Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/46

Hence

$W=-m_{2}{\sqrt {{\frac {\overline {dV}}{dx}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dy}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dz}}{\Bigg |}^{2}}},$

(20)

Hence the second magnet will tend to move towards places of greater resultant force.

The force on the second magnet may be decomposed into a force R, which in this case is always attractive towards the first magnet, and a force H₁ parallel to the axis of the first magnet, where

$R=-3{\frac {m_{1}m_{3}}{r^{4}}}{\frac {4\lambda _{1}^{2}+1}{\sqrt {3\lambda _{1}^{2}+1}}},\quad H_{1}=3{\frac {m_{1}m_{3}}{r^{4}}}{\frac {\lambda _{1}}{\sqrt {3\lambda _{1}^{2}+1}}}.$

(21)

In Fig. XVII, at the end of this volume, the lines of force and equipotential surfaces in two dimensions are drawn. The magnets which produce them are supposed to be two long cylindrical rods the sections of which are represented by the circular blank spaces, and these rods are magnetized transversely in the direction of the arrows.

If we remember that there is a tension along the lines of force, it is easy to see that each magnet will tend to turn in the direction of the motion of the hands of a watch.

That on the right hand will also, as a whole, tend to move towards the top, and that on the left hand towards the bottom of the page.

On the Potential Energy of a Magnet placed in a Magnetic Field.

389.] Let V be the magnetic potential due to any system of magnets acting on the magnet under consideration. We shall call V the potential of the external magnetic force.

If a small magnet whose strength is m, and whose length is ds, be placed so that its positive pole is at a point where the potential is V, and its negative pole at a point where the potential is V' the potential energy of this magnet will be m(V – V'), or, if ds is measured from the negative pole to the positive,

$m{\frac {dV}{ds}}ds.$

(1)

If I is the intensity of the magnetization, and λ, μ, ν its direction-cosines, we may write,

$m\,ds=I\,dxdydz,$

${\text{and }}{\frac {dV}{ds}}=\lambda {\frac {dV}{dx}}+\mu {\frac {dV}{dy}}+\nu {\frac {dV}{dz}},$

and, finally, if A, B, C are the components of magnetization,

$A=\lambda I,\quad B=\mu I,\quad C=\nu I,$