Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/253

We may write this expression in the form

$E=\int {\left(P{\frac {dx}{ds}}+Q{\frac {dy}{ds}}+R{\frac {dz}{ds}}\right)ds},$

(5)

where

$\left.{\begin{aligned}P&=c{\frac {dy}{dt}}-b{\frac {dz}{dt}}-{\frac {dF}{dt}}-{\frac {d\Psi }{dt}},\\Q&=a{\frac {dz}{dt}}-c{\frac {dx}{dt}}-{\frac {dG}{dt}}-{\frac {d\Psi }{dt}},\\R&=b{\frac {dx}{dt}}-a{\frac {dy}{dt}}-{\frac {dH}{dt}}-{\frac {d\Psi }{dt}}.\end{aligned}}\right\}{\begin{matrix}{\text{Equations of }}\\{\text{Electromotive}}\\{\text{Force.}}\end{matrix}}$

(B)

The terms involving the new quantity $\Psi$ are introduced for the sake of giving generality to the expressions for $P$, $Q$, $R$. They disappear from the integral when extended round the closed circuit. The quantity $\Psi$ is therefore indeterminate as far as regards the problem now before us, in which the total electromotive force round the circuit is to be determined. We shall find, however, that when we know all the circumstances of the problem, we can assign a definite value to $\Psi$, and that it represents, according to a certain definition, the electric potential at the point $x,y,z$.

The quantity under the integral sign in equation (5) represents the electromotive force acting on the element $ds$ of the circuit.

If we denote by $T{\mathfrak {F}}$, the numerical value of the resultant of P, Q, and R, and by ε, the angle between the direction of this resultant and that of the element ds, we may write equation (5),

$E=\int {T{\mathfrak {F}}\cos \epsilon \,ds}.$

(6)

The vector $T{\mathfrak {F}}$ is the electromotive force at the moving element ds. Its direction and magnitude depend on the position and motion of ds, and on the variation of the magnetic field, but not on the direction of ds. Hence we may now disregard the circumstance that ds forms part of a circuit, and consider it simply as a portion of a moving body, acted on by the electromotive force $T{\mathfrak {F}}$. The electromotive force at a point has already been defined in Art. 68. It is also called the resultant electrical force, being the force which would be experienced by a unit of positive electricity placed at that point. We have now obtained the most general value of this quantity in the case of a body moving in a magnetic field due to a variable electric system.

If the body is a conductor, the electromotive force will produce a current; if it is a dielectric, the electromotive force will produce only electric displacement.