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

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16
ELEMENTARY THEORY OF MAGNETISM.
[391.

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 , the value of W may be written


(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 , and let it be inclined to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is and dip , then


(10)



(11)


whence W K$ (cos cos0 cos(< 8) + sin <>in 0). (12)

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


(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


(1)


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


&c.,(2)
when

r being the distance of (ξ, η, ζ) from the origin,

(3)

(4)

(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,