Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/79

As the magnetic force increases, however, the magnetization in creases more slowly, and it would appear from experiments described in Chapter VI, that there is a limiting value of the magnetization, beyond which it cannot pass, whatever be the value of the magnetic force.

In the following outline of the theory of induced magnetism, we shall begin by supposing the magnetization proportional to the magnetic force, and in the same line with it.

Definition of the Coefficient of Induced Magnetization.

426.] Let ${\mathfrak {H}}$ be the magnetic force, denned as in Art. 398, at any point of the body, and let ${\mathfrak {J}}$ be the magnetization at that point, then the ratio of ${\mathfrak {J}}$ to ${\mathfrak {H}}$ is called the Coefficient of Induced Magnetization.

Denoting this coefficient by κ, the fundamental equation of induced magnetism is

${\mathfrak {J}}=\kappa {\mathfrak {H}}.$

(1)

The coefficient κ is positive for iron and paramagnetic substances, and negative for bismuth and diamagnetic substances. It reaches the value 32 in iron, and it is said to be large in the case of nickel and cobalt, but in all other cases it is a very small quantity, not greater than 0.00001.

The force ${\mathfrak {H}}$ arises partly from the action of magnets external to the body magnetized by induction, and partly from the induced magnetization of the body itself. Both parts satisfy the condition of having a potential.

427.] Let V be the potential due to magnetism external to the body, let Ω be that due to the induced magnetization, then if U is the actual potential due to both causes

$U=V+\Omega .\,$

(2)

Let the components of the magnetic force ${\mathfrak {H}}$, resolved in the directions of x, y, z, be α, β, γ, and let those of the magnetization ${\mathfrak {J}}$ be A, B, C, then by equation (1),

${\begin{aligned}A=\kappa \alpha ,\\B=\kappa \beta ,\\C=\kappa \gamma .\end{aligned}}$

(3)

Multiplying these equations by dx, dy, dz respectively, and adding, we find

$Adx+Bdy+Cdz=\kappa (\alpha dx+\beta dy+\gamma dz).\,$