Forces acting on a Small Body in the Magnetic Field.—If a small magnet of length ds and pole-strength m is brought into a magnetic field such that the values of the magnetic potential at the negative and positive poles respectively are V1 and V2, the work done upon the magnet, and therefore its potential energy, will be
W = m (V2 − V1) = m dV,
which may be written
| W = m ds | dV | = M | dV | = −MH0 = −vIH0, |
| ds | ds |
where M is the moment of the magnet, v the volume, I the magnetization, and H0 the magnetic force along ds. The small magnet may be a sphere rigidly magnetized in the direction of H0; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have
| dW = −vI dH0 = −v | κ | H0 dH0; |
| 1 + 4⁄3πκ |
whence
| W = − | v | κ | H²0. | |
| 2 | 1 + 4⁄3πκ |
(37)
The mechanical force acting on the sphere in the direction of displacement x is