| F = − | dW | = v | κ | dH²0 | . | |
| dx | 1 + 4⁄3πκ | dx |
(38)
If H0 is constant, the force will be zero; if H0 is variable, the sphere will tend to move in the direction in which H0 varies most rapidly. The coefficient κ / (1 + 4⁄3πκ) is positive for ferromagnetic and paramagnetic substances, which will therefore tend to move from weaker to stronger parts of the field; for all known diamagnetic substances it is negative, and these will tend to move from stronger to weaker parts. For small bodies other than spheres the coefficient will be different, but its sign will always be negative for diamagnetic substances and positive for others;[9] hence the forces acting on any small body will be in the same directions as in the case of a sphere.[10]
Directing Couple acting on an Elongated Body.—In a non-uniform field every volume-element of the body tends to move towards regions of greater or less force according as the substance is paramagnetic or diamagnetic, and the behaviour of the whole mass will be determined chiefly by the tendency of its constituent elements. For this reason a thin bar suspended at its centre of gravity between a pair of magnetic poles will, if paramagnetic, set itself along the line joining the poles, where the field is strongest, and if diamagnetic, transversely to the line. These are the “axial” and “equatorial” positions of Faraday. It can be shown[11] that in a uniform field an elongated piece of any non-crystalline material is in stable equilibrium only when its length is parallel to the lines of force; for diamagnetic substances, however, the directing couple is exceedingly small, and it would hardly be possible to obtain a uniform field of sufficient strength to show the effect experimentally.
Relative Magnetization.—A substance of which the real susceptibility is κ will, when surrounded by a medium having the susceptibility κ′, behave towards a magnet as if its susceptibility were κa = (κ − κ′) / (1 + 4πκ′). Since 1 + 4πκ′ can never be negative, the apparent susceptibility κa will be positive or negative according as κ is greater or less than κ′. Thus, for example, a tube containing a weak solution of an iron salt will appear to be diamagnetic if it is immersed in a stronger solution of iron, though in air it is paramagnetic.[12]
Circular Magnetization.—An electric current i flowing uniformly through a cylindrical wire whose radius is a produces inside the wire a magnetic field of which the lines of force are concentric circles around the axis of the wire. At a point whose distance from the axis of the wire is r the tangential magnetic force is
H = 2ir / a²
(39)
it therefore varies directly as the distance from the axis, where it is zero.[13] If the wire consists of a ferromagnetic metal, it will become “circularly” magnetized by the field, the lines of magnetization being, like the lines of force, concentric circles. So long as the wire (supposed isotropic) is free from torsional stress, there will be no external evidence of magnetism.