Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/48

In this expression l, m, n are the direction-cosines of the axis of the magnet, and K is the magnetic moment of the magnet. If ε is the angle which the axis of the magnet makes with the direction of the magnetic force ${\mathfrak {H}}$, the value of W may be written

$W=-K{\mathfrak {H}}\cos \epsilon .\,$

(9)

If the magnet is suspended so as to be free to turn about a vertical axis, as in the case of an ordinary compass needle, let the azimuth of the axis of the magnet be $\phi$, and let it be inclined $\theta$ to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is $\delta$ and dip $\zeta$, then

$\alpha ={\mathfrak {H}}\cos \zeta \cos \delta ,\quad \beta ={\mathfrak {H}}\cos \zeta \sin \delta ,\quad \gamma ={\mathfrak {H}}\sin \zeta ;$

(10)

$l=\cos \theta \cos \phi ,\quad m=\cos \theta \sin \phi ,\quad n=\sin \theta ;$

(11)

whence $W=-K{\mathfrak {H}}(\cos \zeta \cos \theta \cos(\phi -\delta )+\sin \zeta \sin \theta ).$

(12)

The moment of the force tending to increase $\phi$ by turning the magnet round a vertical axis is

$-{\frac {dW}{d\phi }}=-K{\mathfrak {H}}\cos \zeta \cos \theta \sin(\phi -\delta ).$

(13)

On the Expansion of the Potential of a Magnet in Solid Harmonics.

391.] Let V be the potential due to a unit pole placed at the point (ξ, η, ζ). The value of V at the point x, y, z is

$V=\left\lbrace (\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}\right\rbrace ^{-{\frac {1}{2}}}.$

(1)

This expression may be expanded in terms of spherical harmonics, with their centre at the origin. We have then

$V=V_{0}+V_{1}+V_{2}+\,$&c.,

(2)

when

$V_{0}={\frac {1}{r}},$ r being the distance of (ξ, η, ζ) from the origin,

(3)

$V_{1}={\frac {\xi x+\eta y+\zeta z}{r^{3}}},$

(4)

$V_{2}={\frac {3(\xi x+\eta y+\zeta z)^{2}-(x^{2}+y^{2}+z^{2})(\xi ^{2}+\eta ^{2}+\zeta ^{2})}{2r^{5}}},$

(5)

&c.

To determine the value of the potential energy when the magnet is placed in the field of force expressed by this potential, we have to integrate the expression for W in equation (3) with respect to x, y and z considering ξ, η, ζ as constants.

If we consider only the terms introduced by V₀, V₁ and V₂ the result will depend on the following volume-integrals,