Question

In an ionized medium the phase velocity \[{v_p}\], group velocity ${v_g}$, and speed of light ${v_c}$ are related as;
A) ${v_p} > {v_g} > {v_c}$.
B) ${v_p} = {v_g} = {v_c}$.
C) \[{v_p} < {v_g} < {v_c}\]
D) ${v_p} > {v_c},{v_g} < {v_c}$

Answer 1

Hint: The speed of electromagnetic waves is different in different mediums. In ionized media, the index of refraction is less than unity and is strongly affected by magnetic fields.

Complete answer:
When an electromagnetic wave travels in free space its velocity is equal to c = $3 \times {10^8}m{s^{ - 1}}$. But the same electromagnetic wave if it travels in a medium the velocity changes.
Now what is the ionized phase?

An ionized medium is a medium where ions move freely. In an ionized medium the refractive index is less than unity and it is strongly affected by magnetic fields.
Let’s talk about velocity; the velocity of an electromagnetic wave is different for different mediums. As the refractive index is different for different mediums the deviation that the light wave undergoes is different and so is the velocity.

Now when an electromagnetic wave travels in free space we call the velocity as speed of light and we denote it by c r in this problem ${v_c}$ and similarly there is group velocity and also phase velocity.

What is group velocity?
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes known as the modulation or envelope of the wave propagates through space. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave.
Also group velocity is always less than the speed of light, i.e. ${v_g} < {v_c}$.

What is phase velocity?
The phase velocity of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength $\lambda $ and time period T as

$v = \dfrac{\lambda }{T}$
Hence, we know that for electromagnetic wave travelling in free space velocity is given by;
 ${v_c} = \dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}$ (equation: 1)
Also we know that the phase velocity of light wave is given by;
${v_p} = \dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}\left( {1 - \dfrac{{N{e^2}}}{{{\varepsilon _0}m{\omega ^2}}}} \right)} }}$ (equation: 2)
Comparing equation: 1 and equation: 2, We can conclude that,
${v_p} > {v_c}$
And we know that ${v_c} > {v_g}$

Hence option D is correct.

Note:The refractive index of ionized medium is less than unit; this means that any ionized medium is rarer than free space. The rarer the medium the more is the velocity of electromagnetic waves in that medium. Any other non-ionized medium is denser than free space.