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Problem 16

# Problem 16 - /W W A magnetized plasma is in equilibrium...

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Unformatted text preview: /W W A magnetized plasma is in equilibrium between two conducting plates placed distance A L apart. (See Figure). is The constant magnetic ﬁeld' is g = Bi and the constant plasma mass density nM. A Assume that ideal MHD is applicable. ,9. +142 Pasii-iw— x-Jc'rec (Tn At t = 0, the plate at z = L is very slowly displaced upward. The displacement of the A {at} plate is given by E (t) which is given to be of the form By “very slowly“, we mean L/VA < T, where VA is the Alfvén speed. Assume that {0 < L. From the frozen-in theorem. we may assume that the plasma velocity at z = L, ﬁr(z = L,t), is related to EU) by a,(z = L,t) = dé/dt. By the same token, (a) Based on physical grounds alone, make a sketch of your guess of g = BE + B,(z,t) an tjou oust: If'tbm f; at ruddy W‘mxlu'rkbm att=0,t=r/2,andt>r. «A- 6XMJ §f The rest of this problem is concerned with deriving BJZJ) explicitly from the MHD equations. (b) Write down two general partial differential equations governing {5(2, t) and B,(z,t) for linearized MHD motions between the two conducting plates? (Small amplitude pertur- bations only.) (c) Given T, 5.0, L, and the plasma parameters, use the fact that L / VA << 7' to obtain l f'sfgm‘h‘cani- rider ‘ to WM approximate equations from (b) above.1 [You will “[50 “4‘4 YWV “"4ewrwdmﬂ 74"“ (“2] i A i (d) Solve the equations from (c) to lowest order to obtain an explicit solution for B,(z, t) 0; km" T} for t > 0. The fact that 342,0) = 0 and the boundary conditions on 11, above will be % needed. Make a plot of é,(z,t) vs. 2 for t -> oo in the domain 2 = [0,L). (6) 0% W WV W‘xzﬁteﬂ‘ﬁ‘m ﬁfe/Q63 Wg/‘gfr’mfé >Fﬂm (96% M9 7% Wake 7M me/‘MA7S“M(S)J§ 7L9 §4Aé WWM‘NS‘, ﬂaw/e were BX 491% f [/10 933‘ I W we . W n W acre-2mm? 74 Q? fem” Z14 ﬂ/{Z‘J 748mm 1This problem may be solved exactly, however, an approximate solution is recommended. ...
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