Uniform E and B Fields

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