Page:A Dynamical Theory of the Electromagnetic Field.pdf/13

Mechanical Action between Conductors.

(34) The remaining terms,

${\textstyle {1 \over 2}}{\frac {dL}{dt}}x^{2}+{\frac {dM}{dt}}xy+{\textstyle {1 \over 2}}{\frac {dN}{dt}}y^{2}=W$

(11)

represent the work done in unit of time arising from the variations of $L$, $M$, and $N$, or, what is the same thing, alterations in the form and position of the conducting circuits $A$ and $B$.

Now if work is done when a body is moved, it must arise from ordinary mechanical force acting on the body while it is moved. Hence this part of the expression shows that there is a mechanical force urging every part of the conductors themselves in that direction in which $L$, $M$, and $N$ will be most increased.

The existence of the electromagnetic force between conductors carrying currents is therefore a direct consequence of the joint and independent action of each current on the electromagnetic field. If $A$ and $B$ are allowed to approach a distance $ds$, so as to increase $M$ from $M$ to $M'$ while the currents are $x$ and $y$, then the work done will be

$\left({M'-M}\right)xy,$

and the force in the direction of $ds$ will be

${\frac {dM}{ds}}xy,$

(12)

and this will be an attraction if $x$ and $y$ are of the same sign, and if $M$ is increased as $A$ and $B$ approach.

It appears, therefore, that if we admit that the unresisted part of electromotive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromagnetic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electromagnetic attractions may be proved by mechanical reasoning.

What I have called electromagnetic momentum is the same quantity which is called by Faraday^[1] the electrotonic state of the circuit, every change of which involves the action of an electromotive force, just as change of momentum involves the action of mechanical force.

If, therefore, the phenomena described by Faraday in the ninth Series of his Experimental Researches were the only known facts about electric currents, the laws of Ampère relating to the attraction of conductors carrying currents, as well as those of Faraday about the mutual induction of currents, might be deduced by mechanical reasoning.

In order to bring these results within the range of experimental verification, I shall next investigate the case of a single current, of two currents, and of the six currents in the electric balance, so as to enable the experimenter to determine the values of $L$, $M$, $N$.

1. Experimental Researches, Series I. 60, &c