Question

For the same angle of incidence in media A, B and C the angles of refraction are $20^\circ$, $30^\circ$ and $40^\circ$ respectively which medium will the velocity of light be maximum? Give reso in support of your answer?

Answer 1

Use the formula for reflective index for a medium in terms of velocity of the light a medium, also use the formula for the reflection index for a medium in terms of angle of incidence and angle of reflection. Using these two formulae, derive the equation giving the relation between the velocity of the light in a medium and the angle of refraction.

Formulae used:
The refractive index $\mu$ of a medium is given by
$\mu = \dfrac{c}{v}$ ……. (1)
Here, $c$ is the speed of light in vacuum and $v$ is the speed in the medium.
According to Snell’s law, the refractive index $\mu$ of a medium is given by
$\mu = \dfrac{{\sin i}}{{\sin r}}$ .….. (2)
Here, $i$ is the angle of incidence of the light and $r$ is the angle of reflection of the light.

Complete step by step answer:
It is given that three light rays have the same angle of incidence in the three media A, B and C. But the angle of refraction of the light in the same media are $20^\circ$, $30^\circ$ and $40^\circ$ respectively. We can drive the relation between the angle of reflection and velocity of light in a medium using equations (1) and (2).

Substitute $\dfrac{c}{v}$ for $\mu$ in equation (2)
$\dfrac{c}{v} = \dfrac{{\sin i}}{{\sin r}}$
$\Rightarrow v = \dfrac{{c\sin r}}{{\sin i}}$
From the above equation, we can conclude that the velocity of a light depends on the speed of light, angle refraction of the light and angle of incidence

But for the present question for the present the angle of incidence for all the rays entering the media A, B and C are constant. Also, the speed of light in vacuum is also constant. Hence, the speed of light in a medium is proportional to the angle of refraction of the light ray for that medium.

Therefore, we can conclude that the velocity of the light will be maximum for the medium with the large angle of reflection of the light. The light ray has a maximum angle of refraction for the medium C which has an angle of refraction $40^\circ$.

Hence, for medium C the velocity of light will be maximum.

Note: The students may think that the expression for Snell’s law as ${\mu _1}\sin {\theta _1} = {\mu _2}\sin {\theta _2}$ then why we have no considered the second term of refractive index. But here we have considered the medium from which the light rays enters into media A, B and C are aid medium. The refractive index of the air medium is 1. Hence, the equation for Snell’s law we have used in equation (2) has only one refractive index term.