PHYS 4150 Quiz: homework5Exam
Course CodePHYS 4150
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PHYS4150 Homework 5 due: 10/6/2019
Suppose a plasma of electrons and ions is immersed in an inﬁnite, homogeneous magnetic ﬁeld
whose strength we can change as a function of time. The plasma is then repeatedly taken through
the following four steps:
1. The ﬁeld strength is increased from B(t=0) = B0to αB0on a time-scale slow compared
to an ion cyclotron period, but fast compared to a collision time, i.e.
where τee is the electron collision time and τii is the ion collision time.
2. The plasma is allowed to relax to equilibrium by waiting many collision times (τee,ii ≪
3. The ﬁeld strength is decreased back to B0at the same rate as in step (1).
4. The plasma is again allowed to relax back to equilibrium.
Problem 1 Derive an expression for the plasma energy gain per cycle ∆E
E0. Assume that the
electrons are energy-isotropized faster than ions, but that energy transfer between ions and elec-
trons is slower than the ion collision/isotropization time. (Isotropization is the pitch-angle ran-
domization described in the third problem of homework 3. In other words:
τee ≪τii ≪τei,where
τee is the electron energy isotropization time
τee is the ion energy isotropization time
τie is the electron-ion energy transfer time.
Problem 2 How might you modify steps (1) - (4) to pump energy mainly into electrons, but
not into ions?
Problem 3 How might you modify steps (1) - (4) to pump energy mainly into ions, but not
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