Page:A Dynamical Theory of the Electromagnetic Field.pdf/24

(60) Let $\mu$ be the ratio of the magnetic induction in a given medium to that in air under an equal magnetizing force, then the number of lines of force in unit of area perpendicular to $x$ will be $\mu \alpha$ ($\mu$ is a quantity depending on the nature of the medium, its temperature, the amount of magnetization already produced, and in crystalline bodies varying with the direction).

(61) Expressing the electric momentum of small circuits perpendicular to the three axes in this notation, we obtain the following

Equations of Magnetic Force.

$\left.{\begin{array}{l}\mu \alpha ={\frac {dH}{dy}}-{\frac {dG}{dz}},\\\\\mu \beta ={\frac {dF}{dz}}-{\frac {dH}{dx}},\\\\\mu \gamma ={\frac {dG}{dx}}-{\frac {dF}{dy}}.\end{array}}\right\}$

(B)

Equations of Currents.

(62) It is known from experiment that the motion of a magnetic pole in the electromagnetic field in a closed circuit cannot generate work unless the circuit which the pole describes passes round an electric current. Hence, except in the space occupied by the electric currents,

$\alpha dx+\beta dy+\gamma dz=d\varphi$

(31)

a complete differential of $\varphi$, the magnetic potential.

The quantity $\varphi$ may be susceptible of an indefinite number of distinct values, according to the number of times that the exploring point passes round electric currents in its course, the difference between successive values of $\varphi$ corresponding to a passage completely round a current of strength $c$ being $4\pi c$.

Hence if there is no electric current,

${\ \atop dy}-{\frac {d\beta }{dz}}=0$

but if there is a current $p'$,

$\left.{\begin{array}{lc}&{\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}=4\pi p'.\\\mathrm {Similarly} ,\\&{\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}=4\pi q',\\\\&{\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}=4\pi r'.\end{array}}\right\}$

(C)

We may call these the Equations of Currents.