Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/270

The equations (D) of magnetization, of which the first is

$a=\alpha +4\pi A,\,$

become

${\mathfrak {B}}={\mathfrak {H}}+4\pi {\mathfrak {J}}.\,$

The equations (E) of electric currents, of which the first is

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

become

$4\pi {\mathfrak {G}}=V\nabla {\mathfrak {H}}.\,$

The equation of the current of conduction is, by Ohm's Law,

${\mathfrak {K}}=C{\mathfrak {G}}.\,$

That of electric displacement is

${\mathfrak {K}}={\frac {1}{4\pi }}K{\mathfrak {G}}.\,$

The equation of the total current, arising from the variation of the electric displacement as well as from conduction, is

${\mathfrak {C}}={\mathfrak {K}}+{\dot {\mathfrak {D}}}.\,$

When the magnetization arises from magnetic induction,

${\mathfrak {B}}=\mu {\mathfrak {H}}.\,$

We have also, to determine the electric volume-density,

$e=S\nabla {\mathfrak {D}}.\,$

To determine the magnetic volume-density,

$m=S\nabla {\mathfrak {J}}.\,$

When the magnetic force can be derived from a potential

${\mathfrak {H}}=-\nabla \Omega .\,$