Let's go through the required calculations step-by-step for each part:

Given Data:

• Background gas pressure, $P = 10^{-2} \, \text{Pa}$
• Gas temperature, $T_g = 600 \, \text{K}$
• Electron number density, $n_e = 10^{11} \, \text{cm}^{-3}$ = $10^{17} \, \text{m}^{-3}$
• Electron temperature, $T_e = 4 \, \text{eV}$

A. Debye Radius

The Debye radius ($\lambda_D$) is given by:

$\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$

Where:

• $\varepsilon_0$ is the permittivity of free space, $\varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}$,
• $k_B$ is the Boltzmann constant, $k_B = 1.38 \times 10^{-23} \, \text{J/K}$,
• $T_e$ is the electron temperature in Kelvin. To convert $T_e$ from eV to K, use $T_e (\text{K}) = T_e (\text{eV}) \times 11604$,
• $e$ is the elementary charge, $e = 1.602 \times 10^{-19} \, \text{C}$,
• $n_e$ is the electron number density in $\text{m}^{-3}$.

First, convert the electron temperature to Kelvin: $T_e = 4 \, \text{eV} \times 11604 \, \text{K/eV} = 46416 \, \text{K}$

Now we can calculate $\lambda_D$.

B. Gas Number Density

Use the ideal gas law to find the number density ($n_g$):

$n_g = \frac{P}{k_B T_g}$

Where:

• $P = 10^{-2} \, \text{Pa}$,
• $T_g = 600 \, \text{K}$,
• $k_B = 1.38 \times 10^{-23} \, \text{J/K}$.

C. Elastic Collisional Cross-Section between Electrons and Gas Atoms

The elastic collisional cross-section ($\sigma$) between electrons and hydrogen atoms can be approximated by:

$\sigma \approx 10^{-20} \, \text{m}^2$

This is a typical value for electron-hydrogen collisions. For more precise calculations, detailed quantum mechanical computations would be required.

D. Mean Free Path of Electron-Gas Elastic Collisions

The mean free path ($\lambda$) is given by:

$\lambda = \frac{1}{n_g \sigma}$

Where:

• $n_g$ is the gas number density,
• $\sigma$ is the collisional cross-section.

Let's perform these calculations now.Here are the results for each part of the problem:

A. Debye Radius

The Debye radius ($\lambda_D$) is approximately $4.70 \times 10^{-5} \, \text{m}$ or $47 \, \mu\text{m}$.

B. Gas Number Density

The gas number density ($n_g$) is approximately $1.21 \times 10^{18} \, \text{m}^{-3}$.

C. Elastic Collisional Cross-Section

The cross-section is given as $\sigma \approx 10^{-20} \, \text{m}^2$ (approximated value for electron-hydrogen collisions).

D. Mean Free Path of Electron-Gas Elastic Collisions

The mean free path ($\lambda$) is approximately $82.8 \, \text{m}$.

If you have further questions or need more detailed explanations, let me know!

Here are 5 related questions to expand your understanding:

1. How does the electron temperature affect the Debye radius in a plasma?
2. What happens to the mean free path if the gas pressure increases?
3. How would changing the gas species from hydrogen to helium affect the collisional cross-section?
4. Why is the Debye radius an important parameter in plasma physics?
5. What role does the mean free path play in determining the behavior of plasma?

Tip: The mean free path is inversely proportional to the gas number density; as the gas density increases, the mean free path decreases.