Question

The critical angle going from medium A to medium B is $\theta$. If the speed of light in medium A is $v$, then the speed of light in medium B is:
(A). $\dfrac{v}{\sin \theta }$
(B). $v\sin \theta$
(C). $\dfrac{v}{\tan \theta }$
(D). $v\tan \theta$

Answer 1

Light travels from a medium A to a medium B. Light in a medium undergo total internal reflection when the angle of incidence is greater than the critical angle. The refractive index is the ratio of speed of light in different media. The refractive index is the inverse of sine of critical angle, using both relations, we can calculate speed of light in medium B.
Formulas Used:
${}_{B}{{\mu }_{A}}=\dfrac{1}{\sin \theta }$

Complete answer:
When light travels from denser medium to rarer medium then after refraction at a certain maximum angle for which the angle of refraction is ${{90}^{o}}$, total internal reflection takes place in the medium. This angle is known as the critical angle.
Given that, the critical angle is $\theta$ when light travels from medium A to medium B. This means that medium A is denser than medium B.
The refractive index of medium A with respect to medium B is the ratio of speed of light in medium B to the speed of light in medium A. The refractive index is also related to the critical angle by the following relation,
${}_{B}{{\mu }_{A}}=\dfrac{1}{\sin \theta }$ - (1)
Here, ${}_{B}{{\mu }_{A}}$ depicts the refractive index of medium A with respect to medium B
Also we know that,
${}_{B}{{\mu }_{A}}=\dfrac{{{v}_{B}}}{{{v}_{A}}}$ - (2)
Here, ${{v}_{B}}$ depicts the speed of light in medium B
${{v}_{A}}$ is the speed of light in medium A
From eq (1) and eq (2), we get,
$\begin{aligned} & \dfrac{1}{\sin \theta }=\dfrac{{{v}_{B}}}{{{v}_{A}}} \\\ & \Rightarrow \dfrac{1}{\sin \theta }=\dfrac{{{v}_{B}}}{v} \\\ & \therefore {{v}_{B}}=\dfrac{v}{\sin \theta } \\\ \end{aligned}$
Therefore, in medium B, the velocity of light is $\dfrac{v}{\sin \theta }$.

Hence, the correct option is (A).

Note:
Refractive index is a unitless quantity because it is a ratio of similar units. Also, as the density increases, the refractive index decreases and vice versa. For all angles greater than critical angle, the light undergoes reflection instead of refraction at the boundary between two mediums.