Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/268

CHAPTER XII.

THEORY OF CONJUGATE FUNCTIONS IN TWO DIMENSIONS.

182.] The number of independent cases in which the problem of electrical equilibrium has been solved is very small. The method of spherical harmonics has been employed for spherical conductors, and the methods of electrical images and of inversion are still more powerful in the cases to which they can be applied. The case of surfaces of the second degree is the only one, as far as I know, in which both the equipotential surfaces and the lines of force are known when the lines of force are not plane curves.

But there is an important class of problems in the theory of electrical equilibrium, and in that of the conduction of currents, in which we have to consider space of two dimensions only.

For instance, if throughout the part of the electric field under consideration, and for a considerable distance beyond it, the surfaces of all the conductors are generated by the motion of straight lines parallel to the axis of $z$, and if the part of the field where this ceases to be the case is so far from the part considered that the electrical action of the distant part on the field may be neglected, then the electricity will be uniformly distributed along each gene rating line, and if we consider a part of the field bounded by two planes perpendicular to the axis of $z$ and at distance unity, the potential and the distribution of electricity will be functions of $x$ and $y$ only.

If $\rho \ dx\ dy$ denotes the quantity of electricity in an element whose base is $dx\ dy$ and height unity, and $\rho \ ds$ the quantity on an element of area whose base is the linear element $ds$ and height unity, then the equation of Poisson may be written

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+4\pi \rho =0$.