ACTION AT A DISTANCE OF A PAIR OF ENDS

31

the magnetic properties of bodies-as in what are called magnetometric methods-the magnetisation is equal to the magnetic moment measured divided by the volume.

In case the point in question is neither in the prolongation of the axis nor in the equatorial plane of the bar, the equations for the intensity of the field are less simple than (5) and (6). As regards the shape of the lines of intensity we refer to fig. 7 in which they are represented outside the bar by the dotted lines; the figure in question holds for a short bar; with longer bars, as assumed in the above, the action at a distance proceeds almost exclusively from the geometrical ends, and the lines accordingly diverge radially. In fig. 7 these lines proceed not only from the end surfaces +E and E, but also partially at

least from the neighbouring parts of the sides. The lines either proceed from the positive to the negative end, or they direct their course towards an infinite distance.

§ 23. Mechanical Action of External Fields on Pairs of Ends.— We have already seen in § 21 how in an extraneous field a force is exerted on a single end. The occurrence of single ends is, however, excluded from the nature of the case; we can at most assume that in very long bars one of the ends may be considered singly by neglecting the actions proceeding from the other end, or exerted on it, as being too distant. But, as a matter of fact, in bar magnets, we have always to deal with a

1

Compare § 26, where the lines inside the bar, the lines of magnetisation,' are further discussed.

pair of ends, whether the magnetisation is induced or residual. The area S and the magnetisation 3 being constant, both ends have the strength 3S, though with opposite signs. In an extraneous field, the intensity of which is constant within the space occupied by the bar, and is in the same direction, according to equation (3) § 21, equal and opposite forces are exerted on the ends.

& = ± H (IS)

If the positive axial direction of the bar (§ 19) makes an angle a, with the positive direction of the field, these two forces act at points which are at a distance of L sin a from each other, L again being the length of the bar. They thus exert a torque, the moment of which is given by the equation

or introducing again the magnetic moment M, as in the previous paragraph

The couple therefore in general acts on the pair of ends in such a manner as to tend to draw the bar into the position of stable equilibrium, which corresponds to the value a = 0°. It will make oscillations about this position the period of which 7 is given by the equation

in which K is the moment of inertia of the bar. A position of unstable equilibrium corresponds to the value a 180°.

ment.

These conclusions are most completely confirmed by experiA single force is never exerted on a magnet in a field of constant value, and constant direction, but always a torque ; and this holds not only for bar magnets of constant strength, but also for bodies of any form, and magnetised in any way whatever.

§ 24. Demagnetising Action of a Bar.-It has already been. explained in the case of the divided ring that each of the ends

DEMAGNETISING FACTORS OF CIRCULAR CYLINDERS 33

will also act according to Coulomb's law in the space occupied by the bar itself. This 'self action,' as is at once seen from the plain and feathered arrows in fig. 6, p. 27, will always be opposed to the action of the coil. Both ends exert therefore a demagnetising action on the bar, the intensity of which we will call. This is a minimum at the middle of the bar and increases towards each end; from what has been said, it is proportional at each point to the magnetic strength(3 S) of the bar. Hence this proportionality also holds for the mean value of the vector, which we shall distinguish by a bar over the letter, thus 1.

Let us now consider a circular cylindrical bar with plane ends. The dimensional ratio, by which is meant the ratio of the length to the diameter, we will call m. If we now assume that the cylinder becomes gradually thinner, the length remaining the same, this would correspond to an increase of the ratio m, and to a decrease of the cross-section proportional to that of 1/m2. Hence 5 also decreases, and, from what has been said, in the same proportion as the magnetic strength—that is, proportionally to (J/m2).

If we now consider the quotient H/I—that is, the mean demagnetising intensity per unit of magnetisation—and if, as in equation (1) (§ 17), we denote it with N, we shall call the number thus defined the mean demagnetising factor. It follows, then, from the preceding that Ñ must theoretically be proportional to 1/m2-that is,

I. The demagnetising factor of a circular cylindrical bar is, theoretically, inversely proportional to the square of the ratio of the dimensional ratio.

(9)

If C is the factor of the proportion, then

must be constant.

From an analysis of experiments with cylinders, it is found that this, as a matter of fact, does hold, provided the ratio of the dimensions exceeds 100. C has then the constant value 45. The mean demagnetising factor of a cylinder, whose length is at least 100 times the diameter, may be simply calculated by dividing the square of the dimensional ratio into the number 45.

D

§ 25. Demagnetising Factors of Circular Cylinders.—The experiments mentioned were made with cylinders of varying lengths, but with a given diameter and given material. Owing to the unavoidable heterogeneity of the material, it is better to cut the cylinders gradually shorter from one and the same piece. It would be best of all, retaining the length constant, to gradually turn it down to a smaller diameter, as we have supposed in our theoretical investigation. The curves found thus experimentally for various values of m are then plotted side by side, and the corresponding demagnetising factors are then deduced from the differences of the abscissæ in the manner above described (§ 16).

It appeared that for short cylinders for which m < 100, the value C is not constant but diminishes. In Table I., which, for more convenient comparison with other numbers, is printed on page 41, a general view is given of the mean demagnetising factors found as described. The values C m2 N are also given, as being better fitted for interpolation owing to the slight extent to which they vary.

=

It has further been experimentally established that ferromagnetic prisms or bundles of wire tied together, of any given profile, differ but little from circular cylinders of equal length and equal section. The table furnishes, therefore, a means of reducing the results of experiments with bars and bundles to the proper normal case of endless shapes; in other words, by back shearing, to obtain the normal curve of the material investigated, which is alone characteristic. This is the more necessary as by far the greater number of the existing and, in part, very valuable experiments have been made with bars; and from the nature of the case this will, in the future, also often occur. The results and the curves obtained in this way will also be difficult of direct interpretation, and hence lose much of their value.

§ 26. Short Cylinder. Lines of Magnetisation. We have already seen that for short cylinders (for which m<100) the

1 Such experiments have been made partly by Ewing, Phil. Trans. vol. 176, p. 535, partly by Tanakadaté, Phil. Mag. vol. 26, p. 450, 1888. The theoretical deductions are due to the author (Wied. Ann. vol. 46, p. 497, 1892).

2 Von Waltenhofen, Wien. Ber. vol. 48, part 2, p. 578, 1863. [Extended series of experiments on bundles of wires have lately been undertaken by Ascoli and Lari. R. Acad. dei Lincei, Rome, 1894. H. d. B.]

SHORT CYLINDER.

LINES OF MAGNETISATION

35

principle laid down in § 24 no longer holds, but that their demagnetisation factor is smaller than that principle prescribes; one or more of the assumptions on which it was deduced do not, therefore, hold.

In fact, if a secondary coil is displaced along the cylinder, and the momentary current excited in it on magnetisation is investigated, we find that this does not suddenly cease when the coil is pushed beyond the ends, but becomes smaller at a certain distance from the ends. This distance is relatively more important in the case of a short cylinder than in that of a long one. We conclude from this that the induction is greater in the middle of the bar than towards the ends. This, therefore, is also the case with the magnetisation.

If we examine the action at a distance due to the ends, we shall find it smaller than would correspond to the value of magnetisation in the middle of the bar. At the same time the distribution of the external field is of the same nature as if an action proceeded not only from the geometrical ends, but also from the adjacent parts of the bar: as if in a certain sense there were ends there also. We are, then, thus led to generalise the idea of 'ends.' We no longer consider it as a purely geometrical conception in the sense of meaning by it the terminal faces of the bar; we rather conceive the magnetisation of the short cylinder as if the stronger magnetic condition which exists in the middle of the bar only gradually ended, and therefore shows a great number of magnetic end-elements, each of which exerts its own elementary action at a distance in accordance with Coulomb's law. Just as a positive and a negative end were previously distinguished, so we must now distinguish as positive and negative the end-elements, which are distributed on the corresponding halves of the bar. In this the algebraic sum of the strengths of all the end-elements is always zero, even with a body of any given shape, and magnetised in any given way. This follows, among other things, from the fact adduced in § 23, that an external field of constant value and invariable direction exerts a torque on such a body and never a single force.

Closely connected with this is the course of the lines of magnetisation—that is, of those lines whose tangent at each point coincides in direction with the vector 3, as was established