An introduction to magnetohydrodynamics by P. H Roberts

By P. H Roberts

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3] _ _,r I----.. :_ _; ~ ',_ _ , '---J _ ,_ : '. ,__ I Electromagnetic Induction' __. _~_. '_'~', ----.. 5 below). (c) The displacement of a perfect conductor. The idea that magnetic flux tubes follow a perfect conductor in its motion is a powerful and frequently adopted aid to thought in picturing magnetohydrodynamic processes. It shows that the field B(x) resulting from the displacement ~(xo) of a perfectly conducting fluid containing a field Bo(x o) is independent of the manner or rate at which the displacement ~(xo) is performed, and is uniquely determined, by purely kinematical arguments, from ~(xo) and Bo(xo).

1: 119), and B can be deduced from (52), if B o is known. 46 ,.. \ ' . - , E~ma~~ndu~ ~31 We may give two simple examples of (52). Imagine a flux tube x6 + Y6 < a2 , in which the field Bo = (0, 0, B o) is uniform. Let us subject the tube to the displacement X o -4 x, where x = (AXo, AYo, (zo), (53 and A and ( are positive constants. The term in A corresponds (if A < 1) to a compression of the tube to a radius of aA. The term in ( corresponds (if ( > 1) to a stretching of the tube in the z-direction.

1. e. det D = 1 and P = Po. However (56) remains valid, and the increase in magnetic energy per unit length is given by (57), as before. It may be regarded as the work done (per unit length) against the Maxwell tension along the tube during the stretching process. ) We note here two consequences of (52) which are useful later. 7(d), A (~) = To first order in ~, ~o . grado~-Bodivo~, and by (3), (43), (59) and (1: 130), <5B = curio (~ x Bo). ---- '----- Eleemmragnetre-mductii:Jir"'-' If we recall that u with (12).

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