Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/41

Components of Magnetization.

The magnetization at a point of a magnet (being a vector or directed quantity) may be expressed in terms of its three components referred to the axes of coordinates. Calling these A, B, C

$A=I\lambda ,B=I\mu ,C=I\nu ,\,$

and the numerical value of $I$ is given by the equation (4)

$I^{2}=A2+B^{2}+C^{2}.\,$

(5)

385.] If the portion of the magnet which we consider is the differential element of volume dxdydz, and if $I$ denotes the intensity of magnetization of this element, its magnetic moment is $Idxdydz$. Substituting this for m in equation (3), and remembering that

$\cos \epsilon =\lambda (\xi -x)+\mu (\eta -y)+\nu (\zeta -z),\,$

(6)

where $\xi$, $\eta$, $\zeta$ are the coordinates of the extremity of the vector r drawn from the point (x, y, z), we find for the potential at the point ($\xi$, $\eta$, $\zeta$) due to the magnetized element at (x, y, z),

$\delta V=\left(A(\xi -x)+B(\eta -y)+C(\zeta -z)\right){\frac {1}{r^{3}}}dxdydz.$

(7)

To obtain the potential at the point ($\xi$, $\eta$, $\zeta$) due to a magnet of finite dimensions, we must find the integral of this expression for every element of volume included within the space occupied by the magnet, or

$V=\iiint \left(A(\xi -x)+B(\eta -y)+C(\zeta -z)\right){\frac {1}{r^{3}}}dxdydz.$

(8)

Integrating by parts, this becomes

$V=\iint {A{\frac {1}{r}}dydz}+\iint {B{\frac {1}{r}}dxdz}+\iint {C{\frac {1}{r}}dxdy}$

$-\iiint {{\frac {1}{r}}\left({\frac {dA}{dx}}+{\frac {dB}{dy}}+{\frac {dC}{dz}}\right)dxdydz},$

where the double integration in the first three terms refers to the surface of the magnet, and the triple integration in the fourth to the space within it.

If $l$, m, n denote the direction-cosines of the normal drawn outwards from the element of surface dS, we may write, as in Art. 21, the sum of the first three terms,

$V=\iint {\left(lA+mB+MnC\right){\frac {1}{r}}dS}$,

where the integration is to be extended over the whole surface of the magnet.