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2.15):J¼ðbvjj dsð2:76ÞaWe shall prove that J is invariant in a static, nonuniform B field; the result is alsotrue for a slowly time-varying B field.Before embarking on this somewhat lengthy proof, let us consider an example ofthe type of problem in which a theorem on the invariance of J would be useful. Aswe have already seen, the earth’s magnetic field mirror-traps charged particles,which slowly drift in longitude around the earth (Problem 2.8; see Fig. 2.16).
If themagnetic field were perfectly symmetric, the particle would eventually drift back tothe same line of force. However, the actual field is distorted by such effects as thesolar wind. In that case, will a particle ever come back to the same line of force?Since the particle’s energy is conserved and is equal to 12 mv2⊥ at the turning point,the invariance of μ indicates that jBj remains the same at the turning point.Fig. 2.15 A particlebouncing between turningpoints a and b in a magneticfieldFig. 2.16 Motion of a charged particle in the earth’s magnetic field2.8 Adiabatic Invariants45Fig. 2.17 Proof of theinvariance of JHowever, upon drifting back to the same longitude, a particle may find itself onanother line of force at a different altitude.
This cannot happen if J is conserved. Jdetermines the length of the line of force between turning points, and no two lineshave the same length between points with the same jBj. Consequently, the particlereturns to the same line of force even in a slightly asymmetric field.To prove the invariance of J, we first consider the invariance of vkδs, where δs isa segment of the path along B (Fig. 2.17). Because of guiding center drifts, aparticle on s will find itself on another line of force δs0 after a time Δt. The length ofδs0 is defined by passing planes perpendicular to B through the end points of δs. Thelength of δs is obviously proportional to the radius of curvature:0δs δs¼ 0Rc Rcso that00δs δs Rc Rc¼ΔtδsΔtRcð2:77ÞThe “radial” component of vgc is just0vgc Rc Rc Rc¼RcΔtð2:78ÞFrom Eqs. (2.24) and (2.26), we have12vgc ¼ v∇B þ vR ¼ v⊥ r L2B ∇B mvk Rc Bþq R2c B2B2ð2:79ÞThe last term has no component along Rc.
Using Eqs. (2.78) and (2.79), we canwrite Eq. (2.77) as1 dRc 1 m v2⊥Rcδs ¼ vgc 2 ¼ðB ∇BÞ 23δs dtRc 2 q BRcð2:80ÞThis is the rate of change of δs as seen by the particle. We must now get the rate ofchange of vk as seen by the particle. The parallel and perpendicular energies aredefined by462W1212Single-Particle Motions12ð2:81Þvjj ¼ ½ð2=mÞðW μBÞ1=2ð2:82Þmv2jj þ mv2⊥ ¼ mv2jj þ μB W jj þ W ⊥Thus vk can be writtenHere W and μ are constant, and only B varies. Therefore,_v_ jjμB_μB_11 μB¼¼ 2¼2 W μB2 W jjvjjmvkð2:83ÞSince B was assumed static, B˙ is not zero only because of the guiding centermotion:mv2jj Rc BdB drB_ ¼ ¼ vgc ∇B¼ ∇Bdr dtq R2c B2ð2:84Þ2v_ jjμ ðRc BÞ ∇B1 m v⊥ ðB ∇BÞ Rc¼¼222 q BvjjqRc BR2c B2ð2:85ÞNow we haveThe fractional change in vk δs is1 d1 dδs 1 dvjjvjj δs ¼þvjj δs dtδs dtvjj dtð2:86ÞFrom Eqs.
(2.80) and (2.85), we see that these two terms cancel, so thatvjj δs ¼ constantð2:87ÞThis is not exactly the same as saying that J is constant, however. In taking theintegral of vkδs between the turning points, it may be that the turning points on δs0do not coincide with the intersections of the perpendicular planes (Fig. 2.17).However, any error in J arising from such a discrepancy is negligible becausenear the turning points, vk is nearly zero. Consequently, we have provedJ¼ðbvjj ds ¼ constantð2:88ÞaAn example of the violation of J invariance is given by a plasma heating schemecalled transit-time magnetic pumping.
Suppose an oscillating current is applied tothe coils of a mirror system so that the mirrors alternately approach and withdraw2.8 Adiabatic Invariants47from each other near the bounce frequency. Those particles that have the rightbounce frequency will always see an approaching mirror and will therefore gain vk.J is not conserved in this case because the change of B occurs on a time scale notlong compared with the bounce time.2.8.3The Third Adiabatic Invariant, ФReferring again to Fig.
2.16, we see that the slow drift of a guiding center around theearth constitutes a third type of periodic motion. The adiabatic invariant connectedwith this turns out to be the total magnetic flux Ф enclosed by the drift surface. It isalmost obvious that, as B varies, the particle will stay on a surface such that the totalnumber of lines of force enclosed remains constant. This invariant, Ф, has fewapplications because most fluctuations of B occur on a time scale short comparedwith the drift period.
As an example of the violation of Ф invariance, we can citesome recent work on the excitation of hydromagnetic waves in the ionosphere.These waves have a long period comparable to the drift time of a particle around theearth. The particles can therefore encounter the wave in the same phase each timearound. If the phase is right, the wave can be excited by the conversion of particledrift energy to wave energy.Problems2.13. Derive the result of Problem 2.12b directly by using the invariance of J.ð(a) Let vk ds ’ vk L and differentiate with respect to time(b) From this, get an expression for T in terms of dL/dt. Set dL/dt ¼ 2vm toobtain the answer.2.14.
In plasma heating by adiabatic compression the invariance of μ requires thatKT⊥ increase as B increases. The magnetic field, however, cannot accelerateparticles because the Lorentz force qv B is always perpendicular to thevelocity. How do the particles gain energy?2.15. The polarization drift vp can also be derived from energy conservation. If E isoscillating, the E B drift also oscillates; and there is an energy 12mv2Eassociated with the guiding center motion.
Since energy can be gained froman E field only by motion along E, there must be a drift vp in the E direction.By equating the rate of change of 12 mv2E with the rate of energy gain fromvp · E, find the required value of vp.2.16. A hydrogen plasma is heated by applying a radiofrequency wave withE perpendicular to B and with an angular frequency ω ¼ 109 rad/s.
Theconfining magnetic field is 1 T. Is the motion of (a) the electrons and(b) the ions in response to this wave adiabatic?482Single-Particle Motions2.17. A 1-keV proton with vk ¼ 0 in a uniform magnetic field B ¼ 0.1 T is accelerated as B is slowly increased to 1 T. It then makes an elastic collision with aheavy particle and changes direction so that v⊥ ¼ vk The B-field is thenslowly decreased back to 0.1 T. What is the proton’s energy now?2.18. A collisionless hydrogen plasma is confined in a torus in which externalwindings provide a magnetic field B lying almost entirely in the ϕ direction(Fig. P2.18). The plasma is initially Maxwellian at KT ¼ 1 keV.
At t ¼ 0, B isgradually increased from 1 T to 3 T in 100 μs, and the plasma is compressed.(a) Show that the magnetic moment μ remains invariant for both ions andelectrons.(b) Calculate the temperatures T⊥ and Tk after compression.Fig. P2.182.19. A uniform plasma is created in a toroidal chamber with square cross section,as shown. The magnetic field is provided by a current I along the axis ofsymmetry. The dimensions are a ¼ 1 cm, R ¼ 10 cm. The plasma is Maxwellian at KT ¼ 100 eV and has density n ¼ 1019 m3. There is no electric field.Fig. P2.192.8 Adiabatic Invariants49(a) Draw typical orbits for ions and electrons with vk ¼ 0 drifting in thenonuniform B field.(b) Calculate the rate of charge accumulation (in coulombs per second) onthe entire top plate of the chamber due to the combined v∇B and vRdrifts.
The magnetic field at the center of the chamber is 1 T, and youmay make a large aspect ratio (R a) approximation where necessary.2.20. Suppose the magnetic field along the axis of a magnetic mirror is given byBz ¼ B0(l + α2z2).(a) If an electron at z ¼ 0 has a velocity given by v2 ¼ 3v2jj ¼ 1:5v2⊥ , at whatvalue of z is the electron reflected?(b) Write the equation of motion of the guiding center for the directionparallel to the field.(c) Show that the motion is sinusoidal, and calculate its frequency.(d) Calculate the longitudinal invariant J corresponding to this motion.2.21. An infinite straight wire carries a constant current I in the +z direction.At t ¼ 0, an electron of small gyroradius is at z ¼ 0 and r ¼ r0 withv⊥0 ¼ vk0· (⊥ and k refer to the direction relative to the magnetic field.)(a) Calculate the magnitude and direction of the resulting guiding centerdrift velocity.(b) Suppose that the current increases slowly in time in such a way that aconstant electric field in the z direction is induced.
Indicate on adiagram the relative directions of I, B, E, and vE.(c) Do v⊥ and vk increase, decrease, or remain the same as the currentincreases? Why?Chapter 3Plasmas as Fluids3.1IntroductionIn a plasma the situation is much more complicated than that in the last chapter; theE and B fields are not prescribed but are determined by the positions and motions ofthe charges themselves.
One must solve a self-consistent problem; that is, find a setof particle trajectories and field patterns such that the particles will generate thefields as they move along their orbits and the fields will cause the particles to movein those exact orbits. And this must be done in a time-varying situation. It soundsvery hard, but it is not.We have seen that a typical plasma density might be 1018 ion–electron pairs per3m . If each of these particles follows a complicated trajectory and it is necessary tofollow each of these, predicting the plasma’s behavior would be a hopeless task.Fortunately, this is not usually necessary because, surprisingly, the majority—perhaps as much as 80 %—of plasma phenomena observed in real experimentscan be explained by a rather crude model. This model is that used in fluidmechanics, in which the identity of the individual particle is neglected, and onlythe motion of fluid elements is taken into account.
Of course, in the case of plasmas,the fluid contains electrical charges. In an ordinary fluid, frequent collisionsbetween particles keep the particles in a fluid element moving together. It issurprising that such a model works for plasmas, which generally have infrequentcollisions. But we shall see that there is a reason for this.In the greater part of this book, we shall be concerned with what can be learnedfrom the fluid theory of plasmas.
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