Movement in crossed electric and magnetic fields. III

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 an electric current passes through an ionized gas, electrons, in addition to the speed of their random thermal movement, acquire a small speed directed along electric field. In this case we talk about current drift speed. 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 of units , N- tension magnetic field V Oerstedach . This speed is directed perpendicular to E And N and is superimposed on the thermal velocity of the particles.

L. A. Artsimovich.

Great Soviet Encyclopedia M.: " Soviet encyclopedia", 1969-1978

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DRIFT OF CHARGED PARTICLES

DRIFT OF CHARGED PARTICLES

In plasma, relatively slow directional charge. ch-ts (el-nov and ions) under the influence of decomposition. reasons superimposed on the main (regular or disorderly). For example, basic charging movement h-tsy in a homogeneous magnet. in the absence of collisions - rotation with a cyclotron frequency. The presence of other fields distorts this movement; so, joint electrical and mag. fields leads to the so-called. electric D. z. hours in a direction perpendicular to E and H, with a speed independent of the mass and charge of the particle.

The so-called cyclotron rotation can also be superimposed on it. gradient drift arising due to magnetic inhomogeneity. field and directed perpendicular to H and DH (DH is the field gradient).

D. z. h., distributed unevenly in the medium, can arise due to their thermal movement in the direction of the greatest decrease in concentration (see DIFFUSION) with a speed vD = -Dgradn/n, where gradn is the concentration gradient of n charge. h-ts; D - coefficient diffusion.

In the case where several factors causing D. z. h., for example, electric. field and concentration gradient, drift speeds caused separately by the field, vE and vD add up.

Physical encyclopedic Dictionary. - M.: Soviet Encyclopedia. Chief Editor A. M. Prokhorov. 1983 .

DRIFT OF CHARGED PARTICLES

- relatively slow directional movement of the charger. particles under the influence of decomposition. reasons superimposed on their basis. movement (regular or disorderly). E.g. electrical in k.-l. environment (metals, gases, semiconductors, electrolytes) occurs under the influence of electrical forces. fields and is usually superimposed on the thermal (random) movement of particles. Thermal movement does not form macroscopic. flow, even if the average v this movement is much greater than the drift speed v d. Attitude v d /v characterizes the degree of directionality of charge movement. particles and depends on the type of medium, the type of charged particles and the intensity of the factors causing drift. D. z. hours can also arise when the concentration of charged particles is unevenly distributed ( diffusion), with an uneven distribution of velocities of charged particles ( thermal diffusion).
Drift of charged particles in plasma. For plasmas typically found in a magnetic field. field, characteristic D. z. h. in crossed magnetic and k.-l. other (electric, gravitational) fields. Charge particle located in a homogeneous magnetic field. field in the absence of other forces, describes the so-called. Larmor circle with radius r N=v/ w H=cmv/ZeH. Here N - magnetic tension fields, e, t And v- charge, and particle speed, w H =ZeH/mc - Larmor (cyclotron) frequency. Magn. the field is considered practically uniform if it changes little over a distance of the order of r H . If there is any ext. strength F(electric gravitational, gradient) a smooth shift of the orbit from a stationary state is superimposed on the fast Larmor rotation. speed in a direction perpendicular to the magnet. field and acting force. Drift speed

Since the denominator of the expression contains the charge of the particle, then if F acts equally on ions and electrons, they will drift under the influence of this force in opposite directions (drift current). Drift current carried by particles of a given type: Depending on the type of forces, several are distinguished. types of D. z. including: electric, polarized, gravitational, gradient. Electrical drift is called. D. z. hours in a homogeneous constant electric. field E , perpendicular to the magnetic field (crossed electric and magnetic fields). Electric the field acting in the plane of the Larmor circle accelerates the motion of the particle during that half-period of Larmor rotation when

Rice. 1. Drift of a charged particle in crossed electric and magnetic fields. Magnetic field directed towards the observer. v dE, because the speed component in one direction (downward movement in Fig. 1) is greater than the speed component when moving in the opposite direction (upward movement). Due to different radii rH on different In parts of the particle’s orbit, it is not closed in the direction perpendicular to E and H, i.e., drift occurs in this direction. In the case of electric drift F=ZeE, from here v dE =c/H 2 , i.e. the speed of the electric drift does not depend either on the sign and magnitude of the charge, or on the mass of the particle and is the same for ions and electrons in magnitude and direction. Thus, electric. D. z. h. in mag. the field leads to the movement of the entire plasma and does not excite drift currents. However, forces such as centrifugal force, which in the absence of a magnet. fields act equally on all particles, regardless of their charge, in magnetic. the field is caused not by the drift motion of the plasma as a whole, but by causing electrons and ions to drift in different sides, lead to the appearance of drift currents. acceleration, then their movement occurs as if they were acted upon. When changing electrical fields act on particles in time inertial force, associated with the change (acceleration) of electricity. drift F E =tv dE = ts [N]/N 2 . Using (1), we obtain an expression for the speed of this drift, called polarization, v dr = mc 2 E/ZeH 2 . Polarization direction D. z. hours coincides with the direction of the electric current. fields. Polarization speed drift depends on the sign of the charge, and this leads to the appearance of drift polarization. current In crossed gravitational and mag. fields arises gravitational drift with speed v dG = ts/ZeH 2, Where g- acceleration due to gravity. Because v dG depends on the mass and sign of the charge, then drift currents arise, leading to the separation of charges in the plasma. As a result, gravitational drift motion, instabilities arise. F rр, proportional to the magnetic gradient. fields (so-called gradient D. z. h.). If a particle rotating on a Larmor circle is considered as a “magnet” with magnetic moment

Rice. 2. Gradient drift. The magnetic field increases upward. The drift current is directed to the left.

Gradient drift speed

When a particle moves at a speed v || along a curved line of force (Fig. 3) with a radius of curvature R

drift occurs, which owes its origin to the centrifugal force of inertia mv 2 || /R(so-called centrifugal drift). Speed

Speeds of gradient and centrifugal DZ. h. have opposite directions for ions and electrons, i.e., drift currents arise. Here it is necessary to emphasize that the drifts under consideration are precisely displacements of the centers of Larmor circles (not much different from the displacements of the particles themselves) due to forces perpendicular to the magnetic field. field. For a system of particles (plasma), such a difference is significant. For example, if the particle tempo-pa does not depend on the coordinates, then there is no flow of particles inside the plasma (in full accordance with the fact that the magnetic field does not affect the Maxwellian field), but there is a flow of centers if the magnetic field. the field is inhomogeneous (gradient and centrifugal drift currents).

Rice. 4. Drift of plasma in a toroidal trap. Plasma confinement in a toroidal magnetic trap. Gradient and centrifugal drifts in a torus located horizontally cause vertical drift currents, charge separation and plasma polarization (Fig. 4). The emerging electrical the field forces all the plasma to move towards the outer wall of the torus (the so-called toroidal drift). Lit.: Frank-Kamenetsky D.A., Plasma - the fourth state of matter, 2nd ed., M., 1963: Braginsky S.I., Phenomena in plasma, in: Questions of plasma theory, v. 1, M., 1063: O Raevsky V.N., Plasma on Earth and in Space, K., 1980. S. S. Moiseev.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .

See what “DRIFT OF CHARGED PARTICLES” is in other dictionaries:

Slow (compared to thermal motion) directed movement of charged particles (electrons, ions, etc.) in a medium under external influence, such as electric fields. * * * DRIFT OF CHARGED PARTICLES DRIFT OF CHARGED PARTICLES, slow (according to ... encyclopedic Dictionary

Slow (compared to thermal motion) directed movement of charged particles (electrons, ions, etc.) in a medium under external influence, e.g. electric fields... Big Encyclopedic Dictionary

drift of charged particles- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics: energy in general EN charged particle drift ... Technical Translator's Guide

Relatively slow directed movement of charged particles under the influence of various causes, superimposed on the main movement. So, for example, when an electric current passes through an ionized gas, electrons, in addition to their speed... ... Great Soviet Encyclopedia

Slow (compared to thermal motion) directed movement of charged particles (electrons, ions, etc.) in a medium under external conditions. influence, for example electric fields... Natural science. encyclopedic Dictionary

In electric and magnetic fields, the movement of particles in space under the influence of the forces of these fields. The movements of plasma particles are considered below, although certain provisions are also general for plasma solids(metals, semiconductors). Distinguish... ... Physical encyclopedia

- (Dutch drift). 1) deviation of the ship from straight path. 2) the angle between the direction of movement and the middle of the ship; it depends on the structure of the vessel. 3) the position of the ship under sails positioned so that the ship remains in place slightly tilting... ... Dictionary foreign words Russian language

Partially or fully ionized gas in which the density will put. and deny. the charges are almost identical. When heated strongly, any water evaporates, turning into gas. If you increase the temperature further, the thermal process will sharply intensify... ... Physical encyclopedia

Magnet configurations fields capable long time hold charge particles or plasma in a limited volume. Natural M. l. is, for example, magnetic. Earth's field capturing plasma solar wind and holding it in the form of radiation. layers of the Earth... ... Physical encyclopedia

PROCESSES in plasma are nonequilibrium processes leading to equalization of the spatial distributions of plasma parameters: concentrations, average mass velocity and partial temperatures of electrons and heavy particles. Unlike the P. p. of neutral particles... Physical encyclopedia

First, let's consider the simplest case of the movement of individual charged particles. With a certain approximation, this consideration is applicable to particle flows, when their densities are so small that any interaction between particles can be neglected. For example, for weak beams of electrons or ions in a vacuum, the effect of their own space charge can be ignored.

The motion of an individual charged particle is described as follows general equation:

where M j is the mass of the particle (electron or ion); Z j - charge number (for electron Z e =-1);
- particle speed; But- magnetic field strength; c-speed electromagnetic waves in a vacuum; F- the resultant of all energy forces acting on particles (electric, gravitational, etc.).

The influence of the magnetic field is taken into account for convenience separately from other forces, since it, acting perpendicular to the direction of motion, does not change the energy of the particles.

Equation (6.1) can be solved only in some simple cases. Let's look at some of them, and then move on to the so-called drift approximation.

4.2. Movement of particles in an electric field E 0

IN in this case we write equation (6.1)

(6.2)

where q j is the particle charge.

Depending on the type of field, that is, depending on its coordinates and time, integration (6.2) gives different results. Let us consider some particular examples that will be useful to us for further presentation.

Example 1. Let the field strength be constant both in space and time ( E 0=const). Let's find the trajectory of an ion flying into this electric field at a certain angle θ with an initial speed u 0. (Fig.1)

Integrating (6.2), we obtain

(6.3)

where u 0 x and u 0 y are components initial speed. Eliminating t, we get

(6.5)

This is the equation of a parabola. The motion is similar to the motion of a stone thrown at an angle to the horizontal. This is understandable, since the electric field and the gravitational field are potential.

Example 2. The electric field is uniform in space, but varies in time (for simplicity, we accept the harmonic law of change E 0). An electron flies into the field, the direction of its initial velocity is perpendicular to the direction of the alternating electric field. Let us determine the law of electron motion.

Let's direct the y-axis along the field. Then

(6.7)

Here E m 0 is the amplitude of the electric field strength; ψ is the phase angle of the field at the moment t=0, when the electron begins its movement.

Integrating (6.6), (6.7), we obtain

where u 0 x , u 0 y are the components of the initial velocity of the electron. In our case u 0 y =0.

The movement of the particle is determined by the system

From formulas (6.8), (6.9) it is clear that there is a stationary drift of particles with constant speed, on which a sinusoidal oscillation with amplitude is superimposed (Fig. 2).

This happens, for example, in high-frequency discharges low pressure or at very high frequencies, when the number of elastic collisions of electrons with molecules or ions ν m is much less than the field frequency ω. It is interesting to note that in the ideal approximation (ν m →0) absorption of high-frequency energy does not occur, since the vibrational component of the velocity is shifted in phase with the field by an angle π/2, and the constant in different half-cycles is associated either with the absorption of energy or with its release back to the field.

4.3. Movement of particles in a magnetic field H 0

If all forces except the magnetic field are absent, then we write the equation of motion (6.1) in the form

(6.3)

The solution to this equation depends, as in the case of the electric helmet, on the type of the right side. Let's look at two examples.

Example 1. A particle (electron or ion) flies with a certain speed u j into a uniform constant magnetic field of intensity H 0. It is necessary to determine the law of its movement.

Let us decompose the total velocity of a particle in a magnetic field into two components: u pr- along the field, u lane– perpendicular to it:

From equation (6.12) it follows that

Hence,

that is, the particle moves uniformly along the field. For other components

(6.16)

Vector change rate u the lane is perpendicular to the vector. In this regard, the change in this vector over time can be represented as rotation with a certain angular velocity ω j

The particle rotates uniformly around the direction H 0 with angular velocity ω j, called the cyclotron or Larmor frequency, along a circle with a Larmor radius,

(6.19)

For a positively charged particle angular velocityω j is directed against H 0, for electrons - by vector H 0(Fig. 3). Due to the large difference in the masses of electrons and ions, the radii of their Larmor circles differ from each other by many orders of magnitude.

Periods of revolution along Larmor circles

In addition to rotation, the particle moves translationally at a speed u pr, therefore, its complete movement occurs along a helical line, which winds on power line fields But. Step of this helix

(6.21)

When increasing But, as can be seen from expressions (6.19) and (6.21), the radius of the Larmor circle and the pitch of the helix decrease, but linear speed it does not change.

Cyclotron rotation in a constant uniform magnetic field retains its torque(angular momentum)

where W ⊥ – kinetic energy of cyclotron rotation

Therefore, and

The quantity W ⊥ /H 0 is equal to the magnetic moment of a charge rotating in a magnetic field. In fact, the motion of a charge along a Larmor circle can be considered as circular current

(6.25)

its magnetic moment

where S is the area of ​​the Larmor circle.

Example 2. Now let's consider what happens if a particle flies into a slowly varying (with time) magnetic field.

By such a field we mean a field in which during one revolution around the Larmor circle its radius almost does not change:

Let us show that in this case the magnetic moment approximately retains its value (in this case it is called an adiabatic invariant).

If the magnetic field is a function of time, then, as is known, a vortex electric field arises, the circulation of which closed loop nothing more than electromotive force(e.d.s).

(6.28)

Where E l-electric field strength along the Larmor circle along which integration is performed; φ is the magnetic flux through the area of ​​the Larmor circle.

The change in the energy of cyclotron rotation over time, taking into account expressions (6.24) and (6.27), is equal to

(6.29)

With a slow change in the magnetic field, the value can be taken out of the differentiation sign:

Let us rewrite expression (6.24) in the form

and differentiate it by time:

(6.32)

If we compare this expression obtained earlier directly from energy considerations (6.30), then it immediately becomes obvious that the second term is equal to zero

Magnetic fluxФ, penetrating the cyclotron orbit, also remains unchanged during the movement

. (6.33)

Drifts in magnetic fields

The equation of motion (6.1) can be solved exactly only in simple cases, similar to those already considered. In the presence of a magnetic field, constant in time and uniform in space, and in the absence of electrical and other forces, a movement occurs that consists of two movements - translational along the field and rotational in the transverse plane. If the magnetic field is inhomogeneous, or if some other forces act on the particle besides it, then we will no longer get such a movement. However, in some cases, with a certain approximation, it is possible to reduce the real motion to the rotation of a particle along a Larmor circle, the center of which (the so-called leading center) moves across the magnetic field.

The movement of the leading center across the field is called drift in the magnetic field. In addition, in the presence of a velocity component along the direction of the magnetic field, the center shifts in this direction as well. Such consideration can only be carried out when the influence various forces manifests itself weakly during the period of revolution of the particle in a magnetic field, i.e., in other words, when the adiabaticity conditions (6.27) and (6.34) are satisfied. In this case, the leading center of a charged particle with a magnetic moment μ j moves like a certain particle in a field of force F with kinetic energy W per [see formula (6.26)].

The approximate theory of particle motion in adiabatic systems is called the drift approximation, and the equations that describe the average motion of the leading center and the change in the Larmor radius are called drift equations. Their rigorous derivation is quite complicated. Essentially, it comes down to considering the conditions under which the movement differs little from the movement in permanent fields. Acting forces should not vary greatly over the Larmor radius, in particular, the transverse force F lane should not lead to an excessive increase in the transverse velocities of the particle and the Larmor radius, which would violate the conditions of adiabaticity. The longitudinal force cannot be large F pr. In addition, when considering processes in plasma, when the drift approximation is applicable, the influence of the motion of the particles themselves on the fields in which they move is not taken into account.

Let us first consider drifts in time-constant fields. Equation (6.1) in projections on the Cartesian coordinate axes:

This system can be written in complex form

The solution is not homogeneous equation(6.39) consists of general solution homogeneous equation

which corresponds to cyclotron rotation, and a particular solution

(6.41)

(6.42)

In vector form

This is the speed of drift motion, the origin of which can be clearly explained as follows: during one half of the period of cyclotron rotation, the force acts along the direction of motion of the particle, its speed increases and it must travel a greater distance than during the second half of the period, when the force acts against the motion .

As already mentioned, the drift equation (6.43) describes the average motion of the leading center with approximately constant speed. The fast oscillating motion along the Larmor circle is not taken into account. It should be noted that drift motion (movement of the oscillating center) at first glance has a number of properties that seem to violate the usual ideas about the laws of mechanics. Indeed, a constant force in this case causes not uniformly accelerated, but uniform motion. Later we will see that the electric field does not separate the charges, but forces them to move in one direction, while forces of non-electric origin create electric currents. The fact is that true movement nevertheless, there is a movement along the Larmor circle, which is associated with the selection (and release) of energy and obeys ordinary laws mechanics.

Drift motion is an averaged motion as a consequence of cyclotron rotation in magnetic fields.

Electric drift

Both types of drift in a nonuniform magnetic field depend on the sign of the particles. What differs from them in this respect is electrical drift, i.e., the drift of particles in a magnetic field in the presence of an electric one. Electric drift speed

Really, electric charge is not included in the formula, and with it the dependence of the velocity on the sign of the particles is excluded. Electrical drift for ions and electrons occurs in one direction and at the same speed, despite the large difference in their masses.

It should be borne in mind that formula (6.47) is applicable only at E 0<<Н 0 , иначе скорость дрейфа получается соизмеримой со скоростью света. Весь же наш вы­вод для дрейфовых скоростей сделан исходя из по­стоянства массы частиц, т. е. для нерелятивистских ско­ростей.

We obtained formula (6.47) by substituting the value of the electric force into the general expression (6.43) for the speed of drifts in a magnetic field

However, it can be derived somewhat differently - from the general equation (6.1). This makes sense given some of the useful physical findings obtained.

Let us transform equation (6.1) into a reference system that moves relative to the original (laboratory) coordinate system with constant speed u"D. Particle speed in a moving system u", imuls R". Velocity in laboratory coordinate system

(6.50)

Let's find the change in momentum R:

Where E 0|| And E 0 ⊥,-components of the electric field along and perpendicular to the magnetic field.

Size u"D can be chosen in such a way that two conditions are met:

(6.53)

Conditions (6.52) and (6.53) determine u"D absolutely clear. From condition (6.52) it immediately follows that u"D⊥H 0. Let us multiply the second condition (6.53) vectorially by But:

The term H 0 /c·( u"D N 0)=0 according to condition (6.52). Hence,

(6.55)

those. represents the drift speed. Taking into account (6.53), we write the equation of motion (6.51)

(6.56)

A component has completely fallen out of it E 0per. From this we can conclude that the influence E 0per comes down to creating a drift in a direction perpendicular to the magnetic field. Thus, we obtain uniformly accelerated motion along the field and drift across it. Both movements add up to a parabolic movement (Fig. 8 ). If E 0 lies in the yz plane, then the leading center will not leave this plane. Since the choice of x and y axes is arbitrary, the case shown in Fig. 8 can be considered quite general.

Drift in crossed fields

A special case of electric drift is movement in crossed electric and magnetic fields ( E o ┴H o And u 0pr=0), where u 0pr- initial velocity of the particle along the direction But. Acceleration in direction H 0 absent. The particle moves along a cycloid, normal or shortened, depending on the relationship between the angular velocity ω j and the speed of movement of the center of the circle itself. The latter depends on E 0 and the initial speed u 0 =u 0per along the y axis.

Let us examine in more detail the nature of motion in crossed fields, since this case has a practical purpose, especially for plasma accelerators. Let's look at the motion of an electron and then see how it differs for ions. Fig. 9, and shows what happens if the initial speed u 0 >0. In this case, a Lorentz force arises

directed antiparallel to the x axis. Magnetic force F l is added to the electric force -eE 0. They accelerate the particle together. During the Larmor period τ e it must travel a greater distance than under the action of only one -eE 0. This effect on the particle determines its movement along an elongated cycloid.

In Fig. Figure 9b shows the case corresponding to the initial speed u 0 =0. This produces a normal cycloid. Further, if u0<0и , the cycloid becomes shortened (Fig. 9, c). When both forces are balanced the trajectory remains straight (Fig. 9, d). With a further increase in u 0, the trajectory moves to the right side of the x axis, and the same cycloid shapes are repeated in reverse order - shortened, normal and elongated (Fig. 9, e - g). Distance between successive cycloid vertices

This distance does not depend on the value of the initial speed u 0 .

For ions, the drift is in the same direction, but the rotation occurs in the opposite direction (Fig. 10 - solid lines). It is easy to see that drift in crossed fields occurs along equipotential surfaces of the electric field, since it is directed normal to the electric field.

We want to describe the behavior of one or a few molecules that are different in some way from the vast majority of other gas molecules. We will call “most” molecules “background” molecules, and molecules that differ from them will be called “special” molecules, or (for short) -molecules. A molecule can be special for a number of reasons: it can be, say, heavier than the background molecules. It may also differ from them in chemical composition. Or maybe special molecules carry an electric charge - then it will be an ion against the background of neutral molecules. Due to the unusualness of the masses or charges, the -molecules are subject to forces that differ from the forces between the background molecules. By studying the behavior of molecules, one can understand the basic effects that come into play in many different phenomena. Let us list some of them: diffusion of gases, electric current in a battery, sedimentation, separation using a centrifuge, etc.

Let's start by studying the basic process: a molecule in a gas of background molecules is subject to some special force (this could be gravity or electrical force) and, in addition, more ordinary forces due to collisions with background molecules. We are interested in the general behavior of the molecule. A detailed description of its behavior is continuous rapid impacts and subsequent one after another collisions with other molecules. But if you follow carefully, it becomes clear that the molecule is steadily moving in the direction of the force. We say that drift is superimposed on random motion. But we would like to know how the drift speed depends on the force.

If at some arbitrary point in time we begin to observe the β-molecule, then we can hope that we are right somewhere between two collisions. The molecule will use this time to increase the velocity component along the force, in addition to the speed remaining after all collisions. A little later (on average, after a time) it will again experience a collision and begin to move along a new segment of its trajectory. The starting speed, of course, will be different, but the acceleration from the force will remain unchanged.

To simplify things now, let us assume that after each collision our molecule goes to a completely “free” start. This means that she has no memories of previous accelerations under the influence of force. This assumption would be reasonable if our -molecule were much lighter than the background molecules, but this, of course, is not the case. We'll discuss a more reasonable assumption later.

For now, let us assume that all directions of the velocity of the molecule after each collision are equally probable. The starting velocity is in any direction and cannot make any contribution to the resulting motion, so we will not take into account the starting velocity after each collision. But, in addition to random motion, each -molecule at any moment has an additional speed in the direction of the force, which increases since the time of the last collision. What is the average value of this part of the speed? It is equal to the product of acceleration (where is the mass of the molecule) and the average time elapsed since the last collision. But the average time elapsed since the last collision must be equal to the average time before the next collision, which we have already designated by the letter . The average speed generated by the force is precisely the drift speed; Thus, we came to the relation

This is our basic relationship, the main thing in the entire chapter. When found, all sorts of complications may appear, but the main process is determined by equation (43.13).

Note that drift speed is proportional to force. Unfortunately, a name for constant proportionality has not yet been agreed upon. The coefficient in front of the strength of each variety has its own name. In problems related to electricity, force can be represented as the product of a charge and an electric field: ; in this case, the constant of proportionality between speed and electric field is called “mobility”. Despite possible misunderstandings, we will use the term mobility to refer to the ratio of drift speed to force of any kind. Will write

and call it mobility. From equation (43.13) it follows

Mobility is proportional to the average time between collisions (rare collisions weakly slow down the molecule) and inversely proportional to mass (the greater the inertia, the slower the speed between collisions is gained).

To obtain the correct numerical coefficient in equation (43.13) (and we have it correct), a certain amount of caution is required. To avoid misunderstandings, we must remember that we are using insidious arguments, and they can only be used after careful and detailed study. To show what difficulties there are, although everything seems to be fine, we will again return to those arguments that led to the conclusion of equation (43.13), but these arguments, which look quite convincing, will now lead to an incorrect result (unfortunately, this kind reasoning can be found in many textbooks!).

You can reason like this: the average time between collisions is . After a collision, the particle, having begun to move at a random speed, gains additional speed before the next collision, which is equal to the product of time and acceleration. Since time will pass before the next collision, the particle will gain speed. At the moment of collision this speed is zero. Therefore, the average speed between two collisions is half the final speed, and the average drift speed is . (Wrong!) This conclusion is incorrect, but equation (43.13) is correct, although it would seem that in both cases we reasoned equally convincingly. A rather insidious error crept into the second result: when deriving it, we actually assumed that all collisions are separated from each other by a time of . In fact, some of them occur earlier and others later than this time. Shorter times are more common, but their contribution to the drift speed is small, because in this case the probability of “real pushing forward” is too small. If we take into account the existence of a distribution of free time between collisions, we will see that the factor 1/2 obtained in the second case has nowhere to come from. The error occurred because we, deceived by the simplicity of the arguments, tried too simply to connect the average speed with the average final speed. The relationship between them is not so simple, so it is better to emphasize that we need the average speed on its own. In the first case, we looked for the average speed from the very beginning and found its correct value! Perhaps now you understand why we did not try to find the exact values ​​of all the numerical coefficients in our elementary equations?

Let us return to our assumption that each collision completely erases from the molecule’s memory everything about its previous movement and that after each collision a new start begins for the molecule. Let's assume that our -molecule is a heavy object against a background of lighter molecules. Then one collision is no longer enough to take away from the -molecule its forward-directed impulse. Only a few successive collisions introduce “disorder” into its movement. So, instead of our initial reasoning, let us now assume that after each collision (on average after time ) the molecule loses a certain part of its momentum. We will not explore in detail what such an assumption will lead to. It is clear that this is equivalent to replacing time (the average time between collisions) with another, longer one, corresponding to the average “forgetting time,” i.e., the average time during which a molecule will forget that it once had an impulse directed forward. If we understand this, then we can use our formula (43.15) for cases that are not as simple as the original one.

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 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 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 velocity
, 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
. During 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.
regardless of the trajectory, the drift speed is the same, therefore, the plasma in
fields drifts as a whole in a direction perpendicular to the fields. If the condition of the drift approximation is not met, that is, when
the action of the electric field is not compensated by the action of the magnetic field, so the particle goes into a mode of continuous acceleration (Fig. 3.3). The direction of movement will be a parabola. If the electric field has a longitudinal (along the magnetic field) component, the drift motion is also disrupted, and the charged particle will be accelerated in a direction parallel to the magnetic field. The direction of motion will also be a parabola.

All the conclusions drawn above are correct if instead of 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 the 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 the field change significantly exceeds the Larmor radius:
, which is equivalent to the condition:
. Obviously, this condition is fulfilled the better, the greater the magnetic field strength, since the Larmor radius decreases in inverse proportion to the magnetic field strength. Let us consider some cases that are of general interest, since many types of motion of charged particles in inhomogeneous magnetic fields can be reduced to them.

clause 3.2.1. Drift of charged particles along the plane of a magnetic field jump. Gradient drift.

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 equally directed, but has different magnitudes (see Fig. 3.5), let the right 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 a time equal to the sum of 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 the opposite direction of 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, the 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.
The drift of charged particles in a non-uniform magnetic field of a direct current conductor is associated, first of all, 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 at different points of the trajectory: it spends part of the time in a stronger field, part in a weaker field. Changing the radius of rotation creates drift (Fig. 3.7). A charged particle rotating around a field line can be considered as a magnetic dipole of an equivalent circular current. The expression for the velocity of gradient drift can be obtained from the known expression for the force acting on a magnetic dipole in a non-uniform field:
- diamagnetic force pushing a magnetic dipole out of a strong field, where
,
, Where the component of the kinetic energy of a particle that is transverse to the magnetic field. 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 field line. The direction of drift along the binormal is different for electrons and ions.