Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/265

electromagnetic phenomena depend, is not the same thing as ${\mathfrak {K}}$, the current of conduction, but that the time-variation of ${\mathfrak {D}}$, the electric displacement, must be taken into account in estimating the total movement of electricity, so that we must write,

${\mathfrak {C}}={\mathfrak {K}}+{\dot {\mathfrak {D}}},\quad {\text{(Equation of True Currents.)}}$

(H)

or, in terms of the components,

$\left.{\begin{aligned}u=p+{\frac {df}{dt}},\\v=q+{\frac {dg}{dt}},\\w=r+{\frac {dh}{dt}}.\end{aligned}}\right\}$

(H*)

611.] Since both ${\mathfrak {K}}$ and ${\mathfrak {D}}$ depend on the electromotive force ${\mathfrak {E}}$, we may express the true current & in terms of the electromotive force, thus

${\mathfrak {C}}=\left(C+{\frac {1}{4\pi }}K{\frac {d}{dt}}\right){\mathfrak {E}},$

(I)

or, in the case in which C and K are constants,

$\left.{\begin{aligned}u=CP+{\frac {1}{4\pi }}K{\frac {dP}{dt}},\\v=CQ+{\frac {1}{4\pi }}K{\frac {dQ}{dt}},\\w=CR+{\frac {1}{4\pi }}K{\frac {dR}{dt}}.\end{aligned}}\right\}$

(I*)

612.] The volume-density of the free electricity at any point is found from the components of electric displacement by the equation

$\rho ={\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}.$

(J)

613.] The surface-density of electricity is

$\sigma =lf+mg+nh+l'f'+m'g'+n'h',\,$

(K)

where l, m, n are the direction-cosines of the normal drawn from the surface into the medium in which f, g, h are the components of the displacement, and l', m', n' are those of the normal drawn from the surface into the medium in which they are f', g', h'.

614.] When the magnetization of the medium is entirely induced by the magnetic force acting on it, we may write the equation of induced magnetization,

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

(L)

where μ is the coefficient of magnetic permeability, which may be considered a scalar quantity, or a linear and vector function operating on ${\mathfrak {H}}$, according as the medium is isotropic or not.