Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/45

being in the direction of the axis of m₁, but having its own axis at right angles to that of m₁. If two points, A, B, rigidly connected with m₁ and m₂ respectively, are connected by means of a string T, the system will be in equilibrium, provided T cuts the line m₁m₂ at right angles at a point one-third of the distance from m₁ to m₂.

(4) If we allow the second magnet to turn freely about its centre till it comes to a position of stable equilibrium, W will then be a minimum as regards h₂, and therefore the resolved part of the force due to m₂, taken in the direction of h₁, will be a maximum. Hence, if we wish to produce the greatest possible magnetic force at a given point in a given direction by means of magnets, the positions of whose centres are given, then, in order to determine the proper directions of the axes of these magnets to produce this effect, we have only to place a magnet in the given direction at the given point, and to observe the direction of stable equilibrium of the axis of a second magnet when its centre is placed at each of the other given points. The magnets must then be placed with their axes in the directions indicated by that of the second magnet.

Of course, in performing this experiment we must take account of terrestrial magnetism, if it exists.

Let the second magnet be in a position of stable equilibrium as regards its direction, then since the couple acting on it vanishes, the axis of the second magnet must be in the same plane with that of the first. Hence

$(h_{1}h_{2})=(h_{1}r)+(rh_{2})\,$

(16)

and the couple being

${\frac {m_{1}m_{2}}{r^{3}}}\left(\sin(h_{1}h_{2})-3\cos(h_{1}r)\sin(rh_{2})\right),$

(17)

we find when this is zero

$\tan(h_{1}r)=2\tan(rh_{2}),\,$

(18)

or

$\tan \,H_{1}\,m_{2}\,R=2\tan \,R\,m_{2}\,H_{2}.$

(19)

When this position has been taken up by the second magnet the value of W becomes

$-m_{2}{\frac {dV}{dh_{2}}},$

where h₂ is in the direction of the line of force due to m₁ at m₂.