Page:A Dynamical Theory of the Electromagnetic Field.pdf/22

will be limited sheets, terminating in the electric circuit as their common edge or boundary. The number of these will be equal to the amount of work done on a unit pole in going round the current, and this by the ordinary measurement $=4\pi \gamma$, where $\gamma$ is the value of the current.

These surfaces, therefore, are connected with the electric current as soap-bubbles are connected with a ring in M. Plateau's experiments. Every current $\gamma$ has $4\pi \gamma$ surfaces attached to it. These surfaces have the current for their common edge, and meet it at equal angles. The form of the surfaces in other parts depends on the presence of other currents and magnets, as well on the shape of the circuit to which they belong.

PART III. – GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD.

(53) Let us assume three rectangular directions in space as the axes of $x,y$, and $z$, and let all quantities having direction be expressed by their components in these three directions.

Electrical Currents (p, q, r).

(54) An electrical current consists in the transmission of electricity from one part of a body to another. Let the quantity of electricity transmitted in unit of time across unit of area perpendicular to the axis of $x$ be called $p$, then $p$ is the component of the current at that place in the direction of $x$.

We shall use the letters $p,q,r$ to denote the components of the current per unit of area in the directions of $x,y,z$.

Electrical Displacements (f, g, h).

(55) Electrical displacement consists in the opposite electrification of the sides of a molecule or particle of a body which may or may not be accompanied with transmission through the body. Let the quantity of electricity which would appear on the faces $dy.dz$ of an element $dx,dy,dz$ cut from the body be $f.dy.dz$, then $f$ is the component of electric displacement parallel to $x$. We shall use $f,g,h$ to denote the electric displacements parallel to $x,y,z$ respectively.

The variations of the electrical displacement must be added to the currents $p,q,r$ to get the total motion of electricity, which we may call $p',q',r'$, so that

$\left.{\begin{array}{l}p'=p+{\frac {df}{dt}},\\\\q'=q+{\frac {dg}{dt}},\\\\r'=r+{\frac {dh}{dt}},\end{array}}\right\}$

(A)

Electromotive Force (P, Q, R).

(56) Let P, Q, R represent the components of the electromotive force at any point. Then P represents the difference of potential per unit of length in a conductor