Electromagnetic induction (eddy currents) in a conducting half-space in the absence and presence of inhomogeneities: A new formalism

Two problems are studied. First, a new method is presented for calculating the electromagnetic field in two conjoined conducting half-spaces in the presence of current sources in either or both half-spaces. The method allows the two half-spaces to differ in the conductivity, permeability, and permittivity. The full Maxwell's equations are used; the quasistatic results may be derived as a particular limit. The method is unique in that it depends only on the solution of two variables; the components of the magnetic field Bz, and the current Jz, normal to the interface between the half-spaces. The second problem involves the determination of the fields induced by a current source in one half-space with an arbitrary 3D inhomogeneity in the other. New, coupled integral equations for the fields are written down strictly in terms of Bz, Jz, and the external current source. The same formalism, used to generate the new integral equations, is also shown to yield the standard dyadic volume integral representations. Finally, it is shown that the formalism is a useful way of deriving various asymptotic results. The weak scattering limit (the Born approximation) is derived as an example