r a magnetic intensity, the numerical value of which, 85, is given by the following equation:

in which a is the angle between the direction of the element and the straight line which connects it with the point in question. The direction of the field runs at right angles to the meridianal plane, which can be drawn through this straight line and the element.

B. Straight Conductors.-Let part of a circuit consist of a long straight piece; the other parts may be at a greater distance. In the immediate neighbourhood of the straight piece its influence preponderates, and is exerted as follows: The magnetic field is at each point at right angles to the plane passing through it and the straight piece, and accordingly the lines of intensity are evidently concentric circles. The numerical value of the intensity is given by the equation

in which is the distance of the given point from the conductor-that is, the radius of the corresponding circular line of intensity. This relation, in which the intensity of the field is inversely proportional to the distance from the conductor, is known as 'Biot-Savart's law,' these physicists having first discovered it by experiment.

§ 6. Magnetic Field of a Circular Conductor.-C. Circular Conductor. A plane circular conductor produces in its axis a field in the direction of the axis. Let be the radius of the circle, a the distance along the axis, measured from the centre where it cuts the plane of the circle, z = x2 + 2 is the distance of a point on the axis from the circumference. The numerical value of the field-intensity at the distance x is

This expression has a maximum value at the centre, where x = 0; there we have then

MAGNETIC FIELD OF CIRCULAR CONDUCTORS

7

For points on the axis at such a distance that in comparison with it the linear dimensions of the coil may be neglected we may, in equation (4) put z = x, the distance from the centre of the coil. Further, r2 is the area of the circular conductor, that is the area of the coil S; this, moreover, need not pecessarily be a circle, and with several windings the total area Σ (S) enters into the equation, which may then be written

In this the expression IΣ (S) is put = M, as this quantity determines the action at a distance of the windings. M is called its magnetic moment (comp. § 22).

The action of a circular conductor at points outside the axis can, in general, be only expressed by means of spherical harmonics.

D. Long Coil.-Of special importance is the action at a point in the interior of a uniformly wound bobbin, the length of which is very great in comparison with its cross-section. If the number of turns is n, the length L, the magnetic intensity at all points sufficiently distant from the ends is

It is therefore determined geometrically by n/L—that is, by the number of windings per unit length-and depends neither on the shape nor on the area of the winding. At the ends themselves the value of 1⁄2, as a little consideration will show, is only onehalf the above,

(8).

',

and diminishes rapidly as the distance in the direction of the axis increases, until at greater distances equation (6) again holds. Equation (7) holds also for long closed coils-that is to say, those whose axis forms any closed curve the centroid of the coil, and accordingly has no ends; such coils we shall frequently have to take into consideration.

As regards the relation between the direction of the current in the conductors and the direction of the field pro

duced, this, in cases A, B, C, D, is always the same as that between the direction of clock-motion and the direction from the dial to the works, and is independent of whether the electrical circuit is straight, and the line of magnetic intensity is curved or the reverse.

§ 7. Diamagnetic and Paramagnetic Substances. We have, as has been already stated, always assumed that the medium in which the phenomena take place is a vacuum. But with the exception of certain mixtures specially prepared for this purpose, the magnetic behaviour of all material substances more or less differs from what has been described. In order to investigate this, we introduce any given isotropic substance into a definite place in the magnetic field, and by preference we use it in the form of a sphere in order that, its shape being quite symmetrical, the exploring coil may fit in all directions.

By means of a momentary current induced in the latter, we can then, as above (§ 2), ascertain the magnetic condition of the sphere, and we shall find as follows. The magnetic condition in this case also is a vector, which has exactly the same direction as the intensity would have in a vacuum in the same places. With by far the greater number of substances, the numerical value of that vector, which as before is measured by the quantity of electricity induced per unit surface and unit of resistance, differs only in the fourth or fifth decimal place from the value obtained for a vacuum.2 Two cases may here be distinguished.

In by far the greater number of substances the magnetic vector is very slightly less than the corresponding one in a vacuum; these belong to the group which Faraday called diamagnetic.

If, however, the vector in the substance has a somewhat greater value than in a vacuum, it is classed with what is called the paramagnetic group.

It is usual to say that the magnetic condition in the substance is induced by that which would obtain in the same place in a vacuum, and for the measure of which we have above introduced

1 Faraday, on the magnetic condition of all matter (Exp. Researches, vol. 3, series 20 and 21, 1845).

These small differences can in fact be only qualitatively shown by means of special experimental arrangements; quantitative measurements are out of the question, otherwise than by rather refined methods.

FERROMAGNETIC SUBSTANCES

9

for

the magnetic intensity. There is no adequate reason diverging from this mode of expression, but it must be expressly stated that the process in question must not at all be considered as being explained by assigning to it the term magnetic induction. In fact, so far as we know at present, the magnetic condition in all para- and diamagnetic substances at a given temperature depends exclusively on the intensity of the field in which they are placed, and its value is even proportional to that intensity.

§ 8. Ferromagnetic Substances and Interferric.-It is otherwise with a small number of substances which in this respect claim an exceptional position. The value of the magnetic condition induced in them is not proportional to that of the space which they occupy; in addition to this it depends on the manner in which the substances have acquired their condition at the time considered. Their 'magnetic history' exerts an influence on their behaviour which is principally affected by the periods nearest in time to that in question. The magnetic condition of such bodies lags, as it were, behind the magnetic intensity which induces it. Accordingly at present the whole of the phenomena which belong to this class are included in the name hysteresis (Greek vσTEρέw, to lag or remain behind).1

The phenomenon of magnetic retentiveness, which historically was the first observed, and which was formerly the starting point of all considerations, is only a special case of hysteresis, as the name denotes. Regarded from this point of view, it must be considered as a residue of the action of earlier causes, which may indeed for thousands of years have ceased to exist.

These substances, by the properties which have been mentioned, are apparently different from all others, although it has not hitherto been possible to give any valid reason for this exceptional behaviour. They are accordingly classed in a separate group, the ferromagnetic.2 As, further, their magnetic properties are especially prominent, they have been observed since the most

Our knowledge in this field is due principally to the researches of Warburg and of Ewing.

2 By many authors the terms 'ferromagnetic' and 'paramagnetic' are used pretty indiscriminately. For the present it may be as well to keep the two groups separate. In what follows, we shall usually drop the prefix 'ferro-.' In doing so we follow the ordinary usage, which is not likely to lead to confusion, as we shall deal exclusively with ferromagnetic bodies.

remote times. Their more minute investigation is, however, an acquisition of the modern scientific era.

So far as is at present known, the group in question at ordinary temperatures consists of three metals which are also chemically allied—iron, cobalt, and nickel, together with some of their alloys, and compounds with one another or with other elements, as, for instance, carbon, oxygen, manganese, aluminium, mercury. A sharp delimitation of the ferromagnetic from the paramagnetic group may be made at present, but it is not improbable that in the future this will disappear.1

We shall deal in the sequel with combinations, consisting partly of ferromagnetic substance and partly of such as are not ferromagnetic. From the far more pronounced character of the ferromagnetic part the special nature of the latter does not at all come into play. This may either be paramagnetic or diamagnetic, and may be in any state of aggregation. In most cases in practice it will consist of the surrounding air, but may just as well be supposed to be filled with any other solid, liquid, or gas. We shall frequently call this the interferric,2 and we may regard it with sufficient approximation as magnetically indifferent —that is, as not being different from vacuum.

As regards the ferromagnetic substance, it is tacitly assumed that it is homogeneous, isotropic, and free from any current (§ 54). When the contrary is not expressly mentioned, hysteretic properties will be disregarded; it may, for instance, be assumed that the ferromagnetic substance is exposed to shocks, or to superposed alternating fields, which act in opposition to hysteresis; 3 and it is more particularly assumed that after the cessation of the magnetising influences, no magnetic properties are left behind.

§ 9. Magnetically Indifferent Toroid. We shall now pass to the case of a ring of material indifferent to magnetism, as represented in fig. 1. Let its section be circular and of area S,

' [This discontinuity appears to be already bridged over by the recent valuable researches of P. Curie, Thesis No. 840, Paris, 1895. H. d. B.]

2 German interferricum, a word imitated from the French entrefer which was introduced by Hospitalier, and which has been frequently adopted on the Continent. When the interferric consists of air only, it is usually called the 'air-gap'; the expression in the text is more general.

3 Compare Gerosa and Finzi, Rend. R. Ist. Lomb. vol. 24, p. 149, 1891; Rend. R. Lincei [8], vol. 7, p. 253, 1891; Wied. Beiblätter, vol. 16, p. 329, 1892.