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

 Rh 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)

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

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),

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

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,

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

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