Low‐frequency asymptotics for eddy currents in a conducting half‐space in the absence and presence of inhomogeneities

The low‐frequency asymptotics of an electric field generated inside a conducting half‐space by an external time‐varying current source are derived. The conductivity of the half‐space may be either uniform or it may contain inhomogeneities (i.e., regions in which the conductivity varies). The dependence of the low‐frequency asymptotics on the dimensionality of the current source and the inhomogeneity are exhibited. We display those leading terms in the expansion that are given exactly by the quasistatic approximation. It is shown that in the case of variable conductivity, the asymptotics of the electric fields inside the half‐space are determined by a Fredholm integral equation of the second kind. For the general case, the solution of this integral equation is identical to solving a classical boundary‐value problem of electrostatics. We also show that there is an important class of problems for which the low‐frequency asymptotics are trivial and are given explicitly in terms of the external field via the B...