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The Propagation of Electromagnetic Waves in a Refracting Medium in a Magnetic Field

Published online by Cambridge University Press:  24 October 2008

D. R. Hartree
D. R. Hartree
Affiliation:
Christ's College, Professor of Mathematics, University of Manchester
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Extract

The equations of propagation of electromagnetic waves, simple harmonic in time, in an optically anisotropic stratified medium are obtained from the treatment of the refracted wave as the resultant of the incident wave and wavelets scattered by the elements of volume of the medium, and are reduced to a simple form.

The primitive property of the medium, from which the other optical properties are derived, is the scattering tensor, relating the induced dipole moment per unit volume to the applied electric field.

The relation between the dielectric tensor (corresponding to the dielectric constant of an isotropic medium) and the scattering tensor is obtained.

A medium consisting of classical oscillators in an external magnetic field is then considered, the scattering tensor and dielectric tensor are evaluated for such a medium, and finally a formula for the refractive index is obtained.

For an ionised medium the formula differs from that obtained by Goldstein; the difference is due to the inclusion in the present treatment of a term omitted by Goldstein; the significance of this term is discussed, and its inclusion justified.

Taking this term into account makes an important difference to the properties of the medium for long waves; an example is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 1 , January 1931 , pp. 143 - 162
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Hartree, D. R., Proc. Camb. Phil. Soc. Vol. 25, p. 47 (1929)Google Scholar. This paper will be referred to as I.

† Appleton, E. V., Washington Radio Conference, 1927.Google Scholar

‡ Goldstein, S., Proc. Roy. Soc. Vol. 121, p. 259 (1928).CrossRefGoogle Scholar

§ Darwin, C. G., Trans. Cambs. Phil. Soc. Vol. 23, p. 137 (1924).Google Scholar

* Gibbs, J. W., Vector Analysis [included in Collected Works, Vol. 2].Google Scholar

† Op. cit. § 117.

‡ Op. cit. § 131.

* J. W. Gibbs, op. cit., § 131.

* C. G. Darwin, loc. cit.

* Lorentz, , Theory of Electrons, p. 138.Google Scholar

† Such a contribution is likely to be in the direction of the local dipole moments, i.e. of σ. E. If it were not, it would be necessary to replace the number β by a tensor; for simplicity we disregard this possible complication.

* Op. cit., p. 159.

† Op. cit., § 121.

* We shall only be concerned with forced oscillations in a periodic applied field, and for this case a classical free electron can be treated as an oscillator with k = 0.

† For bound electrons, Lorentz (op. cit., p. 141) gives 2m/r for the value of this coefficient; for free electrons Appleton has shown that the appropriate value is m/r (loc. cit.).

* One way to obtain this result is to put

multiply both sides by expand the product on the right-hand side using (6), and equate coefficients of U,ρρ and U Λρ. This gives three equations from which a, b, c can be found.

* Formula (45) agrees exactly with that given by Appleton (loc. cit.). The following gives the relations between the symbols used here and by Appleton (following Lorentz):

Appleton

Here

* Cf. Lorentz, op. cit., pp. 305—8; Richardson, , Electron Theory of Matter, pp. 71 sqq.Google Scholar

† P=σ. E/4π in the notation of the present paper, if P is in ordinary electrostatic units.

* It is true that if E is the field at the mean position of the oscillating electron, the field on it when displaced is (r. ▽) E, and, even if E were constant in time, this would give a time variation e iket if E were not uniform in space. Such a contribution from the field of the positive charges could only arise from statistical deviations from isotropy or homogeneity in their distribution, and would be likely to be random in direction so that its effect on the scattering by a volume containing many electrons is likely to be very small.

† If the amplitude of the oscillation of the electrons in the ionised medium were large compared with their distance apart, further investigation would be necessary; to this extent, the present treatment is first-order in the amplitude of the incident wave. An estimate gives 10−5 cm. as the order of magnitude of the amplitude of oscillation of the electrons of the Heaviside layer in a strong signal; the mean distance apart of the electrons is of the order 10−2 cm., so in practice the amplitude is small compared with the distance apart.

‡ Loc. cit. Appleton (loc. cit.) retains β but does not specify its value.

* See Goldstein, loc. cit., Fig. 1.

* The dispersion curves for β=0 are very like those in Fig. 1 of Goldstein's paper, since the values of (k H/k 0) and (H 2/H)2 for which the curves are drawn here are close to the values used by him; they have been taken slightly different from his values for convenience in computation.

† It is clear, for example, that the critical frequencies will all be given correctly by the formulae for β = 0, if in place of we use (1−β) but the general improvement of the dispersion formula for β=0 obtained by using this modified value of is not great.

* When 1 + ξ (1 − τ2)≠ 0, we get the same result by putting τx=0, τs=τ in (45); the upper sign in (53) corresponds to the upper or lower sign in (45) according as τ [1 + ξ (1 − τ2)] is greater or less than 0.