By P. H Roberts

ISBN-10: 0582447283

ISBN-13: 9780582447288

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**Extra info for An introduction to magnetohydrodynamics**

**Sample text**

Let C be a material curve X j = Xj(q), where q is a Lagrangian coordinate labelling fluid elements on it. Clearly T j = dxJdq is parallel to the tangent vector at each point of C. Also, by the chain rule, dXj aXj dXoa (124) 11 = dq = aXo a dq = D ja Toa , where To is the initial value of T. <'--' ' - - ', '. , --. -~ , -_ , , . 7] :~ (d) Small displacements. In later chapters of this book, we often have occasion to introduce a displacement field ~ defined by x = x o + ~(xo). (126) The value of any field, resulting from the displacement ~, may be expanded in a power series in ~ in one of two ways.

J(ro/2 Q)exp i [Z (~) t - wt], o}. (22) On examining (21) and (22), we see that for any fixed t the directions of both Band j alternate periodically in z, with wavelength 2n (2Y/fw)t. J-. 2) increasing Z, the distance in which the amplitudes are reduced by a factor of l/e being (211/W)+. , (23) and 3, the time-scale characteristic of the disturbance, is the period of oscillation 2n/w. The length d~ is often called the electromagnetic penetration depth or, sometimes, the electromagnetic skin depth, since practically all the fields and currents are confined to a skin on the surface of the conductor which is a ' few multiples' of d~ thick.

61) Thus, by (2), we must have curl(E+UxB) =0, (62) and U cannot exist unless we can find a single-valued potential * such that E = - UxB - grad <1>. (63) We may observe a difference here between u and U. In a perfect conductor E and B are necessarily orthogonal and the potential gradient does not appear (cf. (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'. *

### An introduction to magnetohydrodynamics by P. H Roberts

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