By P. H Roberts
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Extra resources for An introduction to magnetohydrodynamics
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. The result (64) applies equally to the field in a conductor or an insulator. BIB 2 is arbitrary. This shows that' the velocity of drift of flux tubes relative to the fluid' (and perpendicular to their own length) is j xB 11 U d = U-u = -"2 = - 2(BxcurlB).
First, at a zero of B, the derivatives of B, and in particular curl B, will not vanish in general, but, if they do so, they will vanish to a lower order than B. curl B = 0, it is impossible that B and curl B become parallel as the zero is approached. Thus, we see from (66) that U d is unbounded in any neighbourhood of the zero. The physical reason for this may be traced to the fact that the induced electric field, - u x B, is necessarily zero at any zero of B. Electromagnetic induction is therefore ineffective.
The difference in flux is therefore -/(BI -Bz)(FF/) = -IUn(B 1 -Bz)Dt. The rate of change of flux through r is therefore -lUn(B I - B z ) = I(E1 - E z), by (6), where E denotes the component of E in the direction of J. This agrees with (88). A,I. s~L _ _;ss r'l e , ~ A2 A2 (a) time t (b) time l + lSt Fig. 4. The boundary condition on the electric field at a moving surface. The elementary rectangular circuit A , A 2 C 2 C , is fixed in space, and the surface S intersects it at FG (at time t) and F'G' (at time t + ot).