from which equation it is evident that represents the electrostatic potential.
The principle which is peculiar to Maxwell's theory must now be introduced. Currents of conduction are not the only kind of currents; even in the older theory of Faraday, Thomson, and Mossotti, it had been assumed that electric charges are set in motion in the particles of a dielectric when the dielectric is subjected to an electric field; and the predecessors of Maxwell would not have refused to admit that the motion of these charges is in some sense a current. Suppose, then, that S denotes the total current which is capable of generating a magnetic field: since the integral of the magnetic force round any curve is proportional to the electric current which flows through the gap enclosed by the curve, we have in suitable units curl H = 4πS. In order to determine S, we may consider the case of a condenser whose coatings are supplied with electricity by a conduction-current ι per unit-area of coating. If ± σ denote the surface-density of electric charge on the coatings, we have
i = ∂σ/∂t, and σ = D,
where D denotes the magnitude of the electric displacement D in the dielectric between the coatings; so ι = Ḋ. But since the total current is to be circuital, its value in the dielectric must be the same as the value ι which it has in the rest of the circuit; that is, the current in the dielectric has the value Ḋ. We shall assume that the current in dielectrics always has this value, so that in the general equations the total current must be understood to be ι + Ḋ.
The above equations, together with those which express the proportionality of E to D in insulators, and to ι in conductors, constituted Maxwell's system for a field formed by isotropic bodies which are not in motion. When the magnetic field is due entirely to currents (including both conduction-currents
