Page:A Dynamical Theory of the Electromagnetic Field.pdf/27

Electric Elasticity.

(66) When an electromotive force acts on a dielectric, it puts every part of the dielectric into a polarized condition, in which its opposite sides are oppositely electrified. The amount of this electrification depends on the electromotive force and on the nature of the substance, and, in solids having a structure defined by axes, on the direction of the electromotive force with respect to these axes. In isotropic substances, if $k$ is the ratio of the electromotive force to the electric displacement, we may write the

Equations of Electric Elasticity,

$\left.{\begin{array}{l}P=kf,\\Q=kg,\\R=kh.\end{array}}\right\}$

(E)

Electric Resistance.

(67) When an electromotive force acts on a conductor it produces a current of electricity through it. This effect is additional to the electric displacement already considered. In solids of complex structure, the relation between the electromotive force and the current depends on their direction through the solid. In isotropic substances, which alone we shall here consider, if $\rho$ is the specific resistance referred to unit of volume, we may write the

Equations of Electric Resistance,

$\left.{\begin{array}{l}P=-\rho p,\\Q=-\rho q,\\R=-\rho r.\end{array}}\right\}$

(F)

Electric Quantity.

(68) Let $e$ represent the quantity of free positive electricity contained in unit of volume at any part of the field, then, since this arises from the electrification of the different parts of the field not neutralizing each other, we may write the

Equation of Free Electricity,

$e+{\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}=0$

(G)

(69) If the medium conducts electricity, then we shall have another condition, which may be called, as in hydrodynamics, the

Equation of Continuity,

${\frac {de}{dt}}+{\frac {dp}{dx}}+{\frac {dq}{dy}}+{\frac {dr}{dz}}=0$

(H)

(70) In these equations of the electromagnetic field we have assumed twenty variable