« PreviousContinue »
The intensity of the ray transmitted by this slightly absorbent layer was determined by an electrometer E placed at a quarter wave-length (2-25 metres) from the closed end of the circuit.
This intensity varied periodically with the thickness of the layer, as in the analogous case of a ray of light and a “thin plate"
The layer AB was what Boltzmann has termed a “conducting dielectric,” i. e. both its conducting and dielectric properties were of importance. The theory of a non-conducting plate is much simpler: a problem equivalent to it has recently been treated by Dr. E. H. Barton *, to whose work I shall have several occasions to refer. In the more complex problem I have assumed, we have simply a plane-fronted damped wave-train travelling through an insulating medium and falling at normal incidence on an infinite slab of conducting dielectric. The magnetic permeability of both the plate and the surrounding medium are taken as unity.
The theoretical transmission-curve obtained on these assumptions does not agree well with the experimental one, the divergence being the same in sign and order of magnitude as that noticed by Dr. Barton in his case. We have in fact idealized the experimental facts too far in endeavouring to simplify the analysis: the electrolyte was not at all infinite in extent, and at least one important correction is obviously necessary (section VI.). The experiments can only be regarded as a very rough illustration of the problem, and as giving the raison d'être of this paper.
The general results obtained seem to be of considerable interest. The intensities of the reflected rays, the phasechanges, and so on, for damped wave-trains reflected from such a plate differ from those for steady rays in some cases very considerably.
For convenience and brevity, the surrounding medium is * Preliminary Paper, Proc. Roy. Soc. vol. liv. p. 85; more fully in Thesis for the D.Sc. London, 1894, or Wied. Ann. vol. liii. p. 513, 1894. Final paper, Proc. Roy. Soc., read April 1893 (not yet published).
hereinafter generally referred to as “the dielectric," and the conducting slab as “the plate” or “ electrolyte.”
II. The Phenomena at the First Surface of the Plate. In this section we will deal with the phenomena occurring at the first surface of the “ electrolyte” only.
The direction of propagation of the ray is chosen as the positive direction of the axis of z and the origin is taken indefinitely close to the interface.
The suffix 1 always refers to the dielectric and the suffix 2 to the plate.
Let X, be the electric force in the dielectric and X, that in the plate. Under the limitations we have set ourselvesplane waves and non-conducting dielectric-Maxwell's equations assume the simple forms
where we have abbreviated Maxwell's notation by putting
βι=μKI, β=μοκη, α, =4πμ,(2, (3) My being the magnetic permeability of the plate, K, its dielectric constant, C, its conductivity; m K are the same constants referring to the dielectric, C, being zero. As already stated, we will only take the case of both media being non-magnetic, or
wheret must be not less than zero. p is either wholly imaginary or complex according as the incident ray is steady or damped:
is as yet undetermined ; A, is the amplitude of the incident wave-train, Ai' of the reflected wave-train, and B, of the transmitted wave-train.
Inserting the value of X, given in (4) in equation (2), and carrying out the differentiation, we get
B2p so q is also partly imaginary.
A A, tions. These will be given us by the interface conditions: the conditions namely, that there must be no discontinuous jump in the values of either the tangential electric force or the tangential magnetic force, at any tiine, in passing from the dielectric to the electrolyte. That is, we must have
X,=X2, ? dX dX,
z=0, t=anything. Inserting the values of X, and X, in (6) we get, putting z=0, A1+ ,'=,
A, B, and solving these two equations for
2 VB.. Α, w Bita
If we write for brevity
(8) 1+à' contains
which is partly imaginary; so we must separate the real and imaginary quantities and rationalize. Let Pı be the real part of p, and pa the imaginary part, so that
p=-Pi + pai, and let us write Pi=n cos X,
(9) Pa=n sin x.) In this notation then
P2 logarithmic decrement= 27 P1,
P2 and using the value of q in (5),
Bi 1 a=
The denominator of this expression is
as a further abbreviation we have
a=rea. We can now proceed to rationalize the values of Ay'
and B, AQ given in (8). Inserting our value of ,
o 1-re 2
- ir sin AL
2 Multiplying numerator and denominator by
1+rcos. – ir sin, this becomes
1-qie_2ir sin Ž
0 1 +82 + 2r cos
and multiplying again by 1–72 + 2ir sin
1+7-2r2 cos e
) If then
0 2r sin 2
2 that is, the ray reflected from the first surface of the plate undergoes a phase-change of (-7), the reduction factor or ratio of the amplitude of the reflected to that of the incident ray being
(1+got — 2rcos)? b=
(13) o 1+p? + 2r cos