Completeness of a set of modes connected with the electromagnetic field of a homogeneous sphere embedded in an infinite medium

B.J. Hoenders

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In this paper we consider the mode decomposition of the electromagnetic field connected with a sphere of conductivity σ1, and dielectric and magnetic permeabilities ε1 and µ1 respectively. The sphere is embedded in an infinite medium, characterised by the constants σ2, ε2, and µ2. The continuity requirements of the field across the surface of the sphere lead to a certain set of so called natural modes, which can be calculated explicitly if the radial part of the electric field strength inside the sphere equals zero. The completeness of the radial parts of these modes, which is a set of spherical Bessel functions, is sometimes erroneously deduced from Sturm-Liouville theory. This theory however cannot be used to show the completeness because the continuity conditions of the field lead to a boundary value problem with a boundary condition which explicitly depends on the eigenvalue. The completeness of this set of functions, which is necessary to solve an initial value problem, will be shown. The set of functions will even be shown to be overcomplete. The connection of this problem with many other similar problems occurring in mathematical physics, as well as the physical consequences of the overcompleteness, will be discussed.
Original languageEnglish
Pages (from-to)1815-1832
Number of pages18
JournalJournal of Physics A-Mathematical and General
Issue number9
Publication statusPublished - 1978

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