Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/254

The electromotive force at a point, or on a particle, must be carefully distinguished from the electromotive force along- an arc of a curve, the latter quantity being- the line-integral of the former. See Art. 69.

599.] The electromotive force, the components of which are defined by equations (B), depends on three circumstances. The first of these is the motion of the particle through the magnetic field. The part of the force depending on this motion is expressed by the first two terms on the right of each equation. It depends on the velocity of the particle transverse to the lines of magnetic induction. If ${\mathfrak {G}}$ is a vector representing the velocity, and ${\mathfrak {B}}$ another representing the magnetic induction, then if ${\mathfrak {F}}_{1}$ is the part of the electromotive force depending on the motion,

${\mathfrak {F}}_{1}=V\cdot {\mathfrak {G\,B}},$

(7)

or, the electromotive force is the vector part of the product of the magnetic induction, multiplied by the velocity, that is to say, the magnitude of the electromotive force is represented by the area of the parallelogram, whose sides represent the velocity and the magnetic induction, and its direction is the normal to this parallelogram, drawn so that the velocity, the magnetic induction, and the electromotive force are in right-handed cyclical order.

The third term in each of the equations (B) depends on the time-variation of the magnetic field. This may be due either to the time-variation of the electric current in the primary circuit, or to motion of the primary circuit. Let ${\mathfrak {E}}_{2}$ be the part of the electro motive force which depends on these terms. Its components are

$-{\frac {dF}{dt}},\quad -{\frac {dG}{dt}},{\text{ and }}-{\frac {dH}{dt}},$

and these are the components of the vector, $-{\frac {d{\mathfrak {A}}}{dt}}$ or $-{\dot {\mathfrak {A}}}$.^[1] Hence,

${\mathfrak {E}}_{2}=-{\dot {\mathfrak {A}}},$

(8)

The last term of each equation (B) is due to the variation of the function $\Psi$ in different parts of the field. We may write the third part of the electromotive force, which is due to this cause,

${\mathfrak {E}}_{3}=-\Delta \Psi ,\,$

(9)

The electromotive force, as defined by equations (B), may therefore be written in the quaternion form,

${\mathfrak {E}}=V\cdot {\mathfrak {GB}}-{\dot {\mathfrak {A}}}-\Delta \Psi .$

(10)

1. Implementing Errata from Page 1 of this volume.