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point and the single-primed letters, the observed mean values; then on the average E'>H"E">H'=0, and the mean momentum per unit volume is
The first term on the right leads at once to equations (8)-(10), that is, this term gives, for a fixed potential difference between the plates, the same value for the torque as if the material dielectric were removed. This is the result to which we are led if we follow the usual rule of simply ignoring fine-grained irregularities of field.
7. Local Torque in a Dielectric. The second term on the right in (13) requires special consideration. The general result appears to be that it yields an additional torque which may greatly exceed the one given by (10) but which can have either sign and which has nothing to do with the ordinary electrical properties of the substance but depends directly upon its atomic structure. Indeed, one would expect a torque to exist in many crystals even in the absence of an electric field.
We shall illustrate the possibilities of the case by considering a simple group of electric doublets whose dimensions are small relative to their distance apart.
The mechanical forces upon a given doublet arise in part from the field of its neighbors and of distant charges. This part will depend only upon the strengths and positions of the doublets and of other charges.
A second part arises from the magnetic interaction between the constituent charges of each doublet. If the latter are +e and -e and are separated by a displacement I making an angle † with the x-axis, which we suppose to be the direction of motion, the doublet experiences a torque tending to set its axis at right-angles to the direction of motion. and of magnitude (the magnetic intensity being ẞe sin 5/12).
Let there be N doublets per unit volume with parallel axes. the torque per unit volume is L=NL1,while the resulting polarization is P=Nel, so that the torque per unit volume can also be written
This torque can attain any magnitude independently of P through variation in l. Hence a counterbalancing of it by other effects which depend upon P can occur only as a matter of accident. The denominator N3 is, according to our assumptions, much less than unity.
From this result one seems justified in drawing the broad conclusion that crystals ought, in general, to experience an appreciable torque when moving through the ether. Definite calculations cannot be made without the adoption of some definite atomic theory, but all modern theories suppose electrical separations to exist in the atom which by themselves would produce enormous values of P, and in general one would expect these to give rise to an outstanding torque of the magnitude of L as given by (15) with P a large number and N3 at least less than unity. For instance, if 1023 electrons per unit volume were displaced relative to the atoms a distance of only 10-10 cm. the resulting polarization would be 1023 10-10X4.77-10-10-4,770 electrostatic units. This value of P gives, even with N3 replaced by unity, a value of L which is hundreds of times bigger than the torque per unit volume on Trouton and Noble's condenser; for in the latter case the electric intensity appears not to have exceeded 700 electrostatic units, so that U in (10) divided by the volume would be about 20,000.
Since the occurrence of such a torque would constitute an effect of uniform motion through the ether, Relativity requires that any torque due to this cause must be compensated within the crystal by an equal and opposite torque of different origin (presumably of the same nature as that described below in Sec. 10). Thus the fact that no such torque has ever been observed lends a certain amount of support to Relativity.
To illustrate now the possibilities in an isotropic medium, let us consider the effect of applying an electric field in a direction inclined at an angle to the direction of motion (Fig. 2), and let us suppose the doublets to be oriented irregularly except that their axes are all parallel to the plane defined by these two directions. Let the field displace the positive charge a small distance λ in the direction of the axis and a small distance μ at right angles to it where
λ=aE cos (5-0), μ=bE sin (5-0).
In this case there is an outstanding second-order torque for which, using (14), I find the value
or, introducing P=Ne(a+b)E/2 = (K−1)E/4TM,
where L= torque per unit volume.
This expression is the same as (10) divided throughout by the volume of the condenser, except for the factor in brackets. It is hard to assign a plausible value to the latter factor without assuming some definite theory of atomic structure; but according to our assumptions N3 is small compared with unity, hence the whole bracket ought for a solid dielectric to be comparable with unity and might greatly exceed this value, and it might have either sign. For instance, for N3=1, K=6 (mica) and a=0 (approximately pure rotation of the doublets) the bracket equals +5, while for N31/10, K6 and a=1.5 b the bracket equals -4.
In an actual dielectric, circumstances will no doubt be very different from those of the simple case here treated. Yet we seem justified in concluding that, according to the electromagnetics of Lorentz, a local torque of appreciable magnitude is very likely to act upon a moving polarized dielectric and that this torque might conceivably mask completely the effects of the torques upon the true and apparent charges.
8. The Effect of the Mica in Trouton and Noble's Condenser remains therefore in doubt. It may have caused the null result; more likely, however, being crystalline, it should have greatly increased the effect, perhaps with a reversal of sign.
9. The Fine-Structure of the Charge on the Plates might conceivably have a similar effect. The removal or addition of electrons on the surface might form doublets with axes more or less parallel to the surface, and these, by urging the plates toward a position at right angles to the motion, might mask the main effect. Such an effect could be distinguished in a repetition of the experiment through the circumstance that it would depend only on the charge and not, like the main torque, also upon the difference of potential.
10. The Explanation by Relativity of the null result is a dynamical one and is fully given in Laue's Relativitätstheorie. The torque occurs as a secondary effect superposed upon the far larger electrostatic attraction between the plates: according to Relativity, the intermolecular stresses which balance the latter attraction do not obey Newton's
Third Law but themselves produce a torque which just balances the electromagnetic one. The null result is thus explained, not exactly by the Lorentz-FitzGerald contraction itself, but rather as a consequence of the same cause that produces this contraction.
11. Other Ways of Excape are hard to find. So far as the writer is aware, there is no rival to the Maxwell-Lorentz theory which explains all ordinary phenomena (including Hertzian waves) and also removes the torque on the condenser. Of course, drag of the ether by the earth would do it, but this assumption leads to well-known difficulties. Instead of speculating upon possible modifications of electromagnetic theory it seems more profitable to pass in review the experimental facts upon which the prediction of the torque rests. They are (for vacuum as dielectric):
(1). Moving charged bodies generate a magnetic field. The charged body used in experiments like Rowland's is very similar to either plate of our condenser, the only important difference being that in those experiments the average convection current was closed.
(2). Magnetic fields (of certain kinds, at least) act on moving charged bodies (probably verified only for charged molecules or electrons).
(3). No difference has yet been detected between magnetic fields arising from different causes.
These basic facts lead by very simple reasoning to the predicted torque. As a matter of logic one might attack the sufficiency of the present experimental basis for (2) and (3), but the prospect of discovering any error at this point seems very small. Perhaps the most promising thing to try out would be whether a moving charged conductor really is acted upon by a magnetic field.
12. In Conclusion, the situation may be summed up as follows:
Trouton and Noble's negative result might have been due to either (1) insufficient sensitiveness of their apparatus (Sec. 5), or (2) insufficient distribution of their observations over different times of year (Sec. 5), or (3) a special effect of the mica dielectric (Sec. 7, 8), or (4) a similar special effect in the charged surface of the plates (Sec. 7, 9).
If, however, it be accepted as an experimental fact that an air condenser never experiences a torque due to the earth's motion, then this fact speaks forcibly in favor of Relativity and can hardly be explained on any other basis.
Probably few physicists will refuse today to accept this as an experimental fact, nevertheless the importance of the problem seems to justify a repetition of the experiment, with observations taken on an air condenser, at different times both of day and of year, and with various. distances between the plates.
A SURVEY OF THE STATUS OF THE DETERMINATION OF THE GENERAL PERTURBATIONS OF THE
Appendix to the Report of the Committee on Celestial Mechanics, National Research Council*
*This Committee of the Division of Physical Sciences of the National Research Council consists of the following members: E. W. Brown, Professor of Mathematics, Yale University, Chairman; G. D. Birkhoff, Professor of Mathematics, Harvard University; A. O. Leuschner, Professor of Astronomy, University of California; H. N. Russell, Professor of Astronomy, Princeton University.