Lecture No. 3.
Movement in a non-uniform magnetic field. Drift approximation - conditions of applicability, drift speed. Drifts in a non-uniform magnetic field. Adiabatic invariant. Movement in crossed electric and magnetic fields. General case crossed fields of any strength and magnetic field.
III. Drift motion of charged particles
§3.1. Movement in crossed homogeneous fields.
Let us consider the movement of charged particles in crossed fields
in the drift approximation. The drift approximation is applicable if it is possible to identify a certain constant drift velocity, identical for all particles of the same type, independent of the direction of the particle velocities:
, Where
- drift speed. Let us show that this can be done for the movement of charged particles in crossed
fields. As was shown earlier, the magnetic field does not affect the movement of particles in the direction of the magnetic field. Therefore, the drift speed can only be directed perpendicular to the magnetic one, i.e. let:
, and
, Where
. Equation of motion:
(we still write the multiplier in the GHS). Then for the transverse component of velocity:
, we substitute the expansion in terms of the drift speed:
, i.e.
. Let us replace this equation by two for each component and taking into account
, i.e.,
, we obtain the equation for the drift speed:
. Multiplying vectorially by the magnetic field, we get:
. Taking into account the rule, we get
, where:

- drift speed. (3.1)

.
The drift speed does not depend on the sign of the charge and on the mass, i.e. the plasma shifts as a whole. From relation (3.1) it is clear that when
the drift speed becomes greater than the speed of light, and therefore loses its meaning. And the point is not that it is necessary to take into account relativistic corrections. At
the condition will be violated drift approximation. The condition of the drift approximation for the drift of charged particles in a magnetic field is that the influence of the force causing the drift should be insignificant during the period of revolution of the particle in the magnetic field, only in this case the drift speed will be constant. This condition can be written as:
, from which we obtain the condition for the applicability of drift motion in
fields:
.

To determine possible trajectories charged particles in
fields, consider the equation of motion for the rotating velocity component :
, where
. Let the plane ( x,y) is perpendicular to the magnetic field. Vector rotates with frequency
(electron and ion rotate in different sides) in the plane ( x,y), remaining constant in modulus.

If initial speed particles fall into this circle, then the particle will move along the epicycloid.

Area 2. The circle given by the equation
, corresponds to a cycloid. When rotating the vector the velocity vector at each period will pass through the origin, that is, the velocity will be equal to zero. These moments correspond to points at the base of the cycloid. The trajectory is similar to that described by a point located on the rim of a wheel of radius
. The height of the cycloid is , that is, proportional to the mass of the particle, so the ions will move along a much higher cycloid than electrons, which does not correspond to the schematic representation in Fig. 3.2.

Area 3. The area outside the circle in which
, corresponds to a trochoid with loops (hypocycloid), the height of which
. Loops match negative values velocity components when particles move in the opposite direction.

ABOUT area 4: Point
(
) corresponds to a straight line. If you launched a particle with an initial speed
, then the force of the electric and magnetic force at each moment of time is balanced, so the particle moves rectilinearly. One can imagine that all these trajectories correspond to the movement of points located on a wheel of radius
, therefore, for all trajectories the longitudinal spatial period
. For the period
For all trajectories, mutual compensation of the effects of the electric and magnetic fields occurs. The average kinetic energy of the particle remains constant
. It is important to note again that

All the conclusions drawn above are correct if instead electric force
use arbitrary force , acting on the particle, and
. Drift speed in a field of arbitrary force:

(3.2)

depends on the charge. For example, for gravitational force
:
- speed of gravitational drift.

§3.2. Drift motion of charged particles in a non-uniform magnetic field.

If the magnetic field changes slowly in space, then a particle moving in it will make many Larmor revolutions, winding around the magnetic field line with a slowly changing Larmor radius. We can consider the motion not of the particle itself, but of its instantaneous center of rotation, the so-called leading center. Description of the movement of a particle as the movement of a leading center, i.e. The drift approximation is applicable if the change in the Larmor radius during one revolution is significantly less than the Larmor radius itself. This condition will obviously be satisfied if the characteristic spatial scale of field changes significantly exceeds the Larmor radius:
, which is equivalent to the condition:
. Obviously, this condition is satisfied the better the larger value magnetic field strength, since the Larmor radius decreases in inverse proportion to the magnetic field strength. Let us consider some cases of general interest, since many types of motion of charged particles in inhomogeneous magnetic fields can be reduced to them.

Let us consider the problem of the motion of a charged particle in a magnetic field with a jump, to the left and to the right of the plane of which the magnetic field is uniform and identically directed, but has different sizes(see Fig. 3.5), let there be H 2 > H 1 . When a particle moves, its Larmor circle intersects the shock plane. The trajectory consists of Larmor circles with a variable Larmor radius, as a result of which the particle is “drifted” along the shock plane. As can be seen from Figure 3.5, the drift is perpendicular to the direction of the magnetic field and its gradient, and oppositely charged particles drift in different directions. For simplicity, let the particle intersect the shock plane along the normal. Then in time equal to the sum Larmor half-cycles

Fig.3.5. Gradient drift at the boundary with a jump in the magnetic field.

for the area on the left and right: the particle is displaced along this plane by a length . The drift speed can be defined as . Where  H H 2  H 1  the magnitude of the magnetic field jump, and  H H 2 + H 1  - its average value.

Drift also occurs when the magnetic field to the left and right of a certain plane does not change in magnitude, but changes direction (see Fig. 3.6). To the left and right of the boundary, particles rotate in Larmor circles of the same radius, but with opposite direction rotation. Drift occurs when the Larmor circle intersects the interface plane. Let the particle intersect the layer plane along the normal, then the Larmor circle should be “cut” along

Fig.3.6. Gradient drift when changing the direction of the magnetic field

vertical diameter and then right half should be mirrored upward for the electron and downward for the ion, as shown in Fig. 3.6. In this case, during the Larmor period, the displacement along the layer is obviously two Larmor diameters, so the drift velocity for this case is: .

§3.3. Drift in a direct current magnetic field.
Drift of charged particles in a non-uniform magnetic field straight conductor current is associated primarily with the fact that the magnetic field is inversely proportional to the distance from the current, therefore there will be a gradient drift of a charged particle moving in it. In addition, drift is associated with the curvature of magnetic field lines. Let us consider two components of this force causing drift, and accordingly we obtain two components of drift.
clause 3.3.1. Diamagnetic (gradient) drift.
The mechanism of gradient drift is that the particle has different radii of rotation in different points trajectories: part of the time she spends in a stronger field, part in a more weak field. Changing the radius of rotation creates drift (Fig. 3.7). A charged particle rotating around a field line can be considered as magnetic dipole equivalent circular current. The expression for the velocity of gradient drift can be obtained from famous expression for the force acting on a magnetic dipole in a non-uniform field:
- diamagnetic force that pushes a magnetic dipole out of strong field, Where
,
, Where component transverse to the magnetic field kinetic energy particles. For a magnetic field, as can be shown, the following relation is valid:
, Where R cr- radius of curvature of the force line, - unit normal vector.

The speed of diamagnetic (gradient) drift, where - binormal to the power line. The direction of drift along the binormal is different for electrons and ions.

Lecture No. 3. DRIFT MOTION OF CHARGED PARTICLES Movement in a non-uniform magnetic field. Drift approximation - conditions of applicability, lecture No. 3.
DRIFT MOTION OF CHARGED PARTICLES
Movement in a non-uniform magnetic field. Drift approximation - conditions of applicability,
drift speed. Drifts in a non-uniform magnetic field. Adiabatic invariant.
Movement in crossed electric and magnetic fields.
Motion in crossed homogeneous E H fields.
The drift approximation is applicable if it is possible to distinguish
some constant speed identical for all particles of the same type
drift, independent of the direction of particle velocities. The magnetic field is not
influences the movement of particles in the direction of the magnetic field. Therefore the speed
drift can only be directed perpendicular to the magnetic field.
EH
Vdr c
H2
- drift speed.
Condition for the applicability of drift motion E H
in the fields:
E
V
H
c
To determine the possible trajectories of charged particles in fields, consider
equation of motion for the rotating velocity component:
. q
mu
c
u H

If the condition of the drift approximation is not met, that is, at or at the action of the electric field is not compensated by the action of magnesium

Drift motion of charged particles in a non-uniform magnetic field.

Drift of charged particles along the plane of a magnetic field jump. Gradient drift.

Drift in a direct current magnetic field.

Centrifugal (inertial) drift.

Polarization drift.

10. Toroidal drift and rotational transformation

Lecture No. 3.

Movement in a non-uniform magnetic field. Drift approximation - conditions of applicability, drift speed. Drifts in a non-uniform magnetic field. Adiabatic invariant. Movement in crossed electric and magnetic fields. The general case of crossed fields of any strength and a magnetic field.

III. Drift motion of charged particles

§3.1. Movement in crossed homogeneous fields.

Let us consider the motion of charged particles in crossed fields in the drift approximation. The drift approximation is applicable if it is possible to identify a certain constant drift velocity, identical for all particles of the same type, independent of the direction of the particle velocities:
, Where
- drift speed. Let us show that this can be done for the movement of charged particles in crossed
fields. As was shown earlier, the magnetic field does not affect the movement of particles in the direction of the magnetic field. Therefore, the drift speed can only be directed perpendicular to the magnetic one, i.e. let:
, and
, Where
. Equation of motion:
(we still write the multiplier in the GHS). Then for the transverse component of velocity:
, we substitute the expansion in terms of the drift speed:
, i.e.
. Let us replace this equation by two for each component and taking into account
, i.e.,
, we obtain the equation for the drift speed:
. Multiplying vectorially by the magnetic field, we get:
. Taking into account the rule, we get
, where:

- drift speed. (3.1)

.

The drift speed does not depend on the sign of the charge and on the mass, i.e. the plasma shifts as a whole. From relation (3.1) it is clear that when
the drift speed becomes greater than the speed of light, and therefore loses its meaning. And the point is not that it is necessary to take into account relativistic corrections. At
the drift approximation condition will be violated. The condition of the drift approximation for the drift of charged particles in a magnetic field is that the influence of the force causing the drift should be insignificant during the period of revolution of the particle in the magnetic field, only in this case the drift speed will be constant. This condition can be written as:
, from which we obtain the condition for the applicability of drift motion in
fields:
.

To determine possible trajectories of charged particles in
fields, consider the equation of motion for the rotating velocity component :
, where
. Let the plane ( x,y) is perpendicular to the magnetic field. Vector rotates with frequency
(electron and ion rotate in different directions) in the plane ( x,y), remaining constant in modulus.

If the initial velocity of the particle falls within this circle, then the particle will move along an epicycloid.

Area 2. The circle given by the equation
, corresponds to a cycloid. When rotating the vector the velocity vector at each period will pass through the origin, that is, the velocity will be equal to zero. These moments correspond to points at the base of the cycloid. The trajectory is similar to that described by a point located on the rim of a wheel of radius
. The height of the cycloid is , that is, proportional to the mass of the particle, so the ions will move along a much higher cycloid than electrons, which does not correspond to the schematic representation in Fig. 3.2.

Area 3. The area outside the circle in which
, corresponds to a trochoid with loops (hypocycloid), the height of which
. Loops correspond to negative values ​​of the velocity component when particles move in the opposite direction.

ABOUT area 4: Point
(
) corresponds to a straight line. If you launched a particle with an initial speed
, then the force of the electric and magnetic force at each moment of time is balanced, so the particle moves rectilinearly. One can imagine that all these trajectories correspond to the movement of points located on a wheel of radius
, therefore, for all trajectories the longitudinal spatial period
. For the period
For all trajectories, mutual compensation of the effects of the electric and magnetic fields occurs. The average kinetic energy of the particle remains constant
. It is important to note again that

Rice. 3.2. Characteristic trajectories of particles in
fields: 1) trochoid without loops; 2) cycloid; 3) trochoid with loops; 4) straight.

Drift of charged particles, relatively slow directed movement of charged particles under the influence various reasons, superimposed on the main movement. So, for example, when passing electric current Through the ionized gas, electrons, in addition to the speed of their random thermal movement, acquire a small speed directed along the electric field. In this case we talk about current drift velocity. The second example is D. z. including in crossed fields, when the particle is acted upon by mutually perpendicular electric and magnetic fields. The speed of such drift is numerically equal cE/H, Where With- speed of light, E- electric field strength in GHS system units , N- magnetic field strength in Oerstedach . This speed is directed perpendicular to E And N and is superimposed on the thermal velocity of the particles.

L. A. Artsimovich.

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In astrophysical and thermonuclear problems significant interest represents the behavior of particles in a magnetic field varying in space. Often this change is quite weak, and a good approximation is the solution of the equations of motion by the perturbation method, first obtained by Alfvén. The term "sufficiently weak" means that the distance over which B changes significantly in magnitude or direction is large compared to the radius a of the particle's rotation. In this case, in the zero approximation, we can assume that the particles move in a spiral around the magnetic field lines with a rotation frequency determined by

local magnitude of the magnetic field. In the next approximation, slow changes in the orbit appear, which can be represented as a drift of their leading center (center of rotation).

The first type of spatial field change that we will consider is a change in the direction perpendicular to B. Let there be a gradient of the field magnitude in the direction unit vector, perpendicular to B, so . Then, to a first approximation, the rotation frequency can be written in the form

here is the coordinate in the direction and the expansion is carried out in the vicinity of the origin of coordinates, for which Since B does not change in direction, the movement along B remains uniform. Therefore we will only consider the change lateral movement. Having written it in the form , where is the transverse velocity in a uniform field, a is a small correction, we substitute (12.102) into the equation of motion

(12.103)

Then, keeping only first-order terms, we obtain the approximate equation

From relations (12.95) and (12.96) it follows that in a uniform field the transverse velocity and coordinate are related by the relations

(12.105)

where X is the coordinate of the center of rotation in the unperturbed circular motion(here If in (12.104) we express through then we get

This expression shows that, in addition to the oscillating term, it has a non-zero average value equal to

To determine average size it is enough to take into account that the Cartesian components vary sinusoidally with amplitude a and a phase shift of 90°. Therefore, the average value is affected only by the parallel component, so

(12.108)

Thus, the "gradient" drift velocity is given by

(12.109)

or in vector form

Expression (12.110) shows that for sufficiently small field gradients, when the drift velocity is small compared to orbital speed.

Fig. 12.6. Drift of charged particles due to the transverse gradient of the magnetic field.

In this case, the particle quickly rotates around the leading center, which slowly moves in the direction perpendicular to B and grad B. Drift direction positive particle is determined by expression (12.110). For a negatively charged particle, the drift velocity has opposite sign; this change in sign is associated with the definition of Gradient drift can be qualitatively explained by considering the change in the radius of curvature of the trajectory as the particle moves in regions where the field strength is greater and less than the average. In fig. Figure 12.6 qualitatively shows the behavior of particles with different charge signs.

Another type of field change that leads to drift of the leading center of a particle is the curvature of the field lines. Consider what is shown in FIG. 12.7 two-dimensional field independent of . In fig. 12.7, a shows a uniform magnetic field parallel to the axis. The particle rotates around a line of force in a circle of radius a with speed and simultaneously moves with constant speed along the power line. We will consider this motion as a zero approximation for the motion of a particle in the field with curved field lines shown in Fig. 12.7b, where the local radius of curvature of the field lines R is large compared to a.

Fig. 12.7. Drift of charged particles due to the curvature of field lines. a - in a constant uniform magnetic field, the particle moves in a spiral along the lines of force; b - the curvature of the magnetic field lines causes drift, perpendicular to the plane

The first approximation correction can be found as follows. Since the particle tends to move in a spiral around the field line, and the field line is curved, then for the motion of the leading center this is equivalent to the appearance centrifugal acceleration We can assume that this acceleration occurs under the influence of an effective electric field

(12.111)

as if added to the magnetic field. But, according to (12.98), the combination of such an effective electric field and magnetic field leads to centrifugal drift with a speed

(121,2)

Using the notation, we write the expression for the speed of centrifugal drift in the form

The drift direction is determined vector product, in which R is the radius vector directed from the center of curvature to the particle location. The sign in (12.113) corresponds to positive charge particles and does not depend on the sign of For negative particle the value becomes negative and the drift direction is reversed.

A more accurate, but less elegant derivation of relation (12.113) can be obtained by directly solving the equations of motion. If you enter cylindrical coordinates with the origin of coordinates at the center of curvature (see Fig. 12.7, b), then the magnetic field will only have a -component. It is easy to show that vector equation motion is reduced to the following three scalar equations:

(12-114)

If in the zeroth approximation the trajectory is a spiral with a radius a small compared to the radius of curvature, then in the lowest order. Therefore, from the first equation (12.114) we obtain the following approximate expression: Gaussian plasma particles with temperature have a drift velocity of cm/sec. This means that in a small fraction of a second they will reach the walls of the chamber due to drift. For hotter plasma, the drift speed is correspondingly even greater. One way to compensate for drift in toroidal geometry is to bend the torus into a figure-eight shape. Since the particle usually makes many revolutions inside such closed system, then it passes through regions where both curvature and gradient have various signs, and drifts alternately in various directions. Therefore, at least to first order in, the resulting average drift turns out to be equal to zero. This method of eliminating drift caused by spatial changes in the magnetic field is used in thermonuclear installations stellarator type. Plasma confinement in such installations, in contrast to installations using the pinch effect (see Chapter 10, § 5-7), is carried out using a strong external longitudinal magnetic field.