Hwk2 - Physics 762 Homework#2 Spring ‘11 Dr Drake 1 Show that the Z function satisfies the following differential equation dZ(ξ = −2(1

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Unformatted text preview: Physics 762 Homework #2 Spring ‘11 Dr. Drake 1. Show that the Z function satisfies the following differential equation: dZ (ξ ) = −2(1 + ξZ (ξ )). dξ (1) 2. Calculate the kinetic dispersion relation for ion acoustic waves in a plasma with a Maxwellian distribution of ions. The electrons are also Maxwellian but with a mean velocity Ue = U z . Take the wavevector parallel to Ue . Show that the ˆ general dispersion relation is given by (k, ω ) = 1 − 2 ω k2 ω − kU kDi Z( ) − De Z ( ). 2 2 2k kvti 2k kvte 3. Show that the real frequency ωr for sound waves when Te Ti and U (2) vte is given by 2 ωr = k 2 c2 s 1 , 2 1 + k 2 /kDe (3) where c2 = Te /mi . Keep corrections to obtain the growth rate of the instability, s 3 2 √ ωr kDe ωr − kU Te ωr −ωr /k2 vti 2 2 γ=− π 2 2 ( + e ). 2ωpi k kvte Ti kvti (4) For Ti = 0 show that instability occurs for 2 U 2 (1 + k 2 /kDe ) > c2 . s Hint: you will find that ω k vti and ω − kU vte . (5) ...
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This note was uploaded on 12/29/2011 for the course PHYSICS 762 taught by Professor Drake during the Fall '08 term at Maryland.

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