Lecture notes Plasma physics pdf

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Dr.LeonBurns,New Zealand,Researcher
Published Date:21-07-2017
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Lectures in Plasma Physics This manuscript is the written version of seven two-hour lectures that I gave as a minicourse in Plasma Physics at Illinois Institute of Technology in 1978 and 1980. The course, which was open to all IIT students, served to introduce the basic concept of plasmas and to describe the potentially practical uses of plasma in communication, generation and conversion of energy, and propulsion. A final exam, which involves nomenclature and concepts in plasma physics, is included at the end. It is increasingly evident that the plasma concept plays a basic role in current re- search and development. However, it is not particularly simple for a science or engineering student to obtain a qualitative understanding of the basic properties of plasmas. In a plasma, one must understand and apply the principles of electro- magnetism (and electromagnetic waves in a non-linear diamagnetic medium) dy- namics, thermodynamics, fluid mechanics, statistical mechanics, chemical equi- librium, and atomic collision theory. Furthermore, in plasma one quite frequently encounters non-linear, collective effects, which cannot properly be analyzed with- out sophisticated mathematical techniques. A student typically has not completed the prequisites for a basic course in plasma physics until and unless s/he enters graduate school. For these lectures I have tried to put across the basic physical concepts, without bringing in the elegant formalism, which is irrelevant in this introduction. In giv- ing these lectures, I have assumed that the student has had course work at the level of Physics: Part I and Physics: Part II by Resnick and Halliday, so that s/he is acquainted with the basic concepts of mechanics, wave motion, thermodynam- ics, and electromagnetism. For students of science and engineering who had this sophomore-level physics, I hope that these lectures serve as a proper introduction to this stimulating, rapidly developing and perhaps crucially important field. I have used SI (MKS) units for electromagnetic fields, rather than the Gaussian system that has been more conventional in plasma physics, in order to make the subject more accessible to undergraduate science and engineering students. Porter Wear Johnson Chicago, IllinoisContents 1 Introduction to Plasmas 3 1.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Definition and Characterization of the Plasma State . . . . . . . . 5 2 Single Particle Dynamics 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Orbit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Drift of Guiding Center . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Adiabatic Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Magnetic Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Plasmas as Fluids 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Waves in Plasmas 35 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Electron Plasma Waves . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Ion Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Electromagnetic Waves in Plasma . . . . . . . . . . . . . . . . . 42 4.5 Alfvén Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Diffusion, Equilibrium, Stability 47 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Hydromagnetic Equilibrium . . . . . . . . . . . . . . . . . . . . 53 3Chapter 1 Introduction to Plasmas 1.1 Preliminary In the ancient phlogiston theory there was a classification of the states of matter: i. e., “earth”, “water”, “ air”, and “fire”. While the phlogiston theory had certain basic defects, it did properly enumerate the four states of matter – solid, liquid, gas, and plasma. It is estimated that more than 99% of the matter in the universe exists in a plasma state; however, the significance and character of the plasma state has been recognized only in the twentieth century. The states of solid, liquid, gas, and plasma represent correspondingly increasing freedom of particle motion. In a solid, the atoms are arranged in a periodic crystal lattice; they are not free to move, and the solid maintains its size and shape. In a liquid the atoms are free to move, but because of stong interatomic forces the volume of the liquid (but not its shape) remains unchanged. In a gas the atoms move freely, experiencing occasional collisions with one another. In a plasma, the atoms are ionized and there are free electrons moving about – a plasma is an ionized gas of free particles. Here are some familiar examples of plasmas: 1. Lightning, Aurora Borealis, and electrical sparks. All these examples show that when an electric current is passed through plasma, the plasma emits light (electromagnetic radiation). 2. Neon and fluorescent lights, etc. Electric discharge in plasma provides a rather efficient means of converting electrical energy into light. 3. Flame. The burning gas is weakly ionized. The characteristic yellow color of a wood flame is produced by 579nm transitions (D lines) of sodium ions. 34. Nebulae, interstellar gases, the solar wind, the earth’s ionosphere, the Van Allen belts. These provide examples of a diffuse, low temperature, ionized gas. 5. The sun and the stars. Controlled thermonuclear fusion in a hot, dense plasma provides us with energy (and entropy) on earth. Can we develop a practical scheme for tapping this virtually inexhaustible source of energy? The renaissance in interest and enthusiasm in plasma physics in recent years has occurred because, after decades of difficult developmental work, we are on the verge of doing “break - even” experiments on energy production by controlled fusion. We still have a long way to go from demonstrating scientific feasibility to developing practical power generating facilities. The current optimism among plasma physicists has come about because there does not seem to be any matter of principle that could (somewhat maliciously) prevent controlled thermonuclear fusion in laboratory plasmas. The development of this energy source will require 1 much insight and ingenuity – like the development of sources of electrical energy. Although plasma physics may well hold the key to virtually limitless sources of energy, there are a number of other applications or potential applications of plasma, such as the following: 1. Plasma may be kept in confinement and heated by magnetic and electric fields. These basic features of plasmas could be used to build “plasma guns” that eject ions at velocities up to 100 km/sec. Plasma guns could be used in ion rocket engines, as an example. 2. One could build “plasma motors”, which differ from ordinary motors by having plasma (not metals) as the basic conductor of electricity. These mo- tors could, in principle, be lighter and more efficient than ordinary motors. Similarly, one could develop “plasma generators” to convert mechanical energy into electical energy. The whole subject of direct magnetohydrody- namic production of electical energy is ripe for development. The current “thermodynamic energy converters” (steam plants, turbines, etc.) are noto- riously inefficient in generating electricity. 3. Plasma could be used as a resonator or a waveguide, much like hollow metallic cavities, for electromagnetic radiation. Plasma experiences a whole 1 Napoleon Bonaparte, who was no slouch as an engineer, felt that it would never be practical to transmit electric current over a distance as large as one kilometer.range of electrostatic and electromagnetic oscillations, which one would be able to put to good use. 4. Communication through and with plasma. The earth’s ionosphere reflects low frequency electromagnetic waves (below 1 MHz) and freely transmits high frequency (above 100 MHz) waves. Plasma disturbances (such as solar flares or communication blackouts during re-entry of satellites) are notori- ous for producing interruptions in communications. Plasma devices have es- sentially the same uses in communications, in principle, as semi-conductor devices. In fact, the electrons and holes in a semi-conductor constitute a plasma, in a very real sense. 1.2 Definition and Characterization of the Plasma State A plasma is a quasi-neutral “gas” of charged and neutral particles that exhibits collective behavior. We shall say more about collective behavior (shielding, os- cillations, etc.) presently. The table given below indicates typical values of the density and temperature for various types of plasmas. We see from the table that 16 4 the density of plasma varies by a factor of 10 and the temperature by 10 – this incredible variation is much greater than is possible for the solid, liquid, or gaseous states.Plasma Particle Temp. Debye Plasma Collision Type Density T (K) Length Freq. Freq. 3 /cm (cm) ω (Hz) ν (Hz) P 4 3 4 −4 Interstellar 1 10 10 6× 10 10 Gas 6 6 7 −1 Solar 10 10 1 6× 10 10 Corona 14 4 −4 11 9 Solar Atm. 10 10 10 6× 10 3× 10 Gas Discharge 12 6 −2 10 5 Diffuse Lab 10 10 10 6× 10 10 Plasma 14 6 −4 12 8 Dense Lab 10 10 10 6× 10 5× 10 Plasma 16 8 −3 12 5 Thermonuclear 10 10 10 6× 10 8× 10 Plasma Although plasma does occur under a variety of conditions, it seldom occurs too close to our environment because a gas in equilibrium at STP, say, has essentially no ions. The energy levels of a typical atom are shown. Ionization Energy Excited State V i Ground State Figure 1.1: Energy Levels of a Typical Atom The ionization potential, V , is the energy necessary to ionize an atom. Typ- i ically, V is 10 electron Volts or so. It is relatively improbable for atoms to be i ionized, unless they have a kinetic energy that is comparable in magnitude to this ionization potential. The average kinetic energy of atoms in a gas is of order k T , B where k is Boltzmann’s constant and T is the (Kelvin) temperature. At 300 K, B the factor k T is about 1/40 eV, whereas it is 1 eV at 11600 K. In plasma physics, B it is conventional to express the temperature by giving k T in electron Volts. Note: B −19 1eV = 1.6× 10 Joules.The equilibrium of the ionization reaction of an atom A, + − A→ A + e may be treated by the law of chemical equilibrium. Let n be the number of e electrons per unit volume, n be the number of ions per unit volume (n = n for a i e i neutral plasma), and n the number of neutral particles per unit volume. The ratio n n n i i = K(T) (1.1) n n gives the thermodynamic rate constant K(T). One may compute the quantity K by detailed quantum mechanical analysis. The result is a −V/(k T) i B K(T)= e (1.2) 3 λ where λ is the average DeBroglie wavelength of ions at temperature T , and a is an “orientation factor” which is of order 1. As a rough estimate, p M k T λ≈ h/ b for ions of mass M, where h is Planck’s constant. It is evident from the Table that plasmas exist either at low densities, or else at high temperatures. At low densities the left side of Eq. (1) is small, even if the plasma is fully ionized, and even if the temperature (and consequently the rate constant) is low. It takes a long time for such a plasma to come to equilibrium because the time between collisions is long. However, the probability of recombi- nation collisions is very small, and the diffuse gas remains ionized. By contrast, 6 at high temperatures (10 K or larger) the rate constant is large, and ionization occurs, even in a dense plasma. We shall discuss “quasi-neutral” plasmas, which have equal numbers of elec- trons and ions. The positive and negative particles move freely in a plasma. As a consequence, electric fields are neutralized in a plasma, just as they are in a met- alic conductor. Free charges are “shielded out” by the plasma, as shown in the diagram.− − − − − + Free Positive Charge − − − Battery + + + + + − Free Negative Charge + + + Figure 1.2: Shielding by Plasma However, in the region very close to the free charge, it is not shielded effec- tively and electric fields may exist. The characteristic distance for shielding in a plasma is the Debye length λ . D r k T ε B 0 λ = (1.3) D 2 ne with n charge carriers of charge e per unit volume. In practical units p λ = 740 k T/ncm D B 3 with k T in electron Volts and n as the number per cm . B I shall discuss shielding in a case that is simple to analyze: an infinite sheet of surface charge density σ is placed inside a plasma. We choose a Gaussian surface of cross-sectional area A and length x, as shown. + x + + + E + A - + + + + + Figure 1.3: Gaussian SurfaceWe apply Gauss’s law, I ε E· dS = q (1.4) 0 enc to this surface to obtain   Z x ε A(E (x)− E (0))= q = A σ+ (n − n )edx (1.5) 0 x x enc i e 0 Differentiating, we get dE x ε = e(n − n ) (1.6) 0 i e dx By symmetry, the electric field has only an x-component, which for electric po- tential V(x) is given by E =−dV/dx, so that x 2 d V e =− (n − n ) (1.7) i e 2 dx ε 0 In the plasma the more massive ions and neutral atoms are relatively immobile in comparison with the lighter electrons. The electrons populate the region of high potential according to the Boltzmann formula   eV(x) n (x)= n exp (1.8) e ∞ k T B where n = n is the density of electrons and ions far away. At low potential i ∞ (eV(x)≪ k T ) one has B      eV(x) en ∞ n − n = n 1− exp ≈− V(x) (1.9) i e ∞ k T k T B B Thus 2 2 d V e n 1 ∞ = V(x)= V(x) (1.10) 2 2 dx k T ε λ B 0 D The potential in this case is −x/λ D V(x)= E λe (1.11) 0 The electric field, E(x)= E exp−x/λ , is exponentially shielded by the plasma, 0 D with characteristic distance λ . D The Debye length λ is very small except for the most tenuous of plasmas. D We require that it be much smaller than the size of the plasma. We shall alsorequire that the potential energy of the electrons always be much smaller than k T , a typical electronic kinetic energy. This latter requirement is equivalent to B 3 the condition nλ ≫ 1. In other words, we require that many particles lie within D the shielding volume. Finally, I shall talk about plasma oscillations. We have just seen that static electric fields are shielded out in a plasma. If a positive charge is suddenly brought into a plasma, the negative charges nearby begin to move toward that charge. Be- cause of their inertia, they “overshoot”, creating a net negative charge imbalance. The electrons then move away from that region; and so forth. The characteristic frequency of these oscillations is the plasma frequency ω : P s 2 ne ω = (1.12) P mε 0 In practical units, √ 4 ω = 5× 10 nHz P I shall demonstrate this plasma oscillation by assuming that all of the electrons are somehow removed from an infinite slice of plasma of thickness x, and that twice as many electronc as usual exist in a nearby parallel slice of thickness x, as shown in the figure. + − - + − + − + - − electron electron + − E deficiency + − surplus - + − + − + - − + − x x Figure 1.4: Charged Slabs An electric field is set up between the two plates, as shown. The net (positive) charge per unit area in the electron-deficient slice is σ= nex. From Gauss’s law, nex E = (1.13) ε 0The electric field acts on the electrons in the intermediate region, causing them to move to the left, and causing the slice thickness to decrease. (The acceleration of each electron is x ¨, in fact.) Thus, we obtain from Newton’s second law that 2 ne mx ¨=−eE =− x (1.14) ε 0 Thus, x(t)= x cos(ω t+ δ), so that the charge imbalance oscillates with (angu- 0 P lar) frequency ω . P The particles in a plasma make collisions with one another – the more dense the plasma, the more frequent the collisions. Let ν be the collision frequency, and τ= 1/ν the time between collisions. For the plasma state we require ω τ≫ 1. As P a consequence, it is improbable for the electrons to collide during a single plasma oscillation. Thus, the plasma oscillations are not merely damped out because of interparticle collisions. The electrons undergo these oscillations without any real collisional damping, and such oscillations are characteristic of the plasma state. You will hear more about them. A magnetic field plays an absolutely basic role in plasma physics; it penetrates the plasma, even though the electric field does not. I shall briefly review the motion of a charged particle in a uniform B field. The particle experiences the Lorentz force, F = q v× B By Newton’s second law, a charged particle moves with constant speed in the direction parallel to B, and moves in a uniform circular orbit in the plane perpen- dicular to B. If v is the particle speed in the circle and r is the radius of the ⊥ ⊥ circular orbit, then 2 mv ⊥ qv B= force= mass× acceleration= (1.15) ⊥ r ⊥ The particle moves in a circular orbit with cyclotron frequency ω= v /r = qB/m ⊥ ⊥ The radius of the circular orbit, Larmor radius r = mv /(qB) ⊥ ⊥ depends on the initial speed of the particle in the plane perpendicular to B. The composite motion of the charged particle is a helical spiral motion along one of the lines of magnetic induction. Oppositely charged particles spiral around the field line in the opposite sense.Chapter 2 Single Particle Dynamics 2.1 Introduction This lecture deals exclusively with the motions of a single charged particle in external fields, which may be electrical, magnetic, or gravitational in character. The motions of charged particles in external fields are the full story in cyclotrons, synchrotrons, and linear accelerators – the mving charges themselves produce negligibly small fields. The charged particle beams in such devices aren’t plasmas, since collective effects don’t matter. In fact, nobody knows how to build high- current accelerators in which collective (space charge) effects are large – this is a very costly limitation of accelerator technology. In a plasma, collective effects occur and are likely to be important. For some purposes, we may treat plasma as a collection of individual particles, and sometimes we must handle the plasma as a continuous fluid. The Lorentz force, F = q v× B, causes a charged particle (charge q, mass m) to spiral around a line of force in a uniform magnetic field. In the plane perpendic- ular to the field, the Larmor radius r is determined by the initial transverse speed ⊥ v by the relation ⊥ √ mv 2mE⊥ ⊥ r = = (2.1) ⊥ qB qB 2 with the transverse kinetic energy E = mv /2. In practical units (B in Gauss, ⊥ ⊥ E in electron Volts), ⊥ p r = 3.37 E /Bcm ⊥ ⊥ 132.2 Orbit Theory In the branch of accelerator (plasma) physics known as orbit theory, one assumes that there is present a magnetic field B that is approximately uniform over a particle orbit, so that, to a first approximation, the charged particles spiral around field lines. The charged particles are, however, slightly affected by one or more of the following small perturbative contributions: • A small (time-independent) electric or gravitational field. • A slight non-uniformity in the magnetic field, with its magnitude (B) changing with position). • A slight curvature of the lines of force of B. 2.3 Drift of Guiding Center As a consequence lf any of these small sources of perturbation, the guiding cen- ters of the particle cyclotron orbits experience drifts. These guiding center drifts are not negligible. even though they come from small perturbations, because they often have a large cumulative effect over a long time. It is an excellent approx- imation to think in terms of guiding center drifts with a distance ℓ, over which non-uniformities or small perturbations are important, as very large in compari- son with the Larmor radius r ; (ℓ≫ r ). ⊥ ⊥ We need consider only forces in the plane perpendicular to the uniform B field – those parallel to B simply accelerate the particle along B, and do not affect the transverse motion. Also, note that in a pure magnetic field – be it non-uniform, curved, or whatever – the Lorentz force does no work on the particle: v· F = q v·( v× B)= 0 Consequently, the mechanical energy of the particle does not change. The simplest case to illustrate guiding center drift is that of a constant force F in plane perpendicular to a uniform magnetic induction B. Let the force act in the x-direction, with the B field in the z-direction. The mechanical energy of the 1 2 system, mv − F x, is a constant of the motion. As a consequence, the presence 2 ⊥ of the force F perturbs the circular Larmor orbit; there are greater speeds, and consequently larger radii of curvature (see Eq.(1)) for x 0 than for x 0, v Figure 2.1: Drift of Guiding Center As a consequence, for counterclockwise circulation there is an upward drift of the guiding center for a positive charge, as shown in Figure 2.1. We can analyze this motion quantitatively using Newton’s second law: d v m = F + q v× B (2.2) dt for the x- and y-components: dv x m = F+ qv B y dt dv y m = −qv B (2.3) x dt or   dv qB F x = v + y dt m qB   d F dv qB y v + = = − v (2.4) y x dt qB dt m If we define a velocity u = v + F/(qB)y ˆ, Eqn. (4) represents uniform cir- cular motion with angular velocity ω = qB/m for the velocity vector u. As a consequence, the composite motion is circular motion with angular velocity ω,with respect to a guiding center that is drifting downward with speed F/(qB). Consequently, the drift velocity of the guiding center is F 1 ˆ v =− j = F× B (2.5) D 2 qB qB If the force F is produced by a uniform electric field E , then F = qE and E× B v = (2.6) D 2 B Thus, in a uniform electric field, all charged particles experience an E × B drift, which is independent of the magnitude or sign of their charge, or their mass.  - E - ⊙ B Positive Negative Figure 2.3: Drift of Positive and Negative Charge We shall discuss several other circumstances in which there are guiding center drifts in response to small perturbations to a uniform B field. There is a systematic procedure for analyzing such perturbations, which consists of (i) computing the average residual force on a charged particle as it moves across a circular Larmor orbit, and (ii) using Eq.(5) to compute the drift speed. We illustrate this procedure in discussing Gradient B Drift. Let us assume that the B field has fixed direction, with the magnitude increas- ing as one moves transverse to B, to the left as shown in Figure 2.4. The Larmor radius, r = mv /(qB), varies inversely to B. Thus, the left half of the orbit ⊥ ⊥ should have a larger radius than the right half, as shown. Positive charges should drift upward and negative charges should drift downward, as shown in Figure 2.4. We shall compute the drift speed for positive charges. The magnetic induction is assumed to vary linearly with x:   dB B = B + x z ˆ (2.7) 0 dx⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ - Field Gradient Figure 2.4: Drift in Gradient B Field where B is the magnitude of B at x= 0, and(dB/dx) is its gradient at x= 0. We 0 0 neglect any quadratic variation in x, by assuming the Larmor radius is relatively small. The Lorentz force on a particle in a circular orbit is F = q v× B =−qvBr ˆ (2.8) The force is radially inward, as shown. v  S =  wS F Figure 2.5: Circular Orbit The Lorentz force may be written as dB F =−qvB r ˆ− qv xr ˆ (2.9) 0 dx the first term being responsible for the circular motion, and the second being the residual force. The average residual force over one cycle is   qv dB qv dB 2 ˆ ˆ ˆ F =− hx ii+hxyi j =− r i (2.10) residual ⊥ r dx 2 dx2 1 2 since by symmetryhxyi= 0 andhx i= r . We use (2.10) in Eq. (2.5) to obtain 2 ⊥ the drift speed qV dB ˆ v = j (2.11) D 2 dx In general qv ⊥ v =± (B× gradB) (2.12) D 2 2B with the± signs taken for positive (negative) particles. An additional source of charged particle drift is the curvature of magnetic field lines. We shall discuss the case in which the magnetic field is of constant mag- nitude, with the field lines curved. A charged particle spirals around a particular field line, as shown:  B R 0 Figure 2.6: Spiral around Field Line If the radius of curvature of the field line (approximately circular) is R , the 0 guiding center follows a spiral around the field line, as shown. The motion of the charged particle is most conveniently analyzed in a non-inertial coordinate system with the guiding center at the origin. In that system, the charged particle experiences an inertial centrifugal force mv k F = R (2.13) 0 2 R 0 with v the speed parallel to the field lines. Thus, the curvature drift velocity k comes out to be 2 mv 1 k v = F× B = R × B (2.14) D 0 2 2 qB qB Remark: It is impossible to draw field lines for a B-field that are curved, without havingB to change, as well. Consequently, field gradients are always present whenever the magnetic field lines are curved; we discuss them separately for the sake of simplicity. As an example, let us consider a simple toroidal field. If a total current is distributed uniformly and without “twist” on the surface of the torus, the magnetic induction has only a polar component μ I 0 B = (2.15) 0 2πR 0 where R is the radius of curvature of the field line, The field gradient is 0 μ I R 0 0 gradB=− 2 2π R 0 and the gradB drift is 2 v r R × B mv R × B ⊥ ⊥ 0 0 ⊥ V =± = (2.16) D 2 2 2 2qB R B R B 0 0 The overall drift for this toroidal field is  m R × B 0 2 2 v = V + v /2 (2.17) D ⊥ 2 2 q R B 0 2.4 Adiabatic Drift A particle in a uniform magnetic field spirals along a field line. If the field is slightly non-uniform, the particle either slowly drifts or spirals into regions of space that have different magnetic fields. Under such drifts, the radius of gyration of the orbit changes sufficiently slowly, and there are certain adiabatic invariants associated with the orbit, which do not change. One such adiabatic invariant is the angular momentum action. I L dθ= 2πL k k the integral of the component of angular momentum about B, taken over one gyration. Thus, the angular momentum component taken about the guiding center, and parallel to the field, does not change. 2 2 2 mv m m v 2 ⊥ ⊥ L = mv r = = mv = k ⊥ ⊥ ⊥ ω eB eB 2m E ⊥ = (2.18) e B2 The quantity E = mv /2, the transverse kinetic energy, is the contribution to ⊥ ⊥ the kinetic energy from transverse motion. From (2.18), we see that as the particle spirals or drifts into a region of increasing magnetic field, the transverse speed v ⊥ and energy E both increase. ⊥ As a particle of charge e moves with angular frequency ω in a circular orbit of radius r , one may define an average current I = e/T = eω/(2π), and a magnetic ⊥ moment eω 2 μ = current× area= πr ⊥ 2π 2 2 ev ev eω m E ⊥ 2 ⊥ ⊥ = r = = = (2.19) ⊥ 2 2ω 2 eB B Consequently, the magnetic moment of the “current loop” is also an adiabatic 2 invarian. The flux of magnetic field through the loop, πr B, is an equivalent ⊥ adiabatic invariant. The Larmor radius r does decrease when the magnetic field ⊥ increases, since mv 2 μ ⊥ r = = (2.20) ⊥ eB e v ⊥ B Strong Field Region Weak Field Region Figure 2.7: Spiral Curve withB Changing Slightly As a particle spirals from a weak field to a strong field region, its perpendicular speed v increases. However, since its total kinetic energy ⊥   m 2 2 E = v + v ⊥ k 2

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