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to be an equilibrium state as follows: me(ve \u2212 vi) \u3c4ei = \u2212eE. The current density j induced by the electric \ufb01eld becomes j = \u2212ene(ve \u2212 vi) = e 2ne\u3c4ei me E. The speci\ufb01c electric resistivity de\ufb01ned by \u3b7j = E is (ref.[2]) \u3b7 = me\u3bdei\u2016 nee2 = (me)1/2Ze2 ln\u39b 51.6\u3c01/2\ufffd20 (\u3baTe) \u22123/2 (2.36) = 5.2× 10\u22125Z ln\u39b ( \u3baTe e )\u22123/2 . (\u3a9m) The speci\ufb01c resistivity of a plasma with Te = 1keV, Z = 1 is \u3b7 = 3.3 × 10\u22128 \u3a9m and is slightly larger than the speci\ufb01c resistivity of copper at 20\u25e6C, 1.8×10\u22128 \u3a9m. When a current density of j is induced, the power \u3b7j2 per unit volume contributes to electron heating. This heating mechanism of electron is called Ohmic heating. 2.9 Variety of Time and Space Scales in Plasmas Various kinds of plasma characteristics have been described in this chapter. Characteristic time scales are a period of electron plasma frequency 2\u3c0/\u3a0e, an electron cyclotron period 2\u3c0/\u3a9e, an ion cyclotron period 2\u3c0/|\u3a9i|, electron to ion collision time \u3c4ei, ion to ion collision time \u3c4ii and electron-ion thermal energy relaxation time \u3c4 \ufffdei. Alfv´en velocity vA, which is a propagation velocity of magnetic perturbation, is v2A = B 2/(2\u3bc0\u3c1m) (\u3c1m is mass density)(see chs.5,10). Alfv´en transit time \u3c4H = L/vA is a typical magnetohydrodynamic time scale, where L is a plasma size. In a medium with the speci\ufb01c resistivity \u3b7, electric \ufb01eld di\ufb00uses with the time scale of \u3c4R = \u3bc0L2/\u3b7 (see ch.5). This time scale is called resistive di\ufb00usion time. Characteristic scales in length are Debye length\u3bbD, electron Larmor radius \u3c1\u3a9e, ion Larmor radius \u3c1\u3a9i, electron-ion collision mean free path\u3bbei and a plasma sizeL. The relations between space and time scales are \u3bbD\u3a0e = vTe, \u3c1\u3a9e\u3a9e = vTe, \u3c1\u3a9i|\u3a9i| = vTi, \u3bbei/\u3c4ei \ufffd 31/2vTe, \u3bbii/\u3c4ii \ufffd 31/2vTi, L/\u3c4H = vA, where vTe, vTi are the thermal velocities v2Te = \u3baTe/me, v 2 Ti = \u3baTi/mi. The drift velocity of guiding center is vdrift \u223c \u3baT/eBL = vT(\u3c1\u3a9/L). Parameters of a typical D-T fusion plasma with ne = 1020 m\u22123, \u3baTe = \u3baTi = 10keV, B = 5T, L = 1m are followings: 2\u3c0/\u3a0e = 11.1ps (\u3a0e/2\u3c0 = 89.8GHz) \u3bbD = 74.5\u3bcm 2\u3c0/\u3a9e = 7.1ps (\u3a9e/2\u3c0 = 140GHz) \u3c1\u3a9e = 47.6\u3bcm 2\u3c0/|\u3a9i| = 26ns (|\u3a9i|/2\u3c0 = 38MHz) \u3c1\u3a9i = 2.88mm \u3c4ei = 0.34ms \u3bbei = 25km \u3c4ii = 5.6ms \u3bbii = 9.5km \u3c4 \ufffdei = 0.3 s 19 20 2 Plasma Characteristics \u3c4H = 0.13\u3bcs \u3c4R = 1.2 × 103 s. The ranges of scales in time and space extend to \u3c4R\u3a0e \u223c 1014, \u3bbei/\u3bbD \u223c 1.6 × 108 and the wide range of scales suggests the variety and complexity of plasma phenomena. References [1] D. V. Sivukhin: Reviews of Plasma Physics 4 p.93 (ed. by M. A. Leontovich) Consultant Bureau, New York 1966. [2] L. Spitzer, Jr.: Physics of Fully Ionized Gases, Interscience, New York 1962. [3] T. H. Stix: Plasma Phys. 14, 367 (1972). 20 21 Ch.3 Magnetic Con\ufb01guration and Particle Orbit In this chapter, the motion of individual charged particles in a more general magnetic \ufb01elds is studied in detail. There are a large number of charged particles in a plasma, thus movements do a\ufb00ect the magnetic \ufb01eld. But this e\ufb00ect is neglected here. 3.1 Maxwell Equations Let us denote the electric intensity, the magnetic induction, the electric displacement and the magnetic intensity by E, B, D, and H , respectively. When the charge density and current density are denoted by \u3c1, and j, respectively, Maxwell equations are \u2207×E + \u2202B \u2202t = 0, (3.1) \u2207×H \u2212 \u2202D \u2202t = j, (3.2) \u2207 ·B = 0, (3.3) \u2207 ·D = \u3c1. (3.4) \u3c1 and j satisfy the relation \u2207 · j + \u2202\u3c1 \u2202t = 0. (3.5) Eq.(3.2),(3.4) and (3.5) are consistent with each other due to the Maxwell displacement current \u2202D/\u2202t. From (3.3) the vector B can be expressed by the rotation of the vector A: B = \u2207×A. (3.6) A is called vector potential. If (3.6) is substituted into (3.1), we obtain \u2207× ( E + \u2202A \u2202t ) = 0. (3.7) The quantity in parenthesis can be expressed by a scalar potential \u3c6 and E = \u2212\u2207\u3c6\u2212 \u2202A \u2202t . (3.8) Since any other set of \u3c6\u2032 and A\u2032, A\u2032 = A\u2212\u2207\u3c8, (3.9) \u3c6\u2032 = \u3c6+ \u2202\u3c8 \u2202t (3.10) can also satisfy (3.6),(3.8) with an arbitrary \u3c8, \u3c6\u2032 and A\u2032 are not uniquely determined. When the medium is uniform and isotropic, B and D are expressed by D = \ufffdE, B = \u3bcH . 21 22 3 Magnetic Con\ufb01guration and Particle Orbit \ufffd and \u3bc are called dielectric constant and permeability respectively. The value of \ufffd0 and \u3bc0 in vacuum are \ufffd0 = 107 4\u3c0c2 C2 · s2/kg ·m3 = 8.854× 10\u221212 F/m \u3bc0 = 4\u3c0 × 10\u22127 kg ·m/C2 = 1.257× 10\u22126 H/m 1 \ufffd0\u3bc0 = c2 where c is the light speed in vacuum (C is Coulomb). Plasmas in magnetic \ufb01eld are anisotropic and \ufffd and \u3bc are generally in tensor form. In vacuum, (3.2) and (3.3) are reduced to \u2207×\u2207×A + 1 c2 \u2207\u2202\u3c6 \u2202t + 1 c2 \u22022A \u2202t2 = \u3bc0j, (3.11) \u22072\u3c6+\u2207\u2202A \u2202t = \u2212 1 \ufffd0 \u3c1. (3.12) As \u3c6 and A have arbitrariness of \u3c8 as shown in (3.9),(3.10), we impose the supplementary condition (Lorentz condition) \u2207 ·A+ 1 c \u2202\u3c6 \u2202t = 0. (3.13) Then (3.11),(3.12) are reduced to the wave equations \u22072\u3c6\u2212 1 c2 \u22022\u3c6 \u2202t2 = \u2212 1 \u3b50 \u3c1, (3.14) \u22072A\u2212 1 c2 \u22022A \u2202t2 = \u2212\u3bc0j. (3.15) In derivation of (3.15), a vector relation \u2207× (\u2207× a)\u2212\u2207(\u2207 · a) = \u2212\u22072a is used, which is valid only in (x, y, z) coordinates. The propagation velocity of electromagnetic \ufb01eld is 1/(\u3bc0\ufffd0)1/2 = c in vacuum. When the \ufb01elds do not change in time, the \ufb01eld equations reduce to E = \u2212\u2207\u3c6, B = \u2207×A, \u22072\u3c6 = \u2212 1 \u3b50 \u3c1, \u22072A = \u2212\u3bcj, \u2207 ·A = 0, \u2207 · j = 0. The scalar and vector potentials \u3c6 and A at an observation point P (given by the position vector r) are expressed in terms of the charge and current densities at the point Q (given by r\u2032) by (see \ufb01g.3.1) \u3c6(r) = 1 4\u3c0\ufffd0 \u222b \u3c1(r\u2032) R dr\u2032, (3.16) A(r) = \u3bc0 4\u3c0 \u222b j(r\u2032) R dr\u2032 (3.17) 22 3.2 Magnetic Surface 23 Fig.3.1 Observation point P and the location Q of charge or current. Fig.3.2 Magnetic surface \u3c8 = const., the normal \u2207\u3c8 and line of magnetic force. where R \u2261 r \u2212 r\u2032, R = |R| and dr\u2032 \u2261 dx\u2032d\u2032dz\u2032. Accordingly E and B are expressed by E = 1 4\u3c0\ufffd0 \u222b R R3 \u3c1dr\u2032, (3.18) B = \u3bc0 4\u3c0 \u222b j ×R R3 dr\u2032. (3.19) When the current distribution is given by a current I \ufb02owing in closed loops C, magnetic intensity is described by Biot-Savart equation H = B \u3bc0 = I 4\u3c0 \u222e c s× n R2 ds (3.20) where s and n are the unit vectors in the directions of ds and R, respectively. 3.2 Magnetic Surface A magnetic line of force satis\ufb01es the equations dx Bx = dy By = dz Bz = dl B (3.21) where l is the length along a magnetic line of force (dl)2 = (dx)2 + (dy)2 + (dz)2. The magnetic surface \u3c8(r) = const. is such that all magnetic lines of force lie upon on that surface which satis\ufb01es the condition (\u2207\u3c8(r)) ·B = 0. (3.22) The vector \u2207\u3c8(r) is normal to the magnetic surface and must be orthogonal to B (see \ufb01g.3.2). In terms of cylindrical coordinates (r, \u3b8, z) the magnetic \ufb01eld B is given by Br = 1 r \u2202Az \u2202\u3b8 \u2212 \u2202A\u3b8 \u2202z , B\u3b8 = \u2202Ar \u2202z \u2212 \u2202Az \u2202r , Bz = 1 r \u2202 \u2202r (rA\u3b8)\u2212 1 r \u2202Ar \u2202\u3b8 . (3.23) In the case of axi-symmetric con\ufb01guration (\u2202/\u2202\u3b8 = 0), \u3c8(r, z) = rA\u3b8(r, z) (3.24) 23 24 3 Magnetic Con\ufb01guration and Particle Orbit satis\ufb01es the condition (3.22) of magnetic surface; Br\u2202(rA\u3b8)/\u2202r+B\u3b8 · 0 +Bz\u2202(rA\u3b8)/\u2202z = 0. The magnetic surface in the case of translational symmetry (\u2202/\u2202z = 0) is given by \u3c8(r, \u3b8) = Az(r, \u3b8) (3.25) and the magnetic surface in the case of helical symmetry, in which \u3c8 is the function of r and \u3b8\u2212\u3b1z only, is given by \u3c8(r, \u3b8 \u2212 \u3b1z) = Az(r, \u3b8 \u2212 \u3b1z) + \u3b1rA\u3b8(r, \u3b8 \u2212 \u3b1z) (3.26) where \u3b1 is helical pitch parameter. 3.3 Equation of Motion of a Charged Particle The equation of motion of a particle with the mass m and the charge q in an electromagnetic \ufb01eld E, B is m d2r dt2 = F = q ( E + dr dt ×B ) . (3.27) Since Lorentz force of the second term in the right-hand side of (3.27) is orthogonal to the velocity v, the