§ 176.] Magnitude and Direction of the Force.

239

tangential component; on the other hand, this work is equal to the change of potential VV' from the first point to the second, so that

In like manner, in passing along the radius from the point P at the distance to the point p" at the distance r', calling the radial component F, we have

The value of the resultant, which is the diagonal of the rectangle formed on the two forces Ę and E, is

(3) F

=

=

3 cos2 a + I

sin2 A + I

A being the angle which OP makes with the perpendicular bisector OB of the magnet. The direction of the resultant makes with the tangent an angle I, such that

It will be seen that the force in any given direction varies as the inverse cube of the distance.

For a point A on the axis, a = 0; the tangential component is zero, and we have for the total force along the axis

π

At a point B in the perpendicular bisector, a = ; the force

2

reduces to the tangential component always parallel to the axis, and we have

Two positions such as A and B, one in the prolongation of the axis of the magnet and the other on the perpendicular to it, are called principal positions.

Fig. 158 enables us to form an idea of the field due to an infinitely small magnet. In this case the two points N and s would coincide at the centre of the figure, and the lines of force would be closed curves tangential at this point to the axis AB. The equipotential surfaces are also all tangential at the same point to the plane of symmetry CD.

177. Relative Energy of an Infinitely Small Magnet in a Field. The potential V= w of the infinitely small magnet represents the work done against magnetic force in bringing a unit of positive magnetism from an infinite distance to the point P in the presence of the infinitely small magnet; or, reciprocally, in moving the infinitely small magnet from an infinite distance to its actual position in the presence of a unit magnetic pole at P. It is, therefore, the energy in the given condition of the system formed by this pole and the infinitely small magnet.

If the pole at the point P were equal to m, the energy of the system would be mw. Now this product is the flux of force from the mass m at the point P, which traverses the surface by its positive face. We shall represent by g the flux which traverses a surface, and for convenience of subsequent calculations we shall take q as positive when it enters by the negative face, and negative when it enters by the positive face. We shall accordingly write

If there are other masses, m' m', in the field, we shall have analogous expressions for each of them. The value of the energy

§ 177] Relative Energy of Infinitely Small Magnet. 241

of the infinitely small magnet in the presence of these masses is then

by its

Q being the total flux which passes through the surface negative face. If I is the normal component at the point o of the field, due to the masses m, m', m", we have Q = F.

Any variation which changes into 2 will correspond to an amount of positive work equal to QQ done by the magnetic forces.

Q

CHAPTER XVII.

CONSTITUTION OF MAGNETS.

178. Result of Breaking a Magnetised Bar.-When a magnetised bar-a steel needle, for instance-is broken, each part forms a complete magnet, the two new poles being of the same strength as those of the original magnet, and the axis in the same direction (Fig. 161). The phenomenon can be repeated

as long as the fragments obtained are large enough to admit of examination.

Two important consequences follow from this. The first, that it is impossible to get a detached positive or negative magnetic pole not connected with an equal pole of the opposite kind. The second is that magnetism is a phenomenon the cause of which resides in the molecules of the magnet. If we suppose a magnet split up into elementary particles, each element in its actual condition must be considered as having its two poles and a definite moment. The entire magnet is only the resultant of all these elementary magnets, and the axis, as well as the moment of the resultant magnet, may be obtained by the rule in § 173.

179. Distribution of Magnetism.-It is not possible to determine by experiment the distribution of magnetism in the interior of a magnet, any more than we can determine the distribution of electricity in the interior of a bad conductor. From Gauss's theorem (§ 41), which applies equally to the two cases, the total flux of force which traverses a surface surrounding the magnet would

§ 180.] Particular Cases of Magnetic Distribution.

243

give the total value of the acting mass which it comprises; but we know already that, in the case of a magnet, this mass is zero ($171).

Whatever be the distribution of magnetic masses in the interior of a magnet, it is easy to prove that their action is equivalent to that of two equal magnetic masses of opposite signs distributed on the surface according to a certain law. For, suppose all the magnetic masses to be replaced by fixed electrical masses of the same numerical value (§ 167), and imagine the magnet enclosed within a conducting surface (§ 236). We know that this surface will be covered on the interior with an electrical layer, of equal total mass, but of opposite sign to the sum of the interior masses, which will exert at all external points a force equal and opposite to that of the enclosed masses. Consequently, an equal and similarly distributed layer, of the same sign as the sum of the enclosed masses, would exert externally the same force as they do, and might therefore be substituted for them. Similarly in the case of a magnet. In this case, the total mass of such a layer will be zero-that is to say, it will be composed of two equal layers of opposite signs, separated by a neutral line, one covering the north end and the other the south end of the magnet. The same reasoning will apply to any surface whatever enclosing the magnet.

This shows clearly that the problem of internal distribution is altogether indeterminate.

180. Particular Cases of Magnetic Distribution-Solenoidal Filament. Among the distributions which we may a priori imagine, we shall examine three which present a special interest.

Suppose, in the first case, magnetic elements identical with each other placed in file along any given curve, so that their axes having all the same sense, coincide with this line, the north pole of one being in contact with the south pole of the next. This arrangement is said to be solenoidal, and is spoken of as a solenoidal filament. The filament is neutral throughout its entire length, except at the ends at which there are equal magnetic masses, m and m, of opposite signs. The external action of the filament is therefore reducible to that of these two masses; it is independent of the shape and length of the filament, and depends only on the position of the ends. It vanishes if the filament is closed upon itself: in this case the magnetic state of the filament is not indicated by any external effect.