Unformatted text preview: needed. You can use the cold ion limit. You need to calculate the growth rate for the instability based on the local slope of the electron distribution function, which will vary with time. This diﬀers from the earlier assignment which simply had a drifting Maxwellian. 3. Calculate the ﬁnal electron distribution function after the instability has saturated. What is the minimum velocity of the plateau? The maximum velocity v ? Why was an upper limit given for U above? 4. Calculate the energy density  E k  2 / 8 π in the waves as a function of their phase speed after saturation. Show that the wave amplitude goes to zero at either end of the plateau. Sketch the wave energy versus the phase speed of the wave. Hint: the ﬁnal answer is  E k  2 8 π = 1 3 √ π Δ k k ω 2 ω 2 pi v 2 p ( v pv g ) v 3 te n m e 2 ( v p3 U/ 2) 2 , (2) where v p is the wave phase speed, v g = dω/dk is the group velocity and Δ k = 2 π/L is the mode spacing....
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 Fall '08
 drake
 Energy, Work, Fundamental physics concepts, group velocity, phase velocity, electron distribution function

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