By P. H Roberts
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Extra info for An introduction to magnetohydrodynamics
A 'cylindrical' volume, V o, whose 'curved' surface, L o, is everywhere parallel to the prevailing magnetic field, Bo, and whose' ends' are material surfaces 8 J and 8 2 ; see Fig. 3. ) Since Bo • dS o = 0 everywhere on r o, the integrated flux of Bo across every cross-section, 8 0 , of the tube must [by (3) and the divergence theorem] be the same; thus fB o • dS o = K = where K constant for the tube, (46) is known as the strength of the tube. 52 Fig. 3. A tube of magnetic flux Suppose the contents of Vo are carried at time t to some new volume V by the motion of the fluid; suppose r o be carried to r and 8 0 to 8.
4(d) below] is equal and opposite to the speed at which the fluid motion carries them upward. $ and 0/1. It is important to emphasize, however, that the process of creating flux in V is one of detaching flux tubes from the material within them by electromagnetic diffusion. fE/0/I. And this behaviour is not peculiar to this example. fE 2 ; and when Rm ~ I this is evidently much slower. (d) The velocity of drift offlux tubes through a conductor. The conceptual advantages of picturing lines of force as moving with a fluid are evidently considerable.
13». When 1'/ =1= 0, we have no reason to suppose that E and B will be perpendicular, and the potential term is required to make the necessary adjustment. B = everywhere (see Figs. 7), we need go no further, since (63) is satisfied by taking = constant, and ExB ° U= B 2- +,I'B '1'. --"T'-..... Conductor B Lines of force Insulator (b) (a) Fig. 7. BIB 2 ) is an arbitrary scalar field. Its presence reflects the fact that a (1 - 1) correspondence between lines of force at two different times does not entail a (1-1) correspondence of individual points upon them; the correspondence is necessarily uncertain to the extent of an arbitrary displacement along either line.