac{3}{4}\right)$
Light goes from the denser medium $M_1$ to the rarer medium $M_2$, so total internal reflection is possible.
At the critical angle C: $\mu_1\sin C = \mu_2\sin90^\circ$
$\sin C = \frac{\mu_2}{\mu_1} = \frac{3/2}{2} = \frac{3}{4}$
For total internal reflection the angle of incidence must not be less than the critical angle: $\sin i \geq \frac{3}{4}$
$i \geq \sin^{-1}\left(\frac{3}{4}\right)$