Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/280

Electrokinetic Energy.

634.] We have already, in Art. 578, expressed the kinetic energy

of a system of currents in the form.

where $p$ is the electromagnetic momentum of a circuit, and $i$ is the strength of the current flowing round it, and the, summation extends to all the circuits. But we have proved, in Art. 590, that $p$ may be expressed as a line-integral of the form

where $F$, $G$, $H$ are the components of the electromagnetic momentum, $A$, at the point ($x\,y\,z$) and the integration is to be extended round the closed circuit $s$. We therefore find

If $u$, $v$, $w$ are the components of the density of the current at any point of the conducting circuit, and if $S$ is the transverse section of the circuit, then we may write

and we may also write the volume

and we now find

where the integration is to be extended to every part of space where there are electric currents.

635.] Let us now substitute for $u$, $v$, $w$ their values as given by the equations of electric currents (E), Art. 607, in terms of the components $\alpha$, $\beta$, $\gamma$ of the magnetic force. We then have

where the integration is extended over a portion of space including all the currents.

If we integrate this by parts, and remember that, at a great distance $r$ from the system, $\alpha$, $\beta$, and $\gamma$ are of the order of magnitude $r^{-3}$ , we find that when the integration is extended throughout all space, the expression is reduced to