Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/47

so that the expression (1) for the potential energy of the element of the magnet becomes

$\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz.$

(2)

To obtain the potential energy of a magnet of finite size, we must integrate this expression for every element of the magnet. We thus obtain

$W=\iiint {\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz}$

(3)

as the value of the potential energy of the magnet with respect to the magnetic field in which it is placed.

The potential energy is here expressed in terms of the components of magnetization and of those of the magnetic force arising from external causes.

By integration by parts we may express it in terms of the distribution of magnetic matter and of magnetic potential

$W=\iint {(Al+Bm+Cn)V\,dS}-\iiint V{\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz},$

(4)

where l, m, n are the direction-cosines of the normal at the element of surface dS. If we substitute in this equation the expressions for the surface- and volume-density of magnetic matter as given in Art. 386, the expression becomes

$W=\iint {V\sigma \,dS}+\iiint {V\rho dxdydz}.$[1]

(5)

We may write equation (3) in the form

$W=-\iiint {A\alpha +B\beta +C\gamma )dxdydz}.$

(6)

where α, β, γ are the components of the external magnetic force.

On the Magnetic Moment and Axis of a Magnet.

390.] If throughout the whole space occupied by the magnet the external magnetic force is uniform in direction and magnitude, the components α, β, γ will be constant quantities, and if we write

$\iiint {Adxdydz}=lK,\iiint {Bdxdydz}=mK,\iiint {Cdxdydz}=nK$

(7)

the integrations being extended over the whole substance of the magnet, the value of W may be written

$W=-K(l\alpha +m\beta +n\gamma ).\,$

(8)

1. There is a typo in the second integral in the original: the integration there is over dS