LVII. The Dielectric Properties required for Maxwellian Radiation. By A. PRESS *.

Summary.

UITE apart from the two curl equations symbolizing Maxwell's Electromagnetic Theory, a consideration. of generalized mechanics leads to a differentiation between the μη and k, coefficients for radiant manifestations such as light or Hertzian waves on the one hand, and the μ, and k coefficients for zero frequency conservative systems on the other hand. Indeed, the requirement is that in every case, as experiment shows, the k, and, coefficients should be larger. Then, taking note of Maxwell's curl equations it is proved that the electrical refractive index should not be expected to agree with the optically determined refractive index. The above two considerations necessarily lead to a modification of our concepts of self-induction and capacity coefficients for radiant systems.

The Activity Equations.

In a generalized mechanical system the activity A is defined as the result of multiplying the generalized force by a corresponding generalized velocity. In electrodynamics such force is the electrical intensity E, whereas the generalized velocity is the time-rate of change of generalized displacement D. Thus the activity per per unit volume drdN.dS is given by

with dN as the element of the unit normal N and dS as the element of area normal to N. Thus for a simple sinoidal variation of the quantities involved no real activity can possibly result if the generalized displacement D is taken to be in time phase with the impressed generalized force.

Consequences of Time-Variations Harmonically Considered. For generality let it be assumed that :

E=E1 sin pt + E, cos pt,

D= D1 sin pt + D2 cos pt.

*Communicated by Prof. T. J. Schwatt, Ph.D.

Then we have that

A. =p[{ED1 cos2 pt-E1D, sin2 pt }+(E1D1−ED2)

sin pt. cos pt]. The term involving sin pt, cos pt cannot give rise to any real time integrated activity, since the expression changes sign with the time. Yet by simple trigonometry the term can be transformed, and then

A =p[(E,D1− E1D2), cos2 pt + E1D, cos 2pt

2

+ }(E1D1—E‚D2) sin 2pt]. Whether any real activity results therefore depends on whether the expression (), is greater than zero or not. That is, for real consumption of energy the following inequalities must subsist ::

Indeed with

ED1> E1D2; D1/E1>D2/E2.

ED1-E1D2=0,

or what amounts to the same thing

E1/D1=E2/D2,

there can only be a completely wattless consumption of energy. Thus for true radiation it is seen that the usual expression

D=kE

cannot obtain with k as a real (non-operational) number for the proportionality factor.

For the magnetic case equally and quite independently we must have, with

H=H1 sin pt + H2 cos pt

B=B1 sin pt + B2 cos pt,

that for real watt consumption

μ

Therefore cannot be real but must be time-operationally complex.

Activity Requirements for Progressive Wave of

Displacement Flux.

The inequalities above have been introduced to emphasize the time-phase factors only. A better form would be to set

forth the progressive and stationary wave-components of the generalized displacement. In this manner the radiant energy is localized in the voluminal displacement wave carried forward. Thus let

E = Es+ Ep,

where Es is the stationary wave-component of force (as in a non-radiating condenser action) and Ep represents that component of the impressed force giving rise to the progressive (or radiant) wave-component. In the same way we have

with the understanding that it is E, that produces D., etc. Investigating afresh the activity relations we find

A.=(Es+EP) (dDs/dt+dDx/dt)

It is evident that the terms in () can produce no real

component of loss.

have

With respect to the terms in {} we

{}=p[cos. sin pt. cos pt (Ep. D,+E, . Dp)1

+ sin . (D).Ep cos2 pt-DpE, . sin2 pt)2].

By trigonometry, however, it again turns out that

()2=(EpD ̧— E ̧Dp); . cos2 pt+E ̧Dp.cos 2pt.

3

The real activity therefore depends on whether () gives the following inequalities

Dp De
EpD>EsDp: kp= < =k
Ep E

and similarly for the magnetic case where with

H=H, sin pt + Hp sin (pt +),
B=B, sin pt + B, sin (pt +¥),

H ̧Bp<HpBs ; p=

By

<

Bs

Conceiving then, for convenience, of a new mathematical operator R such that when acting on the activity expression A, it gives the real component of the energy rate consumption, we have

RA.=p sin . (E,D,-E,D,) cos2 pt.

The latter expression can be transformed by means of the relations derived, and it follows:

RA=p sin . EpDp(Es[Ep)(ks/kp−1),

=p sino. E, D,(D./Dp)(1—kp/ks).

For the magnetic case correspondingly we have

The magnitude of the radiation is therefore seen to depend on the relative strength of both fields stationary and progressive at any point. The above separate developments be it noted are quite independent of the two curl equations of Maxwell.

The Differential Equation of Maxwellian Radiation.—If the developed flux coefficients are introduced into the expressions for the two displacement expressions, we have

Substituting the latter in the fundamental curl equations of Maxwell, we really have that

S μηλ

curl E=curl (Es+Ep)=-dB/dt=-d/dt (μ2Hs+μ2Hr), curl II=curl (Is+Hp)=d|dt(Ds+Dp).

Assuming, for example, that the curl (line integral per unit of area) of dP/dt can be separated out into its two components independently, we have

curl Hs=dDs/dt; curl IIp=dDp/dt,

curl Es=dDs/dt; curl Ep=- dBp/dt.

In this manner, therefore, the following two resultant

Relation of Dielectric Constants to Index of Refraction. It is a well-known fact from experiments with light waves for determining the index of refraction that the dielectric constant k always comes out smaller than when determined by the charged condenser method. Since light is a radiation phenomenon, and on the assumption of Maxwell it is electromagnetic, the value of the light velocity e is given by

Yet for the electrical condenser experiments, where radiation effects are not emphasized, to say the least, it is not proper to take the right-hand term as 1/sks. The relative value of the refractive index i should be given by

The work of M. C. Gulton, Comptes Rendus, 1900, cxxx, p. 119, is therefore of exceptional interest. He found that the electrical refractive index for ice, for example, decreased progressively with larger and larger wave lengths, approaching more and more the value found by Fleming for a dry prism of ice when subjected to Hertzian waves. Quite a large table of values of 2 and really k, are given in Fleming's Principles of Wireless Telegraphy.'

Finally, it must be said that since the self-induction coefficient L, and the capacity coefficient C are dependent on the effective, and k, values, our conceptions of the constants are naturally altered in the light of the above inquiries. As for the power factor of an oscillating system, it is seen it must be always less than 50 per cent.

Chevy Chase, Md., U.S.A.

June 7, 1924.