Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/96

sions, is when it is an ellipsoid. We shall therefore apply the method to the case of an ellipsoid.

Let

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1$

(1)

be the equation of the ellipsoid, and let Φ₀ denote the definite integral

$\int _{0}^{\infty }{\frac {d(\phi ^{2})}{\sqrt {(a^{2}+\phi ^{2})(b^{2}+\phi ^{2})(c^{2}+\phi ^{2})}}}.$[1]

(2)

Then if we make

$L=2\pi abc{\frac {d\Phi _{0}}{d(a^{2})}},\quad M=2\pi abc{\frac {d\Phi _{0}}{d(b^{2})}},\quad N=2\pi abc{\frac {d\Phi _{0}}{d(c^{2})}},$

(3)

the value of the potential within the ellipsoid will be

$V_{0}=-{\frac {\rho }{2}}(Lx^{2}+My^{2}+Nz^{2})+{\text{const.}}$

(4)

If the ellipsoid is magnetized with uniform intensity I in a direction making angles whose cosines are l, m, n with the axes of x, y, z, so that the components of magnetization are

$A=Il,\quad B=Im,\quad C=In,$

the potential due to this magnetization within the ellipsoid will be

$\Omega =-I(Llx+Mmy+Nnz).\,$

(5)

If the external magnetizing force is ${\mathfrak {H}}$, and if its components are α, β, γ, its potential will be

$V=Xx+Yy+Zz.\,$

(6)

The components of the actual magnetizing force at any point within the body are therefore

$X-AL,\quad Y-BM,\quad Z-CN.\,$

(7)

The most general relations between the magnetization and the magnetizing force are given by three linear equations, involving nine coefficients. It is necessary, however, in order to fulfil the condition of the conservation of energy, that in the case of magnetic induction three of these should be equal respectively to other three, so that we should have

${\begin{aligned}A&=K_{1}(X-AL)+K_{3}'(Y-BM)+K_{2}'(Z-CN),\\B&=K_{3}'(X-AL)+K_{2}(Y-BM)+K_{1}'(Z-CN),\\C&=K_{2}'(X-AL)+K_{1}'(Y-BM)+K_{3}(Z-CN).\end{aligned}}$

(8)

From these equations we may determine A, B and C in terms of X, Y. Z, and this will give the most general solution of the problem.

The potential outside the ellipsoid will then be that due to the

1. See Thomson and Tait's Natural Philosophy, §522.