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A more refined treatment—the kinetic theory ofplasmas—requires more mathematical calculation than is appropriate for an introductory course. An introduction to kinetic theory is given in Chap. 7.The original version of this chapter was revised. An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-22309-4_11© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_351523 Plasmas as FluidsIn some plasma problems, neither fluid theory nor kinetic theory is sufficient todescribe the plasma’s behavior. Then one has to fall back on the tedious process offollowing the individual trajectories.
Modern computers can do this, although theyhave only enough memory to store the position and velocity components for about 106particles if all three dimensions are involved. Nonetheless, computer simulation playsan important role in filling the gap between theory and experiment in those instanceswhere even kinetic theory cannot come close to explaining what is observed.3.23.2.1Relation of Plasma Physics to OrdinaryElectromagneticsMaxwell’s EquationsIn vacuum:ε0 ∇ E ¼ σð3:1ÞB_ð3:2Þ∇E¼∇B¼0ð3:3Þ∇ B ¼ μ0 j þ ε0 E_ð3:4ÞIn a medium:∇D¼σ∇E¼ð3:5ÞB_ð3:6Þ∇B¼0ð3:7Þ∇ H ¼ j þ D_ð3:8ÞD ¼ εEð3:9ÞB ¼ μHð3:10ÞIn Eqs.
(3.5) and (3.8), σ and j stand for the “free” charge and current densities. The“bound” charge and current densities arising from polarization and magnetizationof the medium are included in the definition of the quantities D and H in terms of εand μ. In a plasma, the ions and electrons comprising the plasma are the equivalentof the “bound” charges and currents. Since these charges move in a complicated way,it is impractical to try to lump their effects into two constants ε and μ. Consequently,in plasma physics, one generally works with the vacuum equations (3.1)–(3.4), inwhich σ and j include all the charges and currents, both external and internal.3.2 Relation of Plasma Physics to Ordinary Electromagnetics53Note that we have used E and B in the vacuum equations rather than theircounterparts D and H, which are related by the constants ε0 and μ0.
This is becausethe forces qE and j B depend on E and B rather than D and H, and it is notnecessary to introduce the latter quantities as long as one is dealing with the vacuumequations.3.2.2Classical Treatment of Magnetic MaterialsSince each gyrating particle has a magnetic moment, it would seem that the logicalthing to do would be to consider a plasma as a magnetic material with a permeability μm. (We have put a subscript m on the permeability to distinguish it from theadiabatic invariant μ.) To see why this is not done in practice, let us review the waymagnetic materials are usually treated.The ferromagnetic domains, say, of a piece of iron have magnetic moments μi,giving rise to a bulk magnetizationM¼1XμV i ið3:11Þper unit volume.
This has the same effect as a bound current density equal tojb ¼ ∇ Mð3:12ÞIn the vacuum equation (3.4), we must include in j both this current and the “free,”or externally applied, current jf:μ0 1 ∇ B ¼ j f þ jb þ ε0 E_ð3:13ÞWe wish to write Eq. (3.13) in the simple form∇ H ¼ j f þ ε0 E_ð3:14Þby including jb in the definition of H. This can be done if we letH ¼ μ0 1 BMð3:15ÞTo get a simple relation between B and H, we assume M to be proportional to B orH:M ¼ χm Hð3:16Þ543 Plasmas as FluidsThe constant χm is the magnetic susceptibility. We now haveB ¼ μ0 ð1 þ χm ÞH μm Hð3:17ÞThis simple relation between B and H is possible because of the linear form ofEq.
(3.16).In a plasma with a magnetic field, each particle has a magnetic moment μα, andthe quantity M is the sum of all these μα’s in 1 m3. But we now haveμα ¼mv2⊥α 1/B2BM/1BThe relation between M and H (or B) is no longer linear, and we cannot writeB ¼ μmH with μm constant. It is therefore not useful to consider a plasma as amagnetic medium.3.2.3Classical Treatment of DielectricsThe polarization P per unit volume is the sum over all the individual moments pi ofthe electric dipoles. This gives rise to a bound charge densityσb ¼∇Pð3:18ÞIn the vacuum equation (3.1), we must include both the bound charge and the freecharge:ε0 ∇ E ¼ σ f þ σ bð3:19ÞWe wish to write this in the simple form∇ D ¼ σfð3:20Þby including σ b in the definition of D.
This can be done by lettingD ¼ ε0 E þ P ε Eð3:21ÞIf P is linearly proportional to E,P ¼ ε0 χ e Eð3:22Þε ¼ ð1 þ χ e Þε0ð3:23Þthen ε is a constant given by3.2 Relation of Plasma Physics to Ordinary Electromagnetics55There is no a priori reason why a relation like Eq. (3.22) cannot be valid in a plasma,so we may proceed to try to get an expression for ε in a plasma.3.2.4The Dielectric Constant of a PlasmaWe have seen in Sect. 2.5 that a fluctuating E field gives rise to a polarizationcurrent jp.
This leads, in turn, to a polarization charge given by the equation ofcontinuity:∂σ pþ ∇ jp ¼ 0∂tð3:24ÞThis is the equivalent of Eq. (3.18), except that, as we noted before, a polarizationeffect does not arise in a plasma unless the electric field is time varying. Since wehave an explicit expression for jp but not for σ p, it is easier to work with the fourthMaxwell equation, Eq. (3.4):∇ B ¼ μ0 j f þ j p þ ε0 E_ð3:25ÞWe wish to write this in the form∇ B ¼ μ0 j f þ εE_ð3:26ÞThis can be done if we letε ¼ ε0 þjpE_ð3:27ÞFrom Eq. (2.67) for jp, we haveε ¼ ε0 þρB2orεR εμ ρc2¼1þ 0 2ε0Bð3:28ÞThis is the low-frequency plasma dielectric constant for transverse motions.
Thequalifications are necessary because our expression for jp is valid only for ω2 ω2cand for E perpendicular to B. The general expression for ε, of course, is verycomplicated and hardly fits on one page.Note that as ρ ! 0, εR approaches its vacuum value, unity, as it should. AsB ! 1, εR also approaches unity. This is because the polarization drift vp thenvanishes, and the particles do not move in response to the transverse electric field.In a usual laboratory plasma, the second term in Eq. (3.28) is large compared withunity. For instance, if n ¼ 1016 m 3 and B ¼ 0.1 T we have (for hydrogen)563 Plasmas as Fluids4π 10μ0 ρc2¼2B71016 1:67 1027ð0:1Þ29 1016¼ 189This means that the electric fields due to the particles in the plasma greatly alter thefields applied externally.
A plasma with large ε shields out alternating fields, just asa plasma with small λD shields out dc fields.Problems3.1 Derive the uniform-plasma low-frequency dielectric constant, Eq. (3.28), byreconciling the time derivative of the equation ∇ D ¼ ∇ ðεEÞ ¼ 0 withthat of the vacuum Poisson equation (3.1), with the help of equations (3.24) and(2.67).3.2 If the ion cyclotron frequency is denoted by Ωc and the ion plasma frequency isdefined byΩ p ¼ ne2 =ε0 M1=2where M is the ion mass, under what circumstances is the dielectric constant εapproximately equal to Ω2p /Ω2c ?3.3The Fluid Equation of MotionMaxwell’s equations tell us what E and B are for a given state of the plasma.
Tosolve the self-consistent problem, we must also have an equation giving theplasma’s response to given E and B. In the fluid approximation, we consider theplasma to be composed of two or more interpenetrating fluids, one for each species.In the simplest case, when there is only one species of ion, we shall need twoequations of motion, one for the positively charged ion fluid and one for thenegatively charged electron fluid. In a partially ionized gas, we shall also need anequation for the fluid of neutral atoms.
The neutral fluid will interact with the ionsand electrons only through collisions. The ion and electron fluids will interact witheach other even in the absence of collisions, because of the E and B fields theygenerate.3.3.1The Convective DerivativeThe equation of motion for a single particle ismdv¼ qðE þ v BÞdtð3:29Þ3.3 The Fluid Equation of Motion57Assume first that there are no collisions and no thermal motions. Then all theparticles in a fluid element move together, and the average velocity u ofthe particles in the element is the same as the individual particle velocity v.The fluid equation is obtained simply by multiplying Eq.
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