choose the origin of the coordinate system in such a way that the cubical region inside which , 5;6 0 be in the interior of the sample, and choose the edge of the cube, 1, to be as large as possible. Substitute the the resulting field into (3.6), and assume, for simplicity, that a == co, µ = µo and 11 is uniform. Then,
[(v, T 2
Thus, the smallest eigenfrequency is necessarily smaller than the latter quantity. Since we chose such that _= 0 outside the sample, the above estimate only depends on the sample's size, and not on the dimensions of the reflecting surface. The estimate shows that, if the wavelength in free space corresponding to the lowest mode is denoted by X,, then, to order of magnitude, Ao/l (l/27r)(µo/eo)~Il-' In the physical situation of Ref. 4 (but not in the
situation of Ref. 2), (l,/2x)(µo/eo)Ill-1 108, i.e.,
independently of the size of the reflecting shield, the lowest modes can b& considered quasistatic (i.e., of essentially infinite vacuum wavelength). To estimate the rate at which energy would leave the region M, in the absence of the reflecting surface, consider the fields due to the currents and charges on the sample alone; neglect all but the magnetic dipole radiation, and let the shield be a sphere of radius ao. The ratio of the energy crossing the shield in one cycle to the energy inside the shield is then
found to be of order (l/ao)3 10-2q.
For the higher modes the rate of radiation is higher. However, it can be shown that if we formally let the speed of light outside the sample tend to infinity, the set of almost unattenuated modes can be extended to an arbitrarily large part of the spectrum.
The author wishes to express his gratitude toward
Professor W. H. J.
Fuchs, for his extremely kind and
patient guidance through the evolution
paper. Thanks also go to Mr.
Goodman for a
number of helpful suggestions that greatly improved
Footnote to page 153 (added in the Internet edition)
At the frequencies of interest in Reference 4 the reflecting surface is not necessary, since an effect
similar to total internal reflection confines the helicon energy to the sample, and essentially no
energy reaches the reflecting shield. As seen in the last paragraphs of the article ("comment on the
reflecting surface S"), at the frequencies in question the fraction of energy reaching the shield
in one cycle would be of order 10-24 (meaning that it would take some 1014 years before most of
the energy leaked out). Nevertheless, the rigorous proof fails without the shield, because to the
mathematics a small leakage of energy is no different from a
substantial one, and the energy leakage at the very high eigenfrequencies (where leakage is substantial) is
as important as the leakage at the low eigenfrequencies (where it is not). Accordingly the proof of this
paper has been split into a "mathematical proof" and a "physical proof". The "mathematical proof", in
Section 2, shows that when there is a shield there generally exists a complete set of oscillatory modes;
the "physical proof", in the last paragraphs of the article, shows that in practice the shield is
unnecessary, since the energy leakage in its absence is negligible.