Page:A Dynamical Theory of the Electromagnetic Field.pdf/29

where $\alpha ,\beta ,\gamma$ are the components of magnetic intensity or the force on a unit magnetic pole, and $\mu \alpha ,\mu \beta ,\mu \gamma$ are the components of the quantity of magnetic induction, or the number of lines of force in unit of area.

In isotropic media the value of $\mu$ is the same in all directions, and we may express the result more simply by saying that the intrinsic energy of any part of the magnetic field arising from its magnetization is

${\frac {\mu }{8\pi }}I^{2}$

per unit of volume, where I is the magnetic intensity.

(72) Energy may be stored up in the field in a different way, namely, by the action of electromotive force in producing electric displacement. The work done by a variable electromotive force, P, in producing a variable displacement, $f$, is got by integrating

$\int Pdf$

from P=0 to the given value of P.

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikisource.org/v1/":): {\displaystyle P=kf} , equation (E), this quantity becomes

$\int kfdf={\frac {1}{2}}kf^{2}={\frac {1}{2}}Pf$

Hence the intrinsic energy of any part of the field, as existing in the form of electric displacement, is

${\frac {1}{2}}\sum (Pf+Qg+Rh)dV$

The total energy existing in the field is therefore

$E=\sum \left\{{\frac {1}{8\pi }}(\alpha \mu \alpha +\beta \mu \beta +\gamma \mu \gamma )+{\frac {1}{2}}(Pf+Qg+Rh)\right\}dV$

(I)

The first term of this expression depends on the magnetization of the field, and is explained on our theory by actual motion of some kind. The second term depends on the electric polarization of the field, and is explained on our theory by strain of some kind in an elastic medium.

(73) I have on a former occasion^[1] attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory.

(74) In speaking of the Energy of the field, however, I wish to be understood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories

1. "On Physical Lines of Force," Philosophical Magazine, 1861-62.