Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/42

If we now introduce two new symbols σ and ρ, denned by the equations

$\sigma =lA+mB+nC\,$,

$\rho =-\left({\frac {dA}{dx}}+{\frac {dB}{dy}}+{\frac {dC}{dz}}\right)$,

the expression for the potential may be written

$V=\iint {{\frac {\sigma }{r}}dS}+\iiint {{\frac {\rho }{r}}dxdydz}.$

386.] This expression is identical with that for the electric potential due to a body on the surface of which there is an electrification whose surface-density is σ, while throughout its substance there is a bodily electrification whose volume-density is ρ. Hence, if we assume σ and ρ to be the surface- and volume-densities of the distribution of an imaginary substance, which we have called 'magnetic matter', the potential due to this imaginary distribution will be identical with that due to the actual magnetization of every element of the magnet.

The surface-density σ is the resolved part of the intensity of magnetization I in the direction of the normal to the surface drawn outwards, and the volume-density ρ is the 'convergence' (see Art. 25) of the magnetization at a given point in the magnet.

This method of representing the action of a magnet as due to a distribution of 'magnetic matter' is very convenient, but we must always remember that it is only an artificial method of representing the action of a system of polarized particles.

On the Action of one Magnetic Molecule on another.

387.] If, as in the chapter on Spherical Harmonics, Art. 129, we make

${\frac {d}{dh}}=l{\frac {d}{dx}}+m{\frac {d}{dy}}+n{\frac {d}{dz}}$,

(1)

where l, m, n are the direction-cosines of the axis h, then the potential due to a magnetic molecule at the origin, whose axis is parallel to h₁, and whose magnetic moment is m₁, is

$V_{1}=-{\frac {d}{dh_{1}}}={\frac {m1}{r^{2}}}\lambda _{1}$,

(2)

where λ₁ is the cosine of the angle between h₁ and r.

Again, if a second magnetic molecule whose moment is m₂ and whose axis is parallel to h₂, is placed at the extremity of the radius vector r, the potential energy due to the action of the one magnet on the other is