Page:A Dynamical Theory of the Electromagnetic Field.pdf/30

it resides in the electrified bodies, conducting circuits, and magnets, in the form of an unknown quality called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and is in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis, as the motion and the strain of one and the same medium.

(75) The conclusions arrived at in the present paper are independent of this hypothesis, being deduced from experimental facts of three kinds:—

1. The induction of electric currents by the increase or diminution of neighbouring currents according to the changes in the lines of force passing through the circuit.

2. The distribution of magnetic intensity according to the variations of a magnetic potential.

3. The induction (or influence) of statical electricity through dielectrics.

We may now proceed to demonstrate from these principles the existence and laws of the mechanical forces which act upon electric currents, magnets, and electrified bodies placed in the electromagnetic field.

PART IV.— MECHANICAL ACTIONS IN THE FIELD.

Mechanical Force on a Moveable Conductor.

(76) We have shown (§§ 34 & 35) that the work done by the electromagnetic forces in aiding the motion of a conductor is equal to the product of the current in the conductor multiplied by the increment of the electromagnetic momentum due to the motion.

Let a short straight conductor of length $a$ move parallel to itself in the direction of $x$, with its extremities on two parallel conductors. Then the increment of the electromagnetic momentum due to the motion of $a$ will be

$a\left({\frac {dF}{dx}}{\frac {dx}{ds}}+{\frac {dG}{dx}}{\frac {dy}{ds}}+{\frac {dH}{dx}}{\frac {dz}{ds}}\right)\delta x$

That due to the lengthening of the circuit by increasing the length of the parallel conductors will be

$-a\left({\frac {dF}{dx}}{\frac {dx}{ds}}+{\frac {dG}{dy}}{\frac {dy}{ds}}+{\frac {dH}{dz}}{\frac {dz}{ds}}\right)\delta x$

The total increment is

$a\delta x\left\{{\frac {dy}{ds}}\left({\frac {dG}{dx}}-{\frac {dF}{dy}}\right)-{\frac {dz}{ds}}\left({\frac {dF}{dz}}-{\frac {dH}{dx}}\right)\right\}$

which is by the equations of Magnetic Force (B), p. 482,

$a\delta x\left({\frac {dy}{ds}}\mu \gamma -{\frac {dz}{ds}}\mu \beta \right)$

Let X be the force acting along the direction of $x$ per unit of length of the conductor, then the work done is $Xa\delta x$.