Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/91

For a nearly solid sphere with a very small spherical hollow,

${\begin{aligned}A_{1}&=-{\frac {2(4\pi \kappa )^{2}}{(3+4\pi \kappa )(3+8\pi \kappa )}}C_{1},\\A_{2}&=-{\frac {4\pi \kappa }{3+4\pi \kappa }}C_{1},\\B_{3}&={\frac {4\pi \kappa }{3+4\pi \kappa }}C_{1}a_{2}^{3}.\end{aligned}}$

(18)

The whole of this investigation might have been deduced directly from that of conduction through a spherical shell, as given in Art. 312, by putting k₁ = (1 + 4πκ)k₂ in the expressions there given, remembering that A_l and A₂ in the problem of conduction are equivalent to C_l + A_l and C_l + A₂ in the problem of magnetic induction.

434.] The corresponding solution in two dimensions is graphically represented in Fig. XV, at the end of this volume. The lines of induction, which at a distance from the centre of the figure are nearly horizontal, are represented as disturbed by a cylindric rod magnetized transversely and placed in its position of stable equilibrium. The lines which cut this system at right angles represent the equipotential surfaces, one of which is a cylinder. The large dotted circle represents the section of a cylinder of a paramagnetic substance, and the dotted horizontal straight lines within it, which are continuous with the external lines of induction, represent the lines of induction within the substance. The dotted vertical lines represent the internal equipotential surfaces, and are continuous with the external system. It will be observed that the lines of induction are drawn nearer together within the substance, and the equipotential surfaces are separated farther apart by the paramagnetic cylinder, which, in the language of Faraday, conducts the lines of induction better than the surrounding medium.

If we consider the system of vertical lines as lines of induction, and the horizontal system as equipotential surfaces, we have, in the first place, the case of a cylinder magnetized transversely and placed in the position of unstable equilibrium among the lines of force, which it causes to diverge. In the second place, considering the large dotted circle as the section of a diamagnetic cylinder, the dotted straight lines within it, together with the lines external to it, represent the effect of a diamagnetic substance in separating the lines of induction and drawing together the equipotential surfaces, such a substance being a worse conductor of magnetic induction than the surrounding medium.