Static image principle for anisotropic‐conducting half‐space problems: PEC and PMC boundaries

Abstract

The image principle for an isotropic half‐space bounded by perfect electric conductor (PEC) or perfect magnetic conductor (PMC) plane is presented in most elementary textbooks on electromagnetics. It is perhaps not so well known that this principle can also be generalized to anisotropic media in the static case, because it is not covered in leading monographs of geoelectromagnetics (Wait, 1982; Negi and Saraf, 1989; Eskola, 1992). The anisotropic image method can be applied to geologic media that exhibit anisotropic electrical conductivity caused by the fractures and fissures in the rock. Such structures are important in the sites for disposal of nuclear waste. The characterization of these structures by electrical geophysical methods is very essential because they form the main paths for groundwater flow. The air‐ground boundary can be treated as a PMC plane representing the nonconducting medium. Otherwise the medium is assumed to be linear (ohmic) and homogeneous in terms of electrical conductivity. The image method presented is also relevant to problems arising in the traditional ore prospecting where a conducting ore body buried in an electrically anisotropic host rock generates secondary electric fields (Asten, 1974; Eloranta, 1988).

Electrostatic Image Theory for an Anisotropic Boundary of an Anisotropic Half-Space

“…Letting ζ y → ∞ (the PEC case), we see that the planar part (now the exponential line charge) vanishes, and we only have a negative point charge shifted laterally from the mirror image point, as in [1]: …”

“…Over the years limited special solutions have been worked out, for example, for an anisotropic halfspace over a perfectly electrically or magnetically conducting surface [1], for an isotropic half-space bounded by an anisotropic surface [2], and for a similarly anisotropic half-space and boundary [3]. Our proposed general solution, in contrast, will work with all physically justified medium parameters, and the known cases are built in the new theory, as will be shown.…”