§ 43. Uniform Distributions. General Laws. Of the distributions referred to above, several are extremely simple in character, and many examples of these will be found in the sequel. A vector is said to be distributed uniformly when at every point throughout the region considered it has the same direction and the same value. We may further distinguish the following cases:—

In a hollow shell bounded by two concentric spherical surfaces there is said to be an accurately (or appreciably) uniform radial distribution when the vector is everywhere in the direction of the radius, and has a strictly (or appreciably) constant value. It is easy to see that such a distribution is lamellar, the equipotential surfaces being those of concentric spheres. On the other hand, the distribution is not solenoidal, but falls infinitely little short of possessing this quality when we come to consider an infinitely thin spherical shell.

In the case of a toroid [anchor-ring (§ 9)], a vector is said to have an accurately (or appreciably) uniform peripheral distribution when the direction of the vector is at each point along the circumference of that circle in which the point would move if the toroid were rotated about its axis of symmetry, the value of the vector being at the same time accurately (or appreciably) constant. This kind of distribution is complex lamellar-solenoidal, since the vector is directed along lines orthogonal to meridian planes' through the axis of the anchorring. When the ring is infinitely thin, the distribution falls infinitely little short of being lamellar.

The analytical conditions for the above distributions are so simple that it seems unnecessary to write them in extenso or to give a full investigation of them.

In conclusion we may adduce some general laws concerning the distribution of a vector :

VI. If a vector satisfies any of the above conditions, its product with a scalar constant satisfies the same conditions.

VII. The superposition of two or more solenoidal or lamellar distributions yields a resultant distribution, which is in turn solenoidal or lamellar, as the case may be.

For it is evident from the above that the derivatives of the components of the vector with respect to the co-ordinates are only

GENERAL LAWS OF UNIFORM DISTRIBUTIONS

57

involved linearly in the equations conditioning these two kinds of distribution. In the case of a complex lamellar distribution this is, however, no longer the case, and consequently the law of superposition does not, in general, hold good for such distributions.

B. Conductors conveying Currents, and Rigid Magnets

§ 44. We must now consider the principal properties of the electromagnetic field which is produced by currents flowing in conductors. This has already been introduced as the foundation for all further treatment, and we have also (§§ 5, 6) given the electromagnetic formula for the special cases of most importance. We must here content ourselves with merely writing down some systems of equations which are not of an elementary character, referring for a fuller treatment of them to those works which deal exhaustively with electromagnetism.'

Consider a conductor of any form, and let & denote the current-flow at a given point (x, y, z) of the conductor-that is, the current per unit area of normal cross-section at the point in question. It will be convenient in this place to introduce the auxiliary vector A (Maxwell's 'vector potential'), having components of the following values at any point distant r from the first-named point, namely

the integration being extended throughout the whole volume of the conductor.

It can then be shown that the components of the electromagnetic field due to the currents in the conductor can be expressed in terms of the derivatives of the auxiliary function It with respect to the co-ordinates; they are, in fact, given by

For example, Maxwell, Treatise, vol. 2, part iv.; Mascart and Joubert, Electricity and Magnetism, vol. 1, part iv.

2 Often called current density,' and expressible in ampères per square centimeter.

the following equations, which hold good for all points, whether within or without the conductor ::

The magnetic intensity arising from the electric current is now distributed solenoidally in all parts of the field without exception; for, by differentiating the above equations, we have identically

J 5x
дх

+

J Hy + მყ

and this is the condition for a solenoidal distribution of the vector [§ 37, equation (4)]. It can also be shown that satisfies the superficial equation of continuity at the surface separating the conductor from the surrounding medium.

§ 45. Magnetic Potential in the Field outside Conductors.— In close connection with the equations of the last paragraph we have the following:

which also hold good for all points throughout the field.

If the point considered lies within the conductor, where C has a finite value, these equations show that the distribution of cannot there be lamellar, since a potential does not exist. But outside the conductor, where there is no current, and where, consequently, C is zero, the terms on the left will vanish, and the terms on the right then express the condition that the vector may be distributed lamellarly (the condition, that is,

MAGNETIC POTENTIAL IN AN OUTSIDE FIELD

59

that its components may be the space-derivatives of a potential). We have, in fact,

[Compare § 39, equation (7).] We accordingly arrive at the following theorem :

VIII. In the region surrounding a conductor (not within the substance of the conductor itself) the magnetic intensity is distributed lamellarly, and has consequently a scalar potential.

This potential of the magnetic intensity we shall denote by T, and we shall call it the magnetic potential. But the region where 5 is lamellar is now no longer simply connected, since it is interrupted by a non-lamellar region at least doubly connected, which includes every separate closed conducting circuit round which a current is flowing. Thus the present case differs from that previously discussed (§ 40). In fact, the line-integral of 1⁄2 taken along any closed curve increases by acyclic constant of integration,' C, for each time that the closed curve embraces the conductor, and it is only along such closed curves as do not embrace the conductor that the line-integral vanishes. In using the term 'cyclic constants' we must, of course, understand that these can only depend on the electric currents embraced by the path along which the line-integral is taken, and not on any purely geometrical relations. We can now determine à priori the value of C by observing that its relation to the current 1 must be one of proportionality, since the same is known from experiment to be true of 5. The completion of the value of C can involve nothing further than the introduction of a numerical factor, and to this again, as in a former case (§ 11), we assign the value 4 π,' following the historical development of the subject, and in conformity with the system of electromagnetic units in general use. Hence, finally,

The introduction of the factor 4 is condemned by several authors, especially by Heaviside. Its elimination, however, could only be effected by remodelling the system of units already adopted, so that this comparatively unimportant simplification would be rather dearly bought; moreover, the factor would probably reappear in another place.

We can easily verify this equation from one of the elementary examples already considered; we shall choose the case of a long straight linear conductor, in the neighbourhood of which the electromagnetic action follows the law of Biot and Savart. The magnetic intensity 5 at a distance from the conductor is [§ 5, equation (3)]

=

2 I

The line of force through the point considered is a circle of circumference 2πr; and consequently the value of the line-integral of 5 along a circuit embracing the conductor once (that is, the value of the cyclic constant C which we have to determine) is

a conclusion identical with (17). We can combine these results in the following fundamental theorem :

IX. Every time the circuit along which we integrate embraces a conductor conveying a current I, the line-integral of magnetic intensity increases by the cyclic constant 4T I, independently of any other variables whatever.

This important general law is abundantly confirmed by experiment, as in the present case is almost self-evident.

§ 46. Action of a Rigid Magnet at External Points.We have already (§ 26) introduced the conception of magnetic end-elements, referring to the present chapter for a fuller mathematical development of the method. We have also given [in § 19, equation (2)] a statement of the elementary law of Coulomb, that the force apparently exerted by a magnetic pole, at any point in its neighbourhood, is given by

where is the magnetic intensity at the distance r from the end, and is directed away from the end when the end is positive, towards the end in the contrary case. S is the cross-sectional area of the bar-magnet, and 3 the vector which we have called the magnetisation (§ 11).