Motion in static, homogeneous situations with ${\bf B} \ne 0$ and other forces

This subsection treats particles moving in a time-invariant and homogeneous plasma subject to magnetic and other forces ${\bf F}$. These other forces include gravity and electric forces. The parallel and perpendicular motions can be split as before:

$$m \frac{d v_{\parallel}}{d t} = F_{\parallel} \ ,$$ (2.11)

$$m \frac{d {\bf v}_{\perp}}{d t} = {\bf F}_{\perp} + q {\bf v}_{\perp} \times {\bf B} \ .$$ (2.12)

The parallel motion has the obvious solution. For the perpendicular motion we assume the form (2.3), i.e., ${\bf v}_{\perp} = {\bf v}_{D} + {\bf v}_{g}$. Substituting in and rearranging leads to

$$m \frac{d {\bf v}_{D}}{d t} + m \frac{d {\bf v}_{g}}{d t} = ... ..._{D} \times {\bf B} \right) + q {\bf v}_{g} \times {\bf B} \ .$$ (2.13)

Cancelling out the terms corresponding to the usual gyromotion, then the solution for a time-invariant drift is given by

$${\bf F}_{\perp} = - q {\bf v}_{D} \times {\bf B} \ .$$ (2.14)

Requiring that ${\bf v}_{D}$ be perpendicular to ${\bf B}$ then leads to the solution

$${\bf v}_{D} = \frac{1}{q}\ \frac{{\bf F} \times {\bf B}}{B^{2}} \ .$$ (2.15)

That is, particles subject to a force with a component perpendicular to the magnetic field will undergo a steady drift perpendicular to both the magnetic field and the perpendicular component of the force. Figure 2.2 shows that this can be understood physically in terms of the force increasing (decreasing) $v_{\perp}$ at the top (bottom) of the orbit relative to the direction of the force, thereby increasing (decreasing) r_L and so the length of the orbit perpendicular to both ${\bf B}$ and ${\bf F}_{\perp}$, and leading to a net drift of the particle in the direction given by Eq. (2.15).

The most common application of (2.135) is when the force is provided by a perpendicular electric field ${\bf E}$. Since ${\bf F}_{\perp} = q {\bf E}_{\perp}$ then, the so-called $E\times B$ drift velocity is then

$${\bf v}_{E\times B} = \frac{{\bf E} \times {\bf B}}{B^2} \ .$$ (2.16)

Very importantly, the $E\times B$ drift velocity is independent of the particle charge. This means that plasma may undergo bulk motion due to an $E\times B$ drift, with no charge separation or build-up of ambipolar electric fields due to particles with different charges or energies moving with different drift velocities.

The solar wind provides a specific illustration of this: usually the solar wind velocity ${\bf v}_{sw}$ is not parallel to the magnetic field ${\bf B}_{sw}$, and it may be asked how the plasma can maintain itself in this state. The way it does this is by setting up and maintaining a ``convection electric field'' ${\bf E}_{sw} = - {\bf v}_{sw} \times {\bf B}_{sw}$ in the plasma. Then the component of the solar wind's velocity perpendicular to the magnetic field is just ${\bf v}_{sw, \perp} = {\bf E}_{sw} \times {\bf B}_{sw} / B^{2}$. The motion of an individual solar wind plasma particle is thus made up of a speed parallel to ${\bf B}_{sw}$, the $E\times B$ drift velocity and the gyromotion.

Exercise 2.3: Show that the situation of a magnetic field perpendicular to a gravitational field ${\bf g}$ leads to a plasma drift with velocity ${\bf v}_{D} = {\bf g} \times {\bf B} / q B^{2}$ that is mass independent but dependent on charge. What are the possible consequences of this charge dependence?