physics 5150 sol'n to hw #7 due: tues, april 25, 2017 1 5150 sol'n to hw #7 due: tues,...

Physics 5150 Sol'n to HW #7 Due: Tues, April 25, 2017

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1. Certainlinearquasilongitudinalwavespropagatingatanarbitraryangleθ

toB0canbeshownusingcommonapproximationstosatisfy,ω2=ωi2+ωe2cos2θ,whereωe,iaretheelectronandionplasmafrequencies.Notethatasθ->90degrees,ωgoestothelowerhybridfrequency,whileatθ=0thewavesbecomeelectronplasma(Langmuir)waves.Forθcloseto90degreesthesewavesaresometimescalledelectrostaticwhistlers.

a) Beginning with cold 2-fluid eqns. and assuming infinitely magnetized electrons

and unmagnetized ions (Ωι = 0 and Ωe → ∞ ), derive this frequency for these longitudinal waves propagating at an arbitrary angle θ.

b) Beginning with the Maxwell matrix eqns. at arbitrary angle θ and making the

same approximations, derive this same frequency from the condition that det M = 0. c) Explain physically why it is sometimes stated that the electrons have an effective

mass me/cos2θ in this approximation. Solution on next three pages:


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2. Morewavestoanalyze

a. Findthemagnitude(relativetoE)andsketchthemotion(relativetoEandBo)ofthewavemagneticfield,B,forthefollowingwavesinthecoldplasmaapproximation:

• Parallel propagating (torsional) Alfven waves • Parallel propagating low frequency whistler waves • Perpendicular propagating lower hybrid waves

b. Findthedirectionandmagnitudeofthefirst-orderwaveelectronandionfluidvelocitiesandthefirst-orderwavedensity(ifitisnon-zero)foreachofthewavesintheaboveproblem.Interpretyourresultsasmuchaspossible.

Part (a) solution on next pages:


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Part (b) solution on next pages:


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3. Assume the magnetic field above a sunspot varies (only in the vertical z-direction) as

B (z) = B0 z

1 + z/a , B0 = 200 G , a = 104 km Assume the density in the region is uniform, with n = 107 cm-3. A left-

circularly-polarized wave is launched vertically upwards from z = 0, with frequency f = 10 kHz.

a) What kind of wave is it? b) What is the wavenumber at z = 0? At z = 105 km? What is the zero-order

WKB wave form at altitudes z in between these? Prove WKB is valid in these regions.

c) What happens at high altitudes, as the local ion cyclotron frequency

decreases and approaches the wave frequency?


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physics 5150 sol'n to hw #7 due: tues, april 25, 2017 1 5150 sol'n to hw #7 due: tues,...

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