Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/81

${\begin{aligned}{\frac {d\Omega }{d\nu }}+{\frac {d\Omega '}{d\nu '}}&=-4\pi \sigma ,{\text{ by Art. 78}},\\&=4\pi {\frac {d\phi }{d\nu }},{\text{ by (7)}},\\&=-4\pi \kappa {\frac {dU}{d\nu }},{\text{ by (4)}},\\&=-4\pi \kappa \left({\frac {dV}{d\nu }}+{\frac {d\Omega }{d\nu }}\right),{\text{ by (2)}}.\end{aligned}}$

We may therefore write the surface-condition

$(1+4\pi \kappa ){\frac {d\Omega }{d\nu }}+{\frac {d\Omega '}{d\nu '}}+4\pi \kappa {\frac {dV}{d\nu }}=0.$

(10)

Hence the determination of the magnetism induced in a homogeneous isotropic body, bounded by a surface S, and acted upon by external magnetic forces whose potential is V, may be reduced to the following mathematical problem.

We must find two functions Ω and Ω' satisfying the following conditions:

Within the surface S, Ω must be finite and continuous, and must satisfy Laplace's equation.

Outside the surface S, Ω' must be finite and continuous, it must vanish at an infinite distance, and must satisfy Laplace's equation.

At every point of the surface itself, Ω = Ω', and the derivatives of Ω, Ω' and V with respect to the normal must satisfy equation (10).

This method of treating the problem of induced magnetism is due to Poisson. The quantity k which he uses in his memoirs is not the same as κ, but is related to it as follows:

$4\pi \kappa (k-1)+3k=0.\,$

(11)

The coefficient κ which we have here used was introduced by J. Neumann.

428.] The problem of induced magnetism may be treated in a different manner by introducing the quantity which we have called, with Faraday, the Magnetic Induction.

The relation between ${\mathfrak {B}}$, the magnetic induction, ${\mathfrak {H}}$, the magnetic force, and ${\mathfrak {J}}$, the magnetization, is expressed by the equation

${\mathfrak {B}}={\mathfrak {H}}+4\pi {\mathfrak {J}}.$

(12)

The equation which expresses the induced magnetization in terms of the magnetic force is

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

(13)