Nonuniform B and E Field

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

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