Geometrical approach to eddy-current systems

The discovery of the constant velocity of propagation of electromagnetic energy led to the description of electromagnetic behaviour in terms of a 4D space having a metric defined by the distance ds, where ds²=dx²+dy²+dz²-c²dt². The authors use a similar approach for eddy-current behaviour. A complex metric is defined which contains a complex velocity and a propagation constant, both of which are dependent on frequency. By considering the interaction of a dissipative eddy-current system with an adjoint system in which energy is being generated, it is possible to produce an invariant 4D volume in terms of the separate electric and magnetic energies. By means of a variational method, upper and lower bounds can be found for both these energies. The propagation constant associates a characteristic length with the penetration of energy into a conductor. This penetration depth classifies conductors as being either thick or thin. In thick conductors the electric and magnetic energies are closely coupled; in thin conductors the coupling is very slight.