Question

A beam of white light is incident on glass air interface from glass to air such that green light just suffers total internet reflection. The colors of the light which will come out to air are
A. Violet, Indigo, Blue
B. All colors except green
C. Yellow, Orange, Red
D. White light

Answer 1

Hint: Basic concept of total internal reflection is used for which the critical angle is inversely proportional to the refractive index. The relation of refractive index with colors is used to solve this question.

Complete step by step answer:
For total internal reflection to take place, the two basic conditions are:
1. The ray should travel from denser to rare medium.
2. The angle of incidence must be greater than the critical angle.
As for green light, total internal reflection takes place.
So, only for those colors of light, total internal reflection will take place corresponding to when the critical angle is less than that of green light.
Now, as we know that
Velocity, $v = $frequency $ \times $ wavelength
$v = f\,\,\lambda $
So, $v\,\, \propto \,\,\lambda \,\,.......\left( 1 \right)$
And the refractive index, $\mu = \dfrac{i}{v}$
i.e. $\mu \,\, \propto \,\,\dfrac{1}{v}.......\left( 2 \right)$
So, from (1) and (2)
$\mu \,\, \propto \,\,\dfrac{1}{\lambda }.......\left( 3 \right)$
Also, $\sin \,\,{i_c} = \dfrac{1}{\mu }$ or ${\sin ^{ - 1}}\dfrac{1}{\mu }$
i.e. ${i_c}\,\, \propto \,\,\dfrac{1}{\mu }.......\left( 4 \right)$
Where ${i_c}$ is critical angle
So, from (3) and (4), we have ${i_c}\,\, \propto \,\,\lambda $
Which means more the wavelength,
More will be critical angle as $\lambda $ (Red) $ > $ $\lambda $ (Green) $ > $ $\lambda $ (Blue)
$ \Rightarrow {i_c}$ (Red) $ > $ ${i_c}$ (Green) $ > $ ${i_c}$ (Blue)
i.e. yellow, orange and red will not suffer Total internal reflection.

So, the correct answer is “Option C”.

Note:
Also by Cauchy’s relation, $\mu $ is less for higher wavelengths. So, the colors having larger wavelength than green will not suffer total internal reflection and come out in air. These colors are yellow, orange and red.