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The RF is applied to anantenna, shown here as one side of a Boswell antenna, which is a version of thebidirectional Nagoya Type III antenna. Helical antennas can couple more10.6Spacecraft Propulsion409Fig. 10.78 Schematic of a helicon thruster [Drawn after C. Charles, J. Phys. D: Appl. Phys.42 (2009) 163001]efficiently to the dominant m ¼ +1 azimuthal mode of the helicon wave, and ringantennas are used for the m ¼ 0 mode. As the plasma leaves the uniform-fieldvolume, the field lines diverge as the B-field expands. The electrons follow thefield lines, and hence their density decreases.
Since the electrons are essentiallyMaxwellian, the plasma potential must decrease with the density according toEq. (3.73). There is therefore an electric field along the field lines shown inFig. 10.78. This E-field accelerates ions; and, if Ti 0, the ion velocity willeventually reach the acoustic velocity cs. At that point, a sheath will form,according to Fig. 8.4, whether or not there is a wall there. This ion-rich sheath in“mid-air” will attract electrons which surround it, forming a double layer which, in1D, shields the E-field from the rest of the plasma.It is possible to calculate where this double layer will form. The ion energy at thesheath edge is ½Mc2s ¼ ½KT e . To accelerate ions to this energy, the potential atthe sheath edge, Vs, must be at least ½KTe/e if Vs 0 in the main plasma.
Hence,the quasineutral density at the sheath edge, ns, is given by Eq. (3.73) asns ¼ n0 expð½Þ:ð10:29ÞSince magnetic flux is conserved in the expansion from r0 to r for each field line,and since electrons are constrained to follow the field lines, the field and densityvary with r as41010r 2Bn0:¼ ¼B0 n0rPlasma Applicationsð10:30ÞThe radius at which a sheath forms is thenrs¼r0 1=2n0¼ e1=4 ¼ 1:28:nsð10:31ÞThus, an ion sheath will form at a position where the field lines have increased theirdistances from the axis by 28%. Ions passing through the potential drop of thissheath are suddenly accelerated to a velocity vex of Eq. (10.22).Helicon thrusters have been tested in space but have not been fully engineered.Nonetheless, helicon discharges are part of the large thruster VASIMR (VariableSpecific Impulse Magnetoplasma Rocket) being built by former astronaut FranklinChang Dı́az for travel to Mars. A diagram of this device is shown in Fig.
10.79.Though helicons are used for ionization, the main power is provided by ICRH (seesection “Heating and Current Drive”).There will not be a more glamorous application of helicons!Fig. 10.79 The VASIMR rocket [Google images]10.710.7Plasmas in Everyday Life411Plasmas in Everyday LifePlasma physics may be an obscure and difficult science, but it is important becausewe see plasmas all the time.
Every time we turn on the television, there are plasmasinside every pixel. Even before that, when we turn on the fluorescent lights, we areusing a plasma. Older computer screens are also lit by fluorescents. The light of dayis caused by the plasma in the sun’s photosphere. Moonlight is a reflection of that.At night, the stars and nebulas can be seen by their plasma light. Without plasma, itwould be very dark.
Auroras are plasmas generated by the solar wind. When we geta shock when touching a doorknob after walking across a rug in winter, the spark isa plasma. Lightning is a larger form of that, coming from clouds. Ball lightning,though, is a slow-moving ball of plasma that no one understands.We conclude with the topic that engendered plasma physics: nuclear fusion,which the general public has never heard of. By about 2050, there will be very littlefossil fuel left; maybe dirty coal, but certainly not oil. Wind, solar, and hydro powerare insufficient for the world’s growing energy needs. Nuclear (fission) energy hasits well known problems. To survive at least a few more centuries, mankind has todevelop fusion power. This will require a workforce trained in plasma physics.
Thecontrol of global warming starts in the classroom.Errata to: Introduction to Plasma Physicsand Controlled FusionFrancis F. ChenErratum to:F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,© Springer International Publishing Switzerland 2016https://doi.org/10.1007/978-3-319-22309-4The book was inadvertently published with the following errors and the same havebeen updated after publication.1. The font of the symbol “nu” was changed to differentiate it from the symbol“vee”.2.
Also, there were typos in vector and scalar notes throughout the book.The updated online versions of these chapters can be found athttps://doi.org/10.1007/978-3-319-22309-4_2https://doi.org/10.1007/978-3-319-22309-4_3https://doi.org/10.1007/978-3-319-22309-4_4https://doi.org/10.1007/978-3-319-22309-4_5https://doi.org/10.1007/978-3-319-22309-4_6https://doi.org/10.1007/978-3-319-22309-4_8https://doi.org/10.1007/978-3-319-22309-4_10The updated online versions of this book can be found athttps://doi.org/10.1007/978-3-319-22309-4© Springer International Publishing Switzerland 2018F.
Chen, Introduction to Plasma Physics and Controlled Fusion,https://doi.org/10.1007/978-3-319-22309-4_11E1Appendix A: Units, Constants and Formulas,Vector RelationsUnitsThe formulas in this book are written in the mks units of the International System(SI). In much of the research literature, however, the cgs-Gaussian system is stillused. The following table compares the vacuum Maxwell equations, the fluidequation of motion, and the idealized Ohm’s law in the two systems:mks-SI∇ D = e(ni ne)∇ E ¼ B_∇B=0∇ H ¼ j þ D_D = E0E B = μ0Hmn dvdt ¼ qnðE þ v BÞ ∇ pE+vB=0cgs-Gaussian∇ E = 4πe(ni ne)c∇ E ¼ B_∇B=0c∇ B ¼ 4π j þ E_E=μ=11mn dvdt ¼ qn E þ c v B ∇ pE þ 1c v B ¼ 0The equation of continuity is the same in both systems.In the Gaussian system, all electrical quantities are in electrostatic units (esu)except B, which is in gauss (emu); the factors of c are written explicitly toaccommodate this exception.
In the mks system, B is measured in tesla (Wb/m2),each of which is worth 104 gauss. Electric fields E are in esu/cm in cgs and V/m inmks. Since one esu of potential is 300 V, one esu/cm is the same as 3 104 V/m.The ratio of E to B is dimensionless in the Gaussian system, so that vE = cE/B. In themks system, E/B has the dimensions of a velocity, so that vE = E/B.
This fact isuseful to keep in mind when checking the dimensions of various terms in anequation in looking for algebraic errors.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 10.1007/978-3-319-22309-4413414Appendix A: Units, Constants and Formulas, Vector RelationsThe current density j = nev has the same form in both systems. In cgs, n and vare in cm3 and cm/s, and e has the value e = 4.8 1010 esu; then j comes out inesu/cm2, where 1 esu of current equals c1 emu or 10/c = 1/(3 109) A.
In mks, n andv are in m3 and m/s, and e has the value e = 1.6 1019 C; then j comes out in A/m2.Most cgs formulas can be converted to mks by replacing B/c by B and 4π by E10 ,where 1/4πE0 = 9 109. For instance, electric field energy density is E2/8π in cgsand E0E2/2 in mks, and magnetic field energy density is B2/8π in cgs and B2/2 μ0 inmks. Here we have used the fact that (E0μ0)1/2 = c = 3 108 m/s.The energy KT is usually given in electron volts. In cgs, one must convert TeV toergs by multiplying by 1.6 1012 erg/eV. In mks, one converts TeV to joules bymultiplying by 1.6 1019 J/eV. This last number is, of course, just the charge e inmks, since that is how the electron volt is defined.Useful Constants and FormulasConstantscVelocity of lighteElectron chargemElectron massMProton massM/m(M/m)1/2KBoltzmann’s constanteVElectron volt1 eVOf temperature KTE0Permittivity of free spaceμ0Permeability of free spaceCross section of H atomπa20Density of neutral atoms at room temperature and1 mTorr pressuremks3 108 m/s1.6 1019 C0.91 1030 kg1.67 1027 kg1837431.38 1023 J/K1.6 1019 J11,600 K8.854 1012 F/m4π 107 H/m0.88 1020 m2193.3 10m3cgs3 1010 cm/s4.8 1010 esu0.91 1027 g1.67 1024 g1837431.38 1016 erg/K1.6 1012 erg11,600 K0.88 1016 cm23.3 1013 cm3Formulas (H) for hydrogenωpPlasma frequencyωcElectron cyclotronfrequencyλDDebye lengthrLLarmor radiusmks 2 1=2neE0 mcgs-Gaussian1=24πne2meBmE0 KT e 1=2ne2mv⊥eBeBmc KT e 1=24πne2mv⊥ ceBHandy formula(n in cm3)pffiffiffif p ¼ 9000 n s1fc = 2.8 GHz/kG740(TeV/n)1/2 cm1=21:4 T evmmðHÞBkG(continued)Appendix A: Units, Constants and Formulas, Vector Relations415Formulas (H) for hydrogenvAAlfvén speedmksBcgs-GaussianBð4πρÞ1=2KT e 1=2McEBvsAcoustic speed (Ti = 0)ðμ0 ρÞ1=2KT e 1=2MvEE B drift speedEBvDDiamagnetic drift speedβMagnetic/plasmapressureKT neB nnKTB2 =2μ0vtheElectron thermal speedνeiElectron–ion collisionfrequencyνeeνiiλeivosc02KT emPeak electron quivervelocityn1=2106 T eV cms ðHÞÞ cm108 EðBV=cmðGÞs0108TBeV R1 cmscKT neB nnKTB2 =8π1=22KT emElectron–electron collision frequencyIon–ion collisionfrequencyCollision mean free pathHandy formula(n in cm3)2:2 1011 pBffiffi cms ðHÞ1=21=25:9 107 T eV cmsωpNDln Λ 1’ 2 106 Zne3=2sT eV’5Z4eE0mω0 m 1=2 T e 3=2MTi106 n ln3=2Λ s1T eVνeeT2 λee λii’ 3:4 1013 n lneVΛ cmðHÞeE0mω0v2osc¼ 7:3I19 λ2μc22I 13 λμv2osc¼ 3:7v2eT eVUseful Vector RelationsA ðB CÞ ¼ B ðC AÞ ¼ C ðA BÞ ðABCÞA ðB CÞ ¼ BðA CÞ CðA BÞA B ðC DÞ ¼ ðA CÞðB DÞ ðA DÞ B CA B ðC DÞ ¼ ABD C ðABCÞD ¼ ðACDÞB ðBCDÞA∇ ðϕAÞ ¼ A ∇ϕ þ ϕ∇ A∇ ðϕAÞ ¼ ∇ϕ A þ ϕ∇ A416Appendix A: Units, Constants and Formulas, Vector Relations A ð∇ BÞ ¼ ∇ A B A ∇ B ðB ∇ ÞA B ð∇ AÞ1ðA ∇ ÞA ¼ ∇ A2 A ð∇ AÞ2∇ ðA BÞ ¼ B ð∇ AÞ A ð∇ BÞ∇ ðA BÞ ¼ A ∇ B B∇ A þ B ∇ A ðA ∇ ÞB∇ ½ðA ∇ ÞA ¼ ðA ∇ Þð∇ AÞ þ ð∇ AÞð∇ AÞ ½ð∇ AÞ ∇ A ∇∇A¼∇ ∇A ∇∇ A∇ ∇ϕ ¼ 0∇ ð∇ A Þ ¼ 0Cylindrical Coordinates (r, θ, z)221 ∂∂ϕ1 ∂ ϕ ∂ ϕrþþ 2∇ ϕ¼r ∂r∂rr ∂θ2 ∂z221 ∂1 ∂∂A θ þ AzðrAr Þ þr ∂rr ∂θ∂z1 ∂Az ∂Aθ∂Ar ∂Az ^1∂1 ∂Ar^r þ^z∇A¼ðrAθ Þ θþr ∂θr ∂rr ∂θ∂z∂z∂r1∂Aθ22^r∇ A ¼ ð∇ ∇ ÞA ¼ ∇ Ar 2 Ar þ 2r∂θ1∂Ar ^þ ∇ 2 A θ 2 Aθ þ 2θ þ ∇ 2 Az^zr∂θ∂Br1 ∂Br∂Br 1þ Aθþ Az Aθ B θðA ∇ ÞB ¼ ^r Arr ∂θr∂r∂z∂Bθ1 ∂Bθ∂Bθ 1þ Aθþ Az Aθ B rþ^θ Arr ∂θr∂r∂z∂Bz1 ∂Bz∂Bzþ ^z Arþ Aθþ Azr ∂θ∂r∂z∇A¼Appendix B: Theory of Waves in a ColdUniform PlasmaAs long as Te = Ti = 0, the waves described in Chap.
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