Page:A Dynamical Theory of the Electromagnetic Field.pdf/11

$R$ the resistance, then $Rx$ will be the resisting force. In steady currents the electromotive force just balances the resisting force, but in variable currents the resultant force $\xi =Rx$ is expanded in increasing the “electromagnetic momentum”, using the word momentum merely to express that which is generated by a force acting during a time, that is, a velocity existing in a body.

In the case of electric currents, the force in action is not ordinary mechanical force, at least we are not as yet able to measure it as common force, but we call it electromotive force, and the body moved is not merely the electricity in the conductor, but something outside the conductor, and capable of being affected by other conductors in the neighbourhood carrying currents. In this it resembles rather the reduced momentum of a driving-point of a machine as influenced by its mechanical connections, than that of a simple moving body like a cannon ball, or water in a tube.

Electromagnetic Relations of two Conducting Circuits

(28) In the case of two conducting circuits, $A$ and $B$, we shall assume that the electromagnetic momentum belonging to $A$ is

$Lx+My,$

and that belonging to $B$,

$Mx+Ny,$

where $L$, $M$, $N$ correspond to the same quantities in the dynamical illustration, except that they are supposed to be capable of variation when the conductors $A$ or $B$ are moved.

Then the equation of the current $x$ in $A$ will be

$\xi =Rx+{\frac {d}{dt}}\left({Lx+My}\right),$

(4)

and that of $y$ in $B$

$\eta =Sy+{\frac {d}{dt}}(Mx+Ny),$

(5)

where $\xi$ and $\eta$ are the electromotive forces, $x$ and $y$ the currents, and $R$ and $S$ the resistances in $A$ and $B$ respectively.

Induction of one Current by another.

(29) Case 1st. Let there be no electromotive force on $B$, except that which arises from the action of A, and let the current of $A$ increase from 0 to the value $x$, then

$Sy+{\frac {d}{dt}}\left({Mx+Ny}\right)=0,$

whence

$Y=\int _{0}^{t}{ydt=-{\frac {M}{S}}}x,$

(6)

that is, a quantity of electricity $Y$, being the total induced current, will flow through $B$ when $x$ rises from 0 to $x$. This is induction by variation of the current in the primary