Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/279

Hence, the electrostatic energy of the whole field will be the same if we suppose that it resides in every part of the field where electrical force and electrical displacement occur, instead of being confined to the places where free electricity is found.

The energy in unit of volume is half the product of the electromotive force and the electric displacement, multiplied by the cosine of the angle which these vectors include.

In Quaternion language it is $-{\frac {1}{2}}S{\mathfrak {E}}{\mathfrak {D}}$.

Magnetic Energy.

632.] We may treat the energy due to magnetization in a similar way. If $A$, $B$, $C$ are the components of magnetization and $\alpha$, $\beta$, $\gamma$ the components of magnetic force, the potential energy of the system of magnets is, by Art. 389,

the integration being extended over the space occupied by magnetized matter. This part of the energy, however, will be included in the kinetic energy in the form in which we shall presently obtain it.

633.] We may transform this expression when there are no electric currents by the following method.

We know that

Hence, by Art. 97, if

as is always the case in magnetic phenomena where there are no currents,

the integral being extended throughout all space, or

Hence, the energy due to a magnetic system

$-{\frac {1}{2}}\iiint (A\alpha +B\beta +C\gamma )\,dx\,dy\,dz$	${}={\frac {1}{8\pi }}\iiint (\alpha ^{2}+\beta ^{2}+\gamma ^{2})\,dx\,dy\,dz$,
	${}=-{\frac {1}{8\pi }}\iiint {\mathfrak {H}}^{2}\,dx\,dy\,dz$.

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