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# Guiding center

The guiding center of a charged particle in a magnetic field is the instantaneous center of the circle tangent to its path. In many practical cases, the motion of the particle can be analyzed as a superposition of a fast gyration about the guiding center and a slow drift of the position of the guiding center.

Given the position $\vec{x}$, velocity $\vec{v}$, and acceleration $\vec{a}$, the position of the guiding center $\vec{x}_\mbox{GC}$ is given by

$\vec{x}_\mbox{GC} = \vec{x} - \frac{\left(\vec{a}\times\vec{v}\right)\times\vec{v}}{v^4}$.

The motion of a charged particle moving perpendicular to a uniform and constant magnetic field in the absence of other forces and fields is simply constant speed on a circular path. This is known as the gyro-orbit. For mass m, charge q, and magnetic field B, the frequency of the circular motion, the gyrofrequency or cyclotron frequency, is

ωc = qB / m

For speed v, the radius of the orbit, called the gyro-radius or Larmor radius, is

rL = v / ωc

If the magnetic field is non-uniform, or if other fields are present, but the additional effects are not too strong, the motion of the particles can be expressed as before as a uniform rotation around the guiding center, but now the guiding center itself moves. Since, by assumption, the motion of the guiding center is slow compared to the gyromotion, it is called a drift.

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## General force drifts

Generally speaking, when there is a force on the particles perpendicular to the magnetic field, then they drift in a direction perpendicular to both the force and the field. If $\vec{F}$ is the force on one particle, then the drift velocity is

$\vec{v}_f = \frac{1}{q} \frac{\vec{F}\times\vec{B}}{B^2}$

The dependence on the charge of the particle implies that the drift direction is opposite for ions as for electrons, resulting in a current. In a fluid picture, it is this current crossed with the magnetic field that provides that force counteracting the applied force.

### Gravitational field

A simple example of a force drift is a plasma in a gravitational field, e.g. the ionosphere. The drift velocity is

$\vec{v}_g = \frac{m}{q} \frac{\vec{g}\times\vec{B}}{B^2}$

Because of the mass dependence, the gravitational drift for the electrons can normally be ignored.

### Electric field

This drift, often called the $\vec{E}\times\vec{B}$ (E-cross-B) drift, is a special case because the force on the particles depends on their charge. As a result, ions (of whatever mass and charge) and electrons both move in the same direction at the same speed, so there is no net current (assuming quasineutrality). In the context of special relativity, in the frame moving with this velocity, the electric field vanishes. The value of the drift velocity is given by

$\vec{v}_E = \frac{\vec{E}\times\vec{B}}{B^2}$

### Nonuniform E

If the electric field is not uniform, the above formula is modified to read

$\vec{v}_E = \left( 1 + \frac{1}{4}r_L^2\nabla^2 \right) \frac{\vec{E}\times\vec{B}}{B^2}$

## Nonuniform B

Guiding center drifts may also result not only from external forces but also from non-uniformities in the magnetic field. It is convenient to express these drifts in terms of the parallel and perpendicular energies

$\epsilon_\| = \frac{1}{2}mv_\|^2$

$\epsilon_\perp = \frac{1}{2}mv_\perp^2$

In that case, the explicit mass dependence is eliminated. If the ions and electrons have similar temperatures, then they also have similar, though oppositely directed, drift velocities.

When a particle moves into a larger magnetic field, the curvature of its orbit becomes tighter, transforming the otherwise circular orbit into a cycloid. The drift velocity is

$\vec{v}_{\nabla B} = \frac{\epsilon_\perp}{qB} \frac{\vec{B}\times\nabla B}{B^2}$

### Curvature drift

In order for a charged particle to follow a curved field line, it needs a drift velocity out of the plane of curvature to provide the necessary centripetal force. This velocity is

$\vec{v}_R = \frac{2\epsilon_\|}{qB}\frac{\vec{R}_c\times\vec{B}}{R_c^2 B}$

### Curved vacuum drift

In the limit of small plasma pressure, Maxwell's equations provide a relationship between gradient and curvature that allows the previous two drifts to be combined as follows

$\vec{v}_R + \vec{v}_{\nabla B} = \frac{2\epsilon_\|+\epsilon_\perp}{qB}\frac{\vec{R}_c\times\vec{B}}{R_c^2 B}$

For a species in thermal equilibrium, $2\epsilon_\|+\epsilon_\perp$ can be replaced by (3 / 2)kBT.

## Polarisation drift

A time-varying electric field also results in a drift given by

$\vec{v}_p = \frac{m}{qB^2}\frac{d\vec{E}}{dt}$

Obviously this drift is different from the others in that it cannot continue indefinitely. Normally an oscillatory electric field results in a polarisation drift oscillating 90 degrees out of phase. Note also the mass dependence. Normally the polarisation drift can be neglected for electrons because of their relatively small mass.

## Diamagnetic drift

The diamagnetic drift is not actually a guiding center drift. A pressure gradient does not cause any single particle to drift. Nevertheless, the fluid velocity is defined by counting the particles moving through a reference area, and a pressure gradient results in more particles in one direction than in the other. The net velocity of the fluid is given by

$\vec{v}_D = -\frac{\nabla p\times\vec{B}}{qn B^2}$

03-10-2013 05:06:04