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Transmission conditions in pre-metric electrodynamics

by S. Kurz and H. Heumann

(Report number 2010-28)

Abstract
The goal of this conceptual paper is to establish a comprehensive derivation of the electromagnetic transmission conditions at moving and deforming interfaces, since they are essential for mathematical modelling and numerical simulation. Transmission conditions are part of pre-metric electrodynamics and can therefore be stated on a differentiable manifold without any additional geometric structure. Electromagnetic fields are represented by differential forms or - alternatively - by fields of Poincaré dual multivectors. To this end, the manifold is equipped with a volume form as additional structure. The paper gives a short introduction about pre-metric electrodynamics, in the four-dimensional relativistic and the 3+1-dimensional setting. Both settings are related diffeomorphically by a so called pre-observer. The transmission conditions are derived from Maxwell’s equations in four dimensions and then decomposed, where various representations in terms of traces of differential forms or restrictions of multivector fields emerge. It is shown that motion and deformation is completely captured by a scalar transverse velocity. This enables immediate generalization to tangentially discontinuous velocity fields. To make the concepts more tangible a simple numerical example is presented. The classical Wilsons experiment in 1913 about the electromagnetic field in a rotating non-conducting cylinder is studied, based on Finite Element analysis. The weak formulation enables assessment of a systematic source of error in the measurement process. This effect can be quantified by numerical experiments.

Keywords: Electromagnetic transmission conditions, Moving and deforming interfaces, Pre-metric electrodynamics

@Techreport{KH10_432,
  author = {S. Kurz and H. Heumann},
  title = {Transmission conditions in pre-metric electrodynamics},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-28.pdf },
  year = {2010}
}

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