Forces on a Magnet in a Uniform Field

We proceed to investigate the forces on a solenoidal magnet in a uniform field.

Let us suppose the magnet held with its axis at right angles to the lines of force, and let l be the distance between its poles, m the strength of each pole, and H the intensity of the field. The north pole is acted on by a force m H at right angles to the axis of the magnet, the south pole by an equal, parallel, but opposite force m H. These two forces constitute a couple; the distance between the lines of action, or arm of the couple, is l, so that the moment of the couple is mlH. If the axis of the magnet be inclined at an angle θ to the lines of force, the arm of the couple will be mlsinθ, and its moment mlHsinθ. In all cases the couple will depend on the product ml.

Definition of magnetic moment of a magnet. - The product of the strength of either pole into the distance between the poles, is called the magnetic moment of a solenoidal magnet. Let us denote it by M; then we see that if the axis of the magnet be inclined at an angle θ to the lines of force, the couple tending to turn the magnet so that its axis shall be parallel to the lines of force is MHsinθ. Thus the couple only vanishes when θ is zero; that is, when the axis of the magnet is parallel to the lines of force.

But, as we have said, the actual bar magnets which we shall use in the experiments described below are not strictly solenoidal, and we must therefore consider the behaviour, in a uniform field, of magnets only approximately solenoidal.

If we were to divide a solenoidal magnet into an infinitely large number of very small, equal, similar, and similarly situated portions, each of these would have identical magnetic properties; each would be a small magnet with a north pole of strength m and a south pole of strength -m.

If we bring two of these elementary magnets together so as to begin to build up, as it were, the original magnet, the north pole of the one becomes adjacent to the south pole of the next; we have thus superposed, a north pole of strength m and a south pole of strength -m; the effects of the two at any distant point being thus equal and opposite, no external action can be observed. We have therefore a magnet equal in length to the sum of the lengths of the other two with two poles of the same strength as those of either.

If however, we were to divide up an actual magnet in this manner, the resulting elementary magnets would not all have the same properties.

We may conceive of the magnet, then, as built up of a number of elementary magnets of equal volume but of different strengths.

Consider two consecutive elements, the north pole of the one of strength m is in contact with the south pole of the other of strength -m' say; we have at the point of junction a north pole of strength m-m', we cannot replace the magnet by centres of repulsive and attractive force at its two ends respectively, and the calculation of its action becomes difficult.

If, however, the magnet be a long bar of well-tempered steel carefully magnetised, it is found that there is very little magnetic action anywhere except near the ends. The elementary magnets of which we may suppose it to consist would have equal strengths until we get near the ends of the magnet, when they would be found to fall off somewhat The action of such a magnet may be fairly represented by that of two equal poles placed close to, but not coincident with, the ends; and we might state, following the analogy of a solenoid, that the magnetic moment of such a magnet was measured by the product of the strength of either pole into the distance between its poles.

We can, however, give another definition of this quantity which will apply with strictness to any magnet The moment of the couple on a solenoidal magnet, with its axis at an angle θ to the lines of magnetic force in a field of uniform intensity H, is, we have seen, MHsinθ, M being the magnetic moment. Thus the maximum couple which this magnet can experience is M H, and the maximum couple which the magnet can be subjected to in a field of uniform force of intensity unity is M.

Now any magnet placed in a uniform field of magnetic force is acted on by a couple, and we may say that for any magnet whatever, the magnetic moment of a magnet is measured by the maximum couple to which the magnet can be subject when placed in a uniform magnetic field of intensity unity.

When the couple is a maximum the magnetic axis of the magnet will be at right angles to the lines of force.

If the angle between the axis of the magnet and the lines of force be θ, the magnetic moment M, and the strength of the field H, the couple will be MHsinθ, just as for a solenoidal magnet.