Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/86

direction, or if they are crowded more in one direction than another, the medium, as Poisson himself shews, will not be isotropic. Poisson therefore, to avoid useless intricacy, examines the case in which each magnetic element is spherical, and the elements are disseminated without regard to axes. He supposes that the whole volume of all the magnetic elements in unit of volume of the substance is k.

We have already considered in Art. 314 the electric conductivity of a medium in which small spheres of another medium are distributed.

If the conductivity of the medium is μ₁, and that of the spheres μ₂, we have found that the conductivity of the composite system is

$\mu =\mu _{1}{\frac {2\mu _{1}+\mu _{2}+2k(\mu _{2}-\mu _{1})}{2\mu _{1}-\mu _{2}-k(\mu _{2}-\mu _{1})}}.$

Putting μ₁ = 1 and μ₂x = ∞, this becomes

$\mu ={\frac {1+2k}{1-k}}.$

This quantity μ is the electric conductivity of a medium consisting of perfectly conducting spheres disseminated through a medium of conductivity unity, the aggregate volume of the spheres in unit of volume being k.

The symbol μ also represents the coefficient of magnetic induction of a medium, consisting of spheres for which the permeability is infinite, disseminated through a medium for which it is unity.

The symbol k, which we shall call Poisson's Magnetic Coefficient, represents the ratio of the volume of the magnetic elements to the whole volume of the substance.

The symbol κ is known as Neumann's Coefficient of Magnetization by Induction. It is more convenient than Poisson's.

The symbol μ we shall call the Coefficient of Magnetic Induction. Its advantage is that it facilitates the transformation of magnetic problems into problems relating to electricity and heat.

The relations of these three symbols are as follows:

${\begin{aligned}&k={\frac {4\pi \kappa }{4\pi \kappa +3}},\quad k={\frac {\mu -1}{\mu +2}}\\&\kappa ={\frac {\mu -1}{4\pi }},\quad \kappa ={\frac {3k}{4\pi (1-k)}},\\&\mu ={\frac {1+2k}{1-k}},\quad \mu =4\pi \kappa +1.\end{aligned}}$

If we put κ = 32, the value given by Thalen's^[1] experiments on

1. Recherches sur les Propriétes Magnétiques du fer, Nova Acta, Upsal, 1863.