# Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/264

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[608.
GENERAL EQUATIONS.

608.] We have now determined the relations of the principal quantities concerned in the phenomena discovered by Örsted, Ampère, and Faraday. To connect these with the phenomena described in the former parts of this treatise, some additional relations are necessary.

When electromotive force acts on a material body, it produces in it two electrical effects, called by Faraday Induction and Conduction, the first being most conspicuous in dielectrics, and the second in conductors.

In this treatise, static electric induction is measured by what we have called the electric displacement, a directed quantity or vector which we have denoted by $\mathfrak{D}$, and its components by f, g, h.

In isotropic substances, the displacement is in the same direction as the electromotive force which produces it, and is proportional to it, at least for small values of this force. This may be expressed by the equation

 $\mathfrak{D} = \frac{1}{4\pi} K \mathfrak{E}, \quad \begin{matrix} \text{(Equation of Electric}\\ \text{Displacement.)} \end{matrix}$ (F)

where K is the dielectric capacity of the substance. See Art. 69.

In substances which are not isotropic, the components f, g, h of the electric displacement $\mathfrak{D}$ are linear functions of the components P, Q, R of the electromotive force $\mathfrak{E}$, .

The form of the equations of electric displacement is similar to that of the equations of conduction as given in Art. 298.

These relations may be expressed by saying that K is, in isotropic bodies, a scalar quantity, but in other bodies it is a linear and vector function, operating on the vector $\mathfrak{E}$, .

609.] The other effect of electromotive force is conduction. The laws of conduction as the result of electromotive force were established by Ohm, and are explained in the second part of this treatise, Art. 241. They may be summed up in the equation

 $\mathfrak{K} = C \mathfrak{E}, \quad \text{Equation of Conductivity.)}$ (G)

where $\mathfrak{E}$ is the intensity of the electromotive force at the point, $\mathfrak{K}$ is the density of the current of conduction, the components of which are p, q, r, and C is the conductivity of the substance, which, in the case of isotropic substances, is a simple scalar quantity, but in other substances becomes a linear and vector function operating on the vector $\mathfrak{E}$. The form of this function is given in Cartesian coordinates in Art. 298.

610.] One of the chief peculiarities of this treatise is the doctrine which it asserts, that the true electric current $\mathfrak{C}$, that on which the