A1  ^?? If we assume that the velocity of the electrons at a certain point is $\vec{v}$, what is the current density $\vec{j}_t$ at that point?

A2  ^?? How to express the total current $\vec{j}_t$ from the conduction current $\vec{j}$ and the time derivative of the polarization density $\partial \vec{P}/\partial t$.

A3  ^?? Show that for the harmonic external field $\tilde{E} e^{-i\omega t}$ with the angular frequency $\omega$, the redefinition of the $\varepsilon$ is
\[\tilde{\varepsilon} = \varepsilon + \frac{i \sigma}{\omega \varepsilon_0}.\]

A4  ^?? Using that for a spherically symmetric function, the Lapplacian could be written as
\[\Delta \Phi = \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d \Phi}{dr} \right), \]
calculate the $\lambda_D$. Express your answer in terms of $\langle n \rangle$ and $T$. Estimate the value of $\lambda_D$ in the gas discharge, where$\langle n \rangle = 10^{16}~\text{м}^{-3}$, $T=10^4~\text{K}$.

Let's calculate the Lapplacian of
\[\Phi(r) = \frac{q}{4\pi \varepsilon_0 r} e^{-r/\lambda_D} \]
\[
\begin{split}
\Delta \Phi &= \frac{q}{4\pi \varepsilon_0 r^2} \frac{d}{dr}
\left( r^2 \left[ -\frac{e^{-r/\lambda_D}}{r\lambda_D} - \frac{e^{-r/\lambda_D}}{r^2} \right] \right)
= -\frac{q}{4\pi \varepsilon_0 r^2} \frac{d}{dr} \left( \left[ \frac{r}{\lambda_D} + 1 \right] e^{-r/\lambda_D}\right) =\\&=
-\frac{q}{4\pi \varepsilon_0 r^2} \left( \frac{1}{\lambda_D} e^{-r/\lambda_D} - \left[ 1 + \frac{r}{\lambda_D} \right] \frac{1}{\lambda_D} e^{-r/\lambda_D} \right) = \frac{q}{4\pi \varepsilon_0 r \lambda_D^2} e^{-r/\lambda_D} = \frac{\Phi(r)}{\lambda_D^2}
\end{split}
\]

Substitution into the equation, introduced in the problem, gives us
\[ \frac{\Phi(r)}{\lambda_D^2} \simeq \frac{e \langle n \rangle}{\varepsilon_0} \left[ 1 + \frac{e \Phi}{k_\text{B} T} - 1 + \frac{e\Phi}{k_\text{B} T}\right] = \frac{2e^2 \langle n \rangle \Phi(r)}{\varepsilon_0 k_\text{B} T}.\]

Finally,
\[\lambda_D = \sqrt{\frac{\varepsilon_0 k_\text{B} T}{2e^2\langle n \rangle}}\]

A5  ^?? Estimate the average number of particles $N_\text{loc}$ whose field is not shielded at the given point in the plasma.

The number $N_\text{loc}$ could be estimated as the number of particles in the sphere with radius equal to the Debye length:
\[ N_\text{loc} = 2 \langle n \rangle \cdot \frac{4}{3} \pi \lambda_D^3 = \frac{8\pi}{3 \langle n \rangle^{1/2}} \left( \frac{\varepsilon_0 k_\text{B}T}{2e^2} \right)^{3/2} \]

A6  ^?? Analyze the motion of the electron in the plasma in the external field $\tilde{E}e^{-i\omega t}$ and find the relative permittivity $\tilde{\varepsilon}$ of the plasma.

We have the second Newton's law $m\vec{a} = -e\tilde{E}e^{-i \omega t}$. A straight integration over time gives us
\[\vec{v} = \vec{v}_0 + \frac{e\tilde{E}}{i m \omega} e^{-i\omega t}, \quad \Rightarrow \quad \vec{j}_t = \frac{ie^2n}{m\omega} \vec{E}_0,\]
where we use that $\vec{v}_0=0$ i.e. it the absence of the $\tilde{E}$ electrons don't move. In the result we have:
\[\tilde{\varepsilon}=1-\frac{e^2n}{m\varepsilon_0 \omega^2}\]

A7  ^?? Find the circular frequency $\omega_p$ of the plasma self-oscillations.

The relative permittivity $\varepsilon$ connects $P$ and $E$:
\[P = \varepsilon_0 (\varepsilon-1) E\]
and if $\varepsilon=0$ it means that the electric field $E$ is completely generated by the local dipoles! We can also note that the relative perimittivity $\tilde{\varepsilon}$ doesn't have an imaginary part, so $\tilde{\varepsilon}=\varepsilon$. Finally, the circular frequency corresponding to zero relative permittivity is:
\[\omega= \sqrt{\frac{m \varepsilon_0}{e^2 n}}.\]