MAXWELL THEORY OF EDDY CURRENTS IN THIN CONDUCTING SHEETS, AND APPLICATIONS TO ELECTROMAGNETIC SHIELDING AND MAGLEV

Using the example of a monopole that is spontaneously generated above a thin conducting sheet, the simplicity and power of Maxwell's 1872 theory of eddy currents in thin conducting sheets is illustrated. This theory employs a receding image construction, with a characteristic recession velocity upsilon = 2/(mu-0-sigma-d), where the sheet has conductivity-sigma and thickness-d. A modern derivation of the theory, employing the magnetic scalar potential, is also presented, with explicit use of the uniqueness theorem. Also discussed are limitations on the theory of which Maxwell, living in a time before the discovery of the electron, could not have been aware. Previous derivations either have not appealed explicitly to the uniqueness theorem, or have employed the now unfamiliar current function, and are therefore either incomplete or inaccessible to the modem reader. After the derivation, two important examples considered by Maxwell are presented-a monopole moving above a thin conducting sheet, and a monopole above a rotating thin conducting sheet (Arago's disk)-and it is argued that the lift force thus obtained makes Maxwell the grandfather, if not the father, of eddy current MAGLEV transportation systems. An energy conservation argument is given to derive Davis's result that, for a magnet of arbitrary size and shape moving parallel to a thin conducting sheet at a characteristic height h, with velocity upsilon, the ratio of drag force to lift force is equal to upsilon-0/upsilon, provided that d much less than delta(c), where delta(c) = square-root 2h/(mu-0-sigma-upsilon). If d much greater than delta(c), the eddy currents are confined to a thickness delta(c), leading to an increase in the dissipation and the drag by a factor of d/delta(c), so that the ratio of drag to lift force becomes proportional to square-root upsilon-0'/upsilon, where upsilon-0' = 2/(mu-0-sigma-h). The case of a monopole fixed in position, but oscillating in strength (such as can be simulated by one end of a long, narrow, ac solenoid), is also treated. This is employed to obtain the results for an oscillating magnetic dipole whose moment is normal to the sheet. A general discussion of electromagnetic induction and electrical conductors, both thick and thin, is given, emphasizing the difference between the high-frequency limit, where flux expulsion occurs and the self-inductance dominates, and the low-frequency limit, where the flux penetrates and the electrical resistance dominates. A discussion of Lenz's law, as a statement about motion, is given. It is argued that the most general form of such a statement of Lenz's law is that induced currents tend to accelerate a conductor in the direction that most effectively decreases the rate of Joule heating. A calculation, in the low-frequency limit, of the drag force on a magnetic dipole falling down a long conducting tube, is also given. This last case can be given a striking demonstration with the newly available neodymium-iron-boron magnets.