Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/263

If we take for the closed curve the parallelogram whose sides are dy and dz, the line-integral of the magnetic force round the parallelogram is

$\left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right)dy\,dz,$

and if u, v, w are the components of the flow of electricity, the current through the parallelogram is

$u\,dy\,dz.$

Multiplying this by 4π, and equating the result to the line- integral, we obtain the equation

$4\pi u={\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}$

with the similar equations

$\left.{\begin{aligned}&4\pi v={\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}},\\&4\pi w={\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}.\\\end{aligned}}\right\}{\begin{matrix}{\text{(Equations of}}\\{\text{Electric Currents.)}}\end{matrix}}$

(E)

which determine the magnitude and direction of the electric currents when the magnetic force at every point is given.

When there is no current, these equations are equivalent to the condition that

$\alpha dx+\beta dy+\gamma dz=-D\Omega ,\,$

or that the magnetic force is derivable from a magnetic potential in all points of the field where there are no currents.

By differentiating the equations (E) with respect to x, y, and z respectively, and adding the results, we obtain the equation

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0,$

which indicates that the current whose components are u, v, w is subject to the condition of motion of an incompressible fluid, and that it must necessarily flow in closed circuits.

This equation is true only if we take u, v and w as the components of that electric flow which is due to the variation of electric displacement as well as to true conduction.

We have very little experimental evidence relating to the direct electromagnetic action of currents due to the variation of electric displacement in dielectrics, but the extreme difficulty of reconciling the laws of electromagnetism with the existence of electric currents which are not closed is one reason among many why we must admit the existence of transient currents due to the variation of displacement. Their importance will be seen when we come to the electro-magnetic theory of light.