Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/416

CHAPTER X.

CONDUCTION IN DIELECTRICS.

325.] We have seen that when electromotive force acts on a dielectric medium it produces in it a state which we have called electric polarization, and which we have described as consisting of electric displacement within the medium in a direction which, in isotropic media,, coincides with that of the electromotive force, combined with a superficial charge on every element of volume into which we may suppose the dielectric divided, which is negative on the side towards which the force acts, and positive on the side from which it acts.

When electromotive force acts on a conducting medium it also produces what is called an electric current.

Now dielectric media, with very few, if any, exceptions, are also more or less imperfect conductors, and many media which are not good insulators exhibit phenomena of dielectric induction. Hence we are led to study the state of a medium in which induction and conduction are going on at the same time.

For simplicity we shall suppose the medium isotropic at every point, but not necessarily homogeneous at different points. In this case, the equation of Poisson becomes, by Art. 83,

${\frac {d}{dx}}\left(K{\frac {dV}{dx}}\right)+{\frac {d}{dy}}\left(K{\frac {dV}{dy}}\right)+{\frac {d}{dz}}\left(K{\frac {dV}{dz}}\right)+4\pi \rho =0$

(1)

where $K$ is the 'specific inductive capacity.'

The 'equation of continuity' of electric currents becomes

${\frac {d}{dx}}\left({\frac {1}{r}}{\frac {dV}{dx}}\right)+{\frac {d}{dy}}\left({\frac {1}{r}}{\frac {dV}{dy}}\right)+{\frac {d}{dz}}\left({\frac {1}{r}}{\frac {dV}{dz}}\right)-{\frac {d\rho }{dt}}=0$

(2)

where $r$ is the specific resistance referred to unit of volume.

When $K$ or $r$ is discontinuous, these equations must be transformed into those appropriate to surfaces of discontinuity.