hwk5-solutions

# hwk5-solutions - Physics 762 Homework#5 Spring ‘11 Dr...

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Unformatted text preview: Physics 762 Homework #5 Spring ‘11 Dr. Drake 1. Calculate the growth rate of the ﬁrehose instability using the fluid equations with a gyrotropic pressure tensor P :PbeJrPMI—bb) {1) Where I is the unit tensor and b : B / B. Assume that the initial magnetic ﬁeld BB is uniform and is in the z direction and that the wavevector k is also along in the z direction. Hint: you will find that you do not need to perturb the pressure. 2. Derive the kinetic dispersion relation for parallel propagating whistler/electron cy— clotron waves in a plasma with an initially uniform magnetic ﬁeld B0 in the z direction and with an initial isotropic Maxwellian distribution, 2 2 k c i g do eivg/vz w (2) 2 i -— ’ Lupe ﬁt); on ks Q where Q = eBo/mec is the electron cyclotron frequency and k is the wavevector in the z direction. The calculation is simpliﬁed if you assume that the wave is right—hand circularly polarized so that the perturbed electric ﬁeld can be written as E 2 as + my. (3) From this polarization it follows that the perturbed distribution is proportional to e”, Where 6 is the polar angle of the velocity vector perpendicular to the magnetic field. Also assume that aim >> fl, which allows you to discard the displacement current in Ampere’s law. #ﬁ‘waﬂaw 75C 9-755 gga Atﬁém‘s ﬁdcwéﬁ‘ﬁ ﬁg ﬂag/mam, MQ75“; Y4” “7%‘6‘ ﬁfe £2956: [pag'ﬂzé’z/n§/ In izjm/(éwwﬁ 3%": 3?ng ELéi’Qi—O) x -: Rt‘ﬁﬁig _ gQLk 3535‘ “EB EX ~LW€533JK§E§ +eki:§ 4— gift? ”K «= rob—é) R: ,33 , f” 133% “fﬁﬂ /[ﬂz:é£_»:’_ _( 'E'zzx/ ﬂééftwéﬂéé‘uﬁ ......... 5’3 6'17, a, M WWW? ~77~7ﬁ W 'R. ...._._..... . _ r , 7» dunk)“. -... ...._ ._ .. _ ﬁmw 1....“ K ‘ I 33 t” X: :2; Exvx+E my Lt“? (was. 94‘:an Emit-:19; 281-522; .2.. ._ W W m m u m w m. w w A A 9.74. ?: v ...
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