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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 9755 gga Atﬁém‘s ﬁdcwéﬁ‘ﬁ ﬁg ﬂag/mam, MQ75“; Y4”
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 Fall '08
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