Page:A Dynamical Theory of the Electromagnetic Field.pdf/12

conductor. When $M$ is positive, the induced current due to increase of the primary current is negative.

Induction of Motion by a Conductor.

(30) Case 2nd. Let $x$ remain constant, and let $M$ change from $M$ to $M'$, then

$Y=-{\frac {M'-M}{S}}x;$

(7)

so that if $M$ is increased, which it will be by the primary and secondary circuits approaching each other, there will be a negative induced current, the total quantity of electricity passed through $B$ being $Y$.

This is induction by the relative motion of the primary and secondary conductors.

Equation of Work and Energy.

(31) To form the equation between work done and energy produced, multiply (1) by $x$ and (2) by $y$, and add

$\xi x+\eta y=Rx^{2}+Sy^{2}+x{\frac {d}{dt}}\left({Lx+My}\right)+y{\frac {d}{dt}}(Mx+My)$

(8)

Here $\xi x$ is the work done in unit of time by the electromotive force $\xi$ acting on the current $x$ and maintaining it, and $\eta y$ is the work done by the electromtoive force $\eta$. Hence the left-hand side of the equation represents the work done by the electromotive forces in unit of time.

Heat produced by the Current.

(32) On the other side of the equation we have, first,

$Rx^{2}+Sy^{2}=H,$

(9)

which represents the work done in overcoming the resistance of the circuits in unit of time. This is converted into heat. The remaining terms represent work not converted into heat. They may be written

${\textstyle {1 \over 2}}{\frac {d}{dt}}\left({Lx^{2}+2Mxy+Ny^{2}}\right)+{\textstyle {1 \over 2}}{\frac {dL}{dt}}x^{2}+{\frac {dM}{dt}}xy+{\textstyle {1 \over 2}}{\frac {dN}{dt}}y^{2}.$

Intrinsic Energy of the Currents.

(33) If $L$,$M$,$N$ are constant, the whole work of the electromotive forces which is not spent against resistance will be devoted to the development of the currents. The whole intrinsic energy of the currents is therefore

${\textstyle {1 \over 2}}Lx^{2}+Mxy+{\textstyle {1 \over 2}}Ny^{2}=E.$

(10)

This energy exists in a form imperceptible to our senses, probably as actual motion, the seat of this motion being not merely the conducting circuits, but the space surrounding them.