Motion with $E_{\parallel} \ne 0$ and homogeneous ${\bf B}$.

Consider time-stationary plasmas with ${\bf E} = {\bf E}_{\parallel} \ne 0$ and a homogeneous magnetic field ${\bf B}$. Then the parallel equation of motion becomes

$$m \frac{d v_{\parallel}}{d t} = q E_{\parallel}\ ,$$ (2.5)

which has the obvious solution $v_{\parallel}(t) = v_{\parallel}(0) + q E_{\parallel} t /m$.

Importantly the motions parallel and perpendicular to the magnetic field are, in general, separable. The velocity perpendicular to the magnetic field, ${\bf v}_{\perp}$, (with ${\bf v} = {\bf v}_{\parallel} + {\bf v}_{\perp}$ and ${\bf v}_{\perp} . {\bf B} = 0$) obeys the equation

$$m \frac{d {\bf v}_{\perp}}{d t} = q {\bf v}_{\perp} \times {\bf B} \ .$$ (2.6)

Differentiating Eq. (2.6) with respect to time and using Eq. (2.1) it is easily shown that

$$m \frac{d {\bf v}_{\perp}}{d t} = - \frac{q^{2} B^{2}}{m} {\bf v}_{\perp} = - m \Omega_{c}^{2} {\bf v}_{\perp}\ .$$ (2.7)

This equation shows simple harmonic motion; the quantity

$$\Omega_{c} = \frac{q B}{m}$$ (2.8)

is the (angular) gyrofrequency or cyclotron frequency of the particle. The gyrofrequency depends on B and the charge and mass of the particle. The gyromotion itself can be constructed by noting that the particle's acceleration is perpendicular to ${\bf B}$ and ${\bf v}_{\perp}$ (Eq. 2.6), and that the sense of rotation depends on the charge.

The gyroperiod T_c is the time for a particle to complete one cyclotron orbit:

$$T = \frac{2 \pi}{\Omega_{c}} \ .$$ (2.9)

Note that the electron gyroperiod is $\sim 2000$ times shorter than the proton gyroperiod.

The gyroradius r_L (or Larmor radius) is the radius of a particle's circular motion about a magnetic field line. By integrating Eq. (2.7) it can be shown that

$$r_{L} = m v_{\perp} / q B = v_{\perp} / \Omega_{c} \ .$$ (2.10)

Moreover it can be shown that the sense of a particle's gyromotion relative to the magnetic field direction depends on the particle's charge, either using Eq. (2.7) or directly using Eq. (2.1): protons gyrate in a clockwise sense and electrons in an anti-clockwise sense (Figure 2.1).

Consider next the current and magnetic field associated with charged particles gyrating about the magnetic field. Inspection quickly shows that these fields are anti-parallel to the background magnetic field ${\bf B}$. Accordingly, plasma particles are diamagnetic.

Exercise 2.1: Construct the gyromotion of a particle in coordinate space and show that the definition (2.10) for r_L is correct.

Exercise 2.2: Demonstrate that Figure 2.1 is correct, with protons and electrons gyrating in opposite screw senses relative to the magnetic field direction, and that plasma particles are diamagnetic.