In an ionized medium, the phase velocity `v_p` group velocity `v_g` and the speed of light are related as

Options

A. `v_p gt v_g gt c `
B. `v_p = v_g = c `
C. `v_p lt v_g lt c `
D. `v_p gt c , v_g lt c `

Answer

To solve the problem regarding the relationship between phase velocity \( v_p \), group velocity \( v_g \), and the speed of light \( c \) in an ionized medium, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Phase Velocity and Group Velocity**:
- Phase velocity \( v_p \) is defined as the speed at which a particular phase of the wave travels. It is given by the formula:
\[ v_p = \frac{\omega}{k} \]
where \( \omega \) is the angular frequency and \( k \) is the wave number.
- Group velocity \( v_g \) is defined as the speed at which the overall envelope shape of the wave's amplitudes (the modulation or the group of waves) travels. It is given by:
\[ v_g = \frac{d\omega}{dk} \]

2. **Speed of Light in a Vacuum**:
- The speed of light \( c \) in a vacuum is defined as:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
where \( \mu_0 \) is the permeability and \( \epsilon_0 \) is the permittivity of free space.

3. **Phase Velocity in an Ionized Medium**:
- In an ionized medium, the phase velocity can be expressed as:
\[ v_p = \frac{1}{\sqrt{1 - \frac{n e^2}{\epsilon_0 m \omega^2}}} \]
where \( n \) is the electron density, \( e \) is the charge of the electron, \( \epsilon_0 \) is the permittivity of free space, \( m \) is the mass of the electron, and \( \omega \) is the angular frequency.

4. **Analyzing the Expression for Phase Velocity**:
- The term \( \frac{n e^2}{\epsilon_0 m \omega^2} \) is a small value when the medium is weakly ionized. Therefore, \( 1 - \frac{n e^2}{\epsilon_0 m \omega^2} < 1 \).
- Consequently, since the denominator is less than 1, \( v_p \) will be greater than \( c \):
\[ v_p > c \]

5. **Group Velocity in an Ionized Medium**:
- The group velocity \( v_g \) in an ionized medium can be derived from the dispersion relation and is generally found to be less than the speed of light:
\[ v_g < c \]

6. **Conclusion**:
- From the above analysis, we can summarize the relationships as follows:
\[ v_p > c > v_g \]

### Final Answer:
In an ionized medium, the relationship between phase velocity \( v_p \), group velocity \( v_g \), and the speed of light \( c \) is:
\[ v_p > c > v_g \]