Maxwell’s Equations

In vacuum:

$$\epsilon_0\nabla\cdot\mathbf E=\sigma$$ $$\nabla\cdot\mathbf E=-\mathbf{\dot B}$$ $$\nabla\cdot\mathbf B=0$$ $$\nabla\times\mathbf B=\mu_0(\mathbf j+\epsilon_0\mathbf{\dot E})$$

In a medium:

$$\nabla\cdot\mathbf D=\sigma$$ $$\nabla\times\mathbf E=-\mathbf{\dot B}$$ $$\nabla\cdot\mathbf B=0$$ $$\nabla\times\mathbf H=\mathbf j+\mathbf{\dot D}$$ $$\mathbf D=\epsilon\mathbf E$$ $$\mathbf B=\mu\mathbf H$$

Classical Treatment of Magnetic Materials

Time-Varying E Field

The electric field

$$\mathbf E=\mathbf E_0 e^{i\omega t}\mathbf{\hat x}$$

The polarization drift

$$\mathbf v_p=\pm\frac{1}{\omega_c B}\frac{dE}{dt}$$

Time Varying B Field

The magnetic field

$$\nabla\times\mathbf E=-\mathbf{\dot B}$$

The magnetic moment is invariant in slowly varying magnetic fields

$$\delta\mu=0$$

The magnetic flux through a Larmor orbit is constant

$$\Phi=B\pi\frac{v_\perp^2}{\omega_c^2}=B\pi\frac{v_\perp^2m^2}{q^2B^2}=\frac{2\pi m}{q^2}\frac{\frac{1}{2}mv_\perp^2}{B}=\frac{2\pi m}{q^2}\mu$$

• $\Phi$ is constant if $\mu$ is constant.

The First Adiabatic Invariant, $\mu$

The Second Adiabatic Invariant, J

The Third Adiabatic Invariant, $\Phi$

Nonuniform B Field

$\nabla B\perp B:\ Grad-B\ Drift$

$$F_y=-qv_x\mathbf B_z(y) =-qv_\perp(cos\omega_ct)[B_0\pm r_L(cos\omega_ct)\frac{\partial B}{\partial y}]$$ $$\overline{\mathbf F}_y=\mp qv_\perp r_L\frac{1}{2}(\partial\mathbf B/\partial y)$$

The guiding center drift velocity

$$\mathbf v_{gc}=\frac{1}{q}\frac{\mathbf F\times\mathbf B}{\mathbf B^2}=\frac{1}{q}\frac{\overline{F_y}}{\lvert\mathbf B\rvert}\hat{\mathbf x}=\mp\frac{v_\perp r_L}{\mathbf B}\frac{1}{2}\frac{\partial\mathbf B}{\partial y}\hat{\mathbf x}$$

$The\ grad-\mathbf B\ drift$

$$\mathbf v_{\nabla B}=\pm\frac{1}{2}v_\perp r_L\frac{\mathbf B\times\nabla\mathbf B}{\mathbf B^2}$$

Curved B: Curvature Drift

The centrifugal force

$$\mathbf F_{cf}=\frac{m{v_{||}}^2}{\mathbf R_c}\hat{\mathbf r}=m{v_{||}}^2\frac{\mathbf R_c}{\mathbf R_c^2}$$

The curvature drift velocity

$$\mathbf v_R=\frac{1}{q}\frac{\mathbf F_{cf}\times\mathbf B}{\mathbf B^2}=\frac{mv_{||}^2}{q\mathbf B^2}\frac{\mathbf R_c\times\mathbf B}{\mathbf R_c^2}$$

The total drift

$$\mathbf v_R+\mathbf v_{\nabla B}=\frac{m}{q}\frac{\mathbf R_c\times\mathbf B}{\mathbf R_c^2\mathbf B^2}(v_{||}^2+\frac{1}{2}v_\perp^2)$$

$\nabla B||B$: Magnetic Mirrors

The magnetic moment of the gyrating particle

$$\mu\equiv\frac{1}{2}mv_{\perp}^2/\mathbf B$$

The average force

$$\overline{\mathbf F}_z=-\mu(\partial\mathbf B_z/\partial z)$$

A plasma trapped between magnetic mirrors

Nonuniform E Field

The electric field

$$\mathbf E\equiv E_0(coskx)\mathbf{\hat{\mathbf x}}$$

The equation of motion

$$m(d\mathbf v/dt)=q[\mathbf{E}(x)+\mathbf v\times\mathbf B]$$

The finite-Larmor-radius effect

$$\mathbf v_E=(1+\frac{1}{4}r_L^2\nabla^2)\frac{\mathbf E\times\mathbf B}{B^2}$$

The conditions: E = 0

The equation of motion

$$m\frac{dv}{dt}=q\mathbf{v}\times\mathbf{B}$$

Cyclotron frequency

$$\omega_c\equiv\frac{\lvert q\rvert\mathbf{B}}{m}$$

$$v_x=v_\perp e^{i\omega_c t}$$ $$v_y=\pm iv_\perp e^{i\omega_c t}$$

combine

The Larmor radius

$$r_L\equiv\frac{v_\perp}{\omega_c}=\frac{mv_\perp}{\lvert q\rvert\mathbf{B}}$$

we get

$$x-x_0=r_Lsin\omega_c t$$ $$y-y_0=\pm r_L cos\omega_c t$$

• guiding center ($x_0, y_0$)
• plasmas are diamagnetic

The conditions: finite E

The equation of motion

$$m\frac{d\mathbf v}{dt}=q(\mathbf E+\mathbf v\times\mathbf B)$$

$$v_x=v_\perp e^{i\omega_c t}$$ $$v_y=\pm iv_\perp e^{i\omega_c t}-\frac{\mathbf E_x}{\mathbf B}$$

• The usual circular Larmor gyration
• A drift of the guiding center

• The three-dimensional orbit in space is therefore a slanted helix with changing pitch.

The conditions: gravitational field

$$\mathbf v_g=\frac{m}{q}\frac{\mathbf g\times\mathbf B}{\mathbf B^2}$$

• The magnitude of $v_g$ is usually negligible
• But an effective gravitational force due to centrifugal force is not negligible

Reference Book
Introduction to Plasma Physics and Controlled Fusion by Francis F. Chen

Our Universe is made of 69% dark energy, 27% dark matter, 1% normal matter.

What is plasma?

Plasma is also called the “fourth state of matter”. Solid is heated to become a liquid, liquid is heated to become a gas.
Upon further heating, the gas is ionized into a plasma. Since plasma usually exists only in a vacuum, we need to pump the air out of a vacuum chamber in the laboratory.

The Definition of Plasma

A plasma is a quasineutral gas of charged and neutral particles which exhibits collective behavior.

• The Coulomb force between A and B diminishes as $1/r^2$.
• However, for a given solid angle($\Delta$r/r = constant), the volume of plasma in B that can affect A increases as $r^3$.

The Saha Equation

$$\frac{n_i}{n_n}\approx2.4\times10^{21}\frac{T^{3/2}}{n_i}e^{-U_i/KT}$$ $$n_i\ is\ the\ density\ of\ ionized\ atoms$$ $$n_n\ is\ the\ density\ of\ neutral\ atoms$$ $$K\ is\ Boltzmann's\ constant$$ $$U_i\ is\ the\ ionization\ energy\ of\ the\ gas$$

Physical meaning

• When temperature is raising, the whole value is increasing exponentially with $U_i/KT$.

• The higher value of $n_i$, the lower recombination rate of ionized atoms.

The Maxwellian Distribution

The one-dimensional Maxwellian distribution

$$f(u)=Aexp(-\frac{1}{2}mu^2/KT)$$ $$fdu\ is\ the\ number\ of\ particles\ per\ m^3\ with\ velocity\ between\ u\ and\ u+du$$ $$\frac{1}{2}mu^2\ is\ the\ kinetic\ energy$$

Boltzmann’s constant K

$$K=1.38\times10^{-23}J/^{\circ}K$$

The particles density n

$$n=\int_{-\infty}^{\infty}f(u)du$$

The constant A is related to the density n

$$A=n(\frac{m}{2\pi KT})^{1/2}$$

The average kinetic energy is $\frac{1}{2}KT$

The three-dimensional Maxwellian distribution

$$f(u,v,w)=A_3exp[-\frac{1}{2}m(u^2+v^2+w^2)/KT]$$

Reference Book
Introduction to Plasma Physics and Controlled Fusion by Francis F. Chen