Question

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

Answer 1

Use the formula for refractive index of a medium in terms of velocity of the light in a medium. Also use the formula for the refractive index of a medium in terms of the angle of incidence and angle of refraction. 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 of a medium is given by
$\mu = \dfrac{c}{v}$ …… (1)
Here, $\mu$ is the refractive index of the medium, $c$ is the speed of light in vacuum and $v$ is the speed of light in the medium.
According to Snell’s law the refractive index 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 refraction of the light.

Complete step by step answer:
It is given that the three light has same angle of incidence in the three media A, B and C. But the angle of refraction for the light in the same media are $20^\circ$, $30^\circ$ and $40^\circ$ respectively.We can derive the relation between the angle of refraction and he velocity of the light in a medium using equations (1) and (2).
Substitute $\dfrac{c}{v}$ for $\mu$ in equation (2).
$\dfrac{c}{v} = \dfrac{{\sin i}}{{\sin r}}$
Rearrange the above equation for the velocity of the light in a medium.
$v = \dfrac{{c\sin r}}{{\sin i}}$
From the above equation, we can conclude that the velocity of a light depends on the speed of light, angle of incidence of the light and the angle of refraction of the light.But the speed of the light is constant and angle of incidence for the present case in the question are also constant for media A, B and C.Hence, the speed of light in a medium depends only on the angle of refraction of the light for this question and the speed of light and angle of refraction (sine of the angle) are proportional to each other.
$v \propto \sin r$
Therefore, we can conclude that the velocity of the light will be maximum for the medium with a larger angle of refraction for the light.

The medium C has the largest angle of refraction of light which is $40^\circ$.Hence, for medium C the velocity of light will be maximum.

Note: The students may get confused that Snell’s law for refraction involves two terms of refractive index. But here we have considered that the first medium in which the angles of incidence are the same for all media is air and hence the one term of refractive index of air has value 1.