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Solution of Electromagnetic Scattering Problems as Power Series in the Ratio (Dimension of Scatterer)/Wavelength
1.Lord Rayleigh, Phil. Mag. 44, 28 (1897).
2.C. T. Tai, Trans. Inst. Radio Engrs. PGAP‐1, p. 13 (February, 1952).
3.The case where the external medium is characterized by con‐stants can be deduced by replacing ε, μ, k of the text by respectively, and multiplying all electric fields (including the exciting field) by and all magnetic fields by
3.3a The coefficients of the powers of kl will, of course, be functions of ε, μ (as well as of other dimensionless constants entering into the problem). A consideration of the way in which these constants enter into the successive terms of the series shows that, if |ε|, μ are not of the order unity, the rapidity of convergence of the series is determined by the magnitude of rather than of kl itself‐i.e., the wavelength in the body, rather than the wave‐length in vacuum, must be considered.
4.J. A. Stratton, Electromagnetic Theory (McGraw‐Hill Book Company, Inc, New York, 1941), p. 466. Our expressions are slightly different from Stratton’s but are easily deducible from his. Note that our direction for n is the opposite to Stratton’s. The outgoing wave condition at infinity is required to deduce Eqs. (2.19) and (2.20) for an external region, though this fact is not mentioned by Stratton. In applying Eqs. (2.19) and (2.20), it will be assumed, where necessary, that the field point is not actually on S; but the final results are always valid for this limiting case.
5.By a well‐known theorem in potential theory, it follows from Eq. (2.40) that if is also an external harmonic function. This cannot be assumed in advance, however, and is in general not the case.
6.In any case, the problem is reduced to finding the appropriate solutions of Laplace’s equation. We might, alternatively, assume solutions for containing adjustable constants and then use (2.41), (2.42) directly.
7.We can also dispense with the condition (2.36b) for by putting where is an internal harmonic function and is given by an expression similar to that for in (2.39), using internal instead of external fields.
8.It might be thought that it would be sufficient to make |ε| (or, to be more precise, the imaginary part of ε)→∞, and that the results would then be independent of μ. This must actually be the case; but our method of expansion in powers of k makes it impossible to see what the limit would be: it would be necessary to sum the series before proceeding to the limit and then expand again in powers of k. The procedure given here obviates the necessity for summing the series.
9.Reference 4, pp. 555, 557.
10.To apply Stokes’ theorem in the usual form, we imagine a small hole made in 5. The integral taken over the remainder of S is then equal to a line integral taken round the rim of the hole. Making the hole shrink to zero, the required result follows.
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