Plasma Physics: Brief Overview

13.1 Debye Shielding in Plasmas

A plasma screens electric fields over the Debye length:

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

For $n_e = 10^{18}$ m$^{-3}$, $T = 10^4$ K: $\lambda_D = 7.4 \times 10^{-5}$ m $= 74\,\mu$M.

The plasma frequency:

$\omega_p = \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}}$

For $n_e = 10^{18}$ m$^{-3}$: $\omega_p = 5.64 \times 10^{10}$ rad/s, $f_p = 8.98$ GHz. EM waves with $\omega < \omega_p$ cannot propagate (evanescent).

13.2 Plasma Oscillations

Small displacements of the electron cloud create restoring forces, leading to Langmuir waves:

$\omega_{\text{Langmuir} = \omega_p\left(1 + \frac{3k_BT}{2m_e}\frac{k^2}{\omega_p^2}\right)^{-1/2}}$

At long wavelengths ($k \to 0$): $\omega \to \omega_p$ (undamped). With ion motion: the ion-acoustic wave has $\omega^2 = k^2 c_s^2/(1 + k^2\lambda_D^2)$ where $c_s = \sqrt{k_BT/m_i}$.

Worked Examples

Example 1: Gauss”s law

Problem. A uniformly charged sphere of radius $R$ has total charge $Q$. Find $E$ inside and outside.

Solution. Outside ($r > R$): $\oint E \cdot dA = Q/\varepsilon_0 \implies E \cdot 4\pi r^2 = Q/\varepsilon_0 \implies E = \frac{Q}{4\pi\varepsilon_0 r^2}$. Inside ($r < R$): enclosed charge $= Q(r/R)^3$. $E = \frac{Qr}{4\pi\varepsilon_0 R^3}$.

$\blacksquare$

Example 2: Poynting vector

Problem. An EM wave has $E_0 = 100 \mathrm{ V/m}$ in vacuum. Find the average Poynting vector magnitude.

Solution. ${\langle S \rangle = \frac{E_0^2}{2\mu_0 c} = \frac{100^2}{2 \times 4\pi \times 10^{-7} \times 3 \times 10^8} = \frac{10000}{754} \approx 13.3 \mathrm{ W/m}^2}$.

$\blacksquare$

Common Pitfalls

• Confusing Gauss’s law applications. Gauss’s law is most useful for systems with high symmetry (spherical, cylindrical, planar). Fix: Choose a Gaussian surface matching the symmetry; the flux through the surface equals the enclosed charge divided by $\varepsilon_0$.
• Wrong Maxwell equation sign. Faraday’s law has a negative sign: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$. Fix: The minus sign reflects Lenz’s law — the induced EMF opposes the change in flux.
• Confusing $\vec{D}$and $\vec{E}$, $\vec{H}$and $\vec{B}$. $\vec{D} = \varepsilon_0\vec{E} + \vec{P}$; $\vec{H} = \vec{B}/\mu_0 - \vec{M}$. Fix: In vacuum: $\vec{D} = \varepsilon_0\vec{E}$, $\vec{H} = \vec{B}/\mu_0$.

Summary

• Maxwell’s equations: Gauss’s law, Gauss’s law for magnetism, Faraday’s law, Ampère-Maxwell law.
• Gauss’s law: $\oint \vec{E} \cdot d\vec{A} = Q_{\text{enc}}/\varepsilon_0$.
• EM waves: $E_0 = cB_0$; $c = 1/\sqrt{\mu_0\varepsilon_0}$; Poynting vector $\vec{S} = \vec{E} \times \vec{H}/\mu_0$.
• Boundary conditions: tangential $E$ and normal $B$ are continuous across interfaces.

Cross-References