Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/126

On Specific Inductive Capacity.

83.] In the preceding investigation of surface-integrals I have adopted the ordinary conception of direct action at a distance, and have not taken into consideration any effects depending on the nature of the dielectric medium in which the forces are observed.

But Faraday has observed that the quantity of electricity induced by a given electromotive force on the surface of a conductor which bounds a dielectric is not the same for all dielectrics. The induced electricity is greater for most solid and liquid dielectrics than for air and gases. Hence these bodies are said to have a greater specific inductive capacity than air, which is the standard medium.

We may express the theory of Faraday in mathematical language by saying that in a dielectric medium the induction across any surface is the product of the normal electric force into the coefficient of specific inductive capacity of that medium. If we denote this coefficient by $K$, then in every part of the investigation of surface-integrals we must multiply $X$, $Y$, and $Z$ by $K$, so that the equation of Poisson will become

${\frac {d}{dx}}\cdot \,K{\frac {dV}{dx}}+{\frac {d}{dy}}\cdot \,K{\frac {dV}{dy}}+{\frac {d}{dz}}\cdot \,K{\frac {dV}{dz}}+4\pi \,\rho =0$.

At the surface of separation of two media whose inductive capacities are $K_{1}$ and $K_{2}$ , and in which the potentials are $V_{1}$ and $V_{2}$ the characteristic equation may be written

$K_{2}{\frac {dV_{2}}{dv}}-K_{1}{\frac {dV_{1}}{dv}}+4\pi \,\rho =0$;

where $v$ is the normal drawn from the first medium to the second, and $\sigma$ is the true surface-density on the surface of separation; that is to say, the quantity of electricity which is actually on the surface in the form of a charge, and which can be altered only by conveying electricity to or from the spot. This true electrification must be distinguished from the apparent electrification $\rho '$, which is the electrification as deduced from the electrical forces in the neighbourhood of the surface, using the ordinary characteristic equation

${\frac {dV_{2}}{dv}}-{\frac {dV_{1}}{dv}}+4\pi \,\rho '=0$.

If a solid dielectric of any form is a perfect insulator, and if its surface receives no charge, then the true electrification remains zero, whatever be the electrical forces acting on it.