Question

An initial planer cylindrical beam travels in a medium of refractive index ${\mu _o}\left( I \right) = {\mu _o} + {\mu _2}\left( I \right)$, where ${\mu _o}$ and ${\mu _2}$are positive constants and I is the intensity of light beam. The intensity of the beam is decreasing with increasing radius. The speed of the light in the medium is
(A). Minimum on the axis of the beam
(B). the same everywhere in the beam
(C). directly proportional to the intensity
(D). maximum on the axis of the beam

Answer 1

The speed of the light is inversely proportional to the refractive index of the medium in which the beam travels. The refractive index of the medium depends upon the distance from the axis. With the help of these two relations we can calculate the speed of the light.
Formula used:
The refractive index is
${\mu _o}\left( I \right) = {\mu _o} + {\mu _2}\left( I \right)$
Intensity of light is
$I = \dfrac{K}{{{r^2}}}$

Complete answer:
For calculating the speed of the light, we have to find out the relation between the refractive index and the distance from the axis of the light.
We know that the refractive index of the medium is given by the following expression:
${\mu _o}\left( I \right) = {\mu _o} + {\mu _2}\left( I \right) \cdots \cdots \left( 1 \right)$
And the intensity of the light is
$I = \dfrac{K}{{{r^2}}} \cdots \cdots \left( 2 \right)$
From the above two relations
${\mu _o}\left( I \right) = {\mu _o} + {\mu _2}\left( {\dfrac{K}{{{r^2}}}} \right) \cdots \cdots \left( 3 \right)$
We know that speed of the light is given by the following expression:
$V = \dfrac{C}{{{\mu _o}\left( I \right)}} \cdots \cdots \left( 4 \right)$
From equation (3) and (4) we get the speed of the light as
$V = \dfrac{C}{{{\mu _o} + {\mu _2}\left( {\dfrac{K}{{{r^2}}}} \right)}} \cdots \cdots \left( 5 \right)$
From the equation (5) we are seeing that when the value of (r) increases then the speed of the light (V) also increases and when the value of (r) decreases then the speed of the light also decreases.
So it is clear that the speed of the light increases when we go outside from the axis. Thus, speed of the light is minimum on the axis of the beam.

Therefore, option A is correct.

Note:
In this problem, we have to determine correctly the relation in between velocity of the light and the distance from the axis of the beam. Then correctly gives the relation between the speed of the light and the distance from the axis of the beam.