to the direction of magnetization, and equal to § 1rI. When it is
desired to have a uniform magnet with definitely situated poles, it
it usual to employ one having the form of an ovoid, or elongated
ellipsoid of revolution, instead of a rectangular or cylindrical bar.
If the magnetization is parallel to the major axis, and the lengths
of the major and minor axes are 2a and 2c, the poles are situated at
a distance equal to § a from the centre, and the magnet will behave
externally like a simple solenoid of length § a. The internal force
F is opposite to the direction of the magnetization; and equal to
N I, where N is a coefficient depending on y on the ratio of the axes.
The moment = § 1rac'I = - § 1rac“FN.

The distribution of magnetism and the position of the poles in magnets of other shapes, such as cylindrical or rectangular bars, cannot be specified by any general statement, though approximate determinations may be obtained experimentally in individual cases.1 According to F. W. G. Kohlrauschz the distance between the poles of a cylindrical magnet the length of which is from 10 to 30 times the diameter, is sensibly equal to five-sixths of the length of the bar. This statement, however, is only approximately correct, the distance between the poles depending upon the intensity of the magnetization.3 In general, the greater the ratio of length to section, the more nearly will the poles approach the end of the bar, and the more nearly uniform will be the magnetization. For most practical purpose a knowledge of the exact position of the poles is of no importance; the magnetic moment, and therefore the mean magnetization, can always be determined with accuracy.

Magnetic Induction or Magnetic Flux.-When magnetic force acts on any medium, whether magnetic, diamagnetic or neutral, it produces within it a phenomenon of the nature of a fiux or flow called magnetic induction (Maxwell, loc. cit., § 428). Magnetic induction, like other fluxes such as electrical, thermal or fluid currents, is defined with reference to an area; it satisfies the same conditions of continuity as the electric current does, and in isotropic media it depends on the magnetic force just as the electric current depends on the electromotive force. The magnitude of the flux produced by a given magnetic force differs in different media. In a uniform magnetic field of unit intensity formed in empty space the induction or magnetic fiux across an area of I square centimetre normal to the direction of the field is arbitrarily taken as the unit of induction. Hence if the induction per square centimetre at any point is denoted by B, then in empty space B is numerically equal to H; moreover in isotropic media both have the same direction, and for these reasons it is often sald that in empty space (and practically in air and other nonmagnetic substances) B and H are identical. Inside a magnetized body, B is the force that would be exerted on a unit pole if placed in a narrow crevasse cut in the body, the walls of the crevasse bein perpendicular to the direction of the magnetization (Maxwell, §§ 399, 6o4); and its numerical value, being artly due to the free magnetism on the walls, is generally very digerent from that of H. In the case of a straight uniformly magnetized bar the direction of the magnetic force due to the poles of the magnet is from the north to the south pole outside the magnet, and from the south to the north inside. The magnetic fiux r square centimetre at any point (B, B, or 23) is briefly called tlifinduction, or, especially by electrical engineers, the flux-density. The direction of magnetic induction may be indicated by lines of induction; a line of induction is always a closed curve, though it may possibly extend to and return from infinity. Lines of induction drawn through every point in the contour of a small surface form a re-entrant tube bounded by lines of induction; such a tube is called a tube of induction. The cross section of a tube of induction may vary in different parts, but the total induction across any section is everywhere the same. A special meaning has been assigned to the term “ lines of induction." Suppose the whole space in which induction exists to be divided up into unit tubes, such that the surface integral of the induction over any cross-section of a tube is equal to unity, and along the axis of each tube let a line of induction be drawn. These axial lines constitute the system of lines of induction which are so often referred to in the specification of a field. Where the induction is high the lines will be crowded together; where it is weak they will be widely separated, the number per square centimetre crossing a normal surface at any point being always equal to the numerical value of B. The induction may therefore be specified as B lines per square centimetre. The direction of the induction is also of course indicated by the direction of the lines, which thus serve to map out space in a convenient manner. Lines of induction are frequently but inaccurately spoken of as lines of force.

When induction or magnetic fiux takes place in a ferromagnetic metal, the metal becomes magnetized, but the magnetization at any point is proportional not to B, but to B-H. The factor of proportionality will be I-41r, so that

The principal theoretical investi ations are summarized in Mascart and ]oubert's Electricity andg Magnetism, i. 391-398 and ii. 646-657. The case of a long iron bar has been experimentally studied with great care by C. G. Lamb, Proc. Phys. Soc., 1899, 16, 509.

Wied. Ann., 1884, 22, 411.

See C. G. Lamb, loc. cit. p. 518.

l

I= (B"H)/41m (24)

or B =H~-4-/rI. (25)

Unless the path of the induction is entirely inside the metal, free magnetic poles are developed at those parts of the metal where induction enters and leaves, the polarity being south at the entry and north at the exit of the flux. These free poles produce a magnetic field which is superposed upon that arising from other sources. The resultant magnetic field, therefore, is compounded of two fields, the one being due to the poles, and the other to the external causes which would be operative in the absence of the magnetized metal. The intensity (at any point) of the field due to the magnetization may be denoted by Hr, that of the external field by Ho, and that of the resultant field by H. In certain cases, as, for instance, in an iron ring wrapped uniformly round with a coil of wire through which a current is passing, the induction is entirely within the metal; there are, consequently, no free poles, and the ring, though magnetized, constitutes a poleless magnet. Magnetization is usually regarded as the direct effect of the resultant magnetic force, which is therefore often termed the magnetizing force.

Permeability and Susceptibility.-The ratio B/H is called the permeability of the medium in which the induction is taking place, and is denoted by p.. The ratio I /H is called the susceptibility of the magnetized substance, and is denoted by tc. Hence B = uH and I=xH. (26)

Also l»¢=% 31"= =1'l'41|'K» (27)

and x =% (28)

Since in empty space B has been assumed to be numerically equal to H, it fol ows that the permeability of a vacuum is equal to 1. The permeability of most material substances differs very slightly from unity, being a little greater than I in paramagnetic and a little less in diamagnetic substances. In the case of the ferromagnetic metals and some of their alloys and compounds, the permeability has generally a much higher value. Moreover, it is not constant, being an apparently arbitrary function of H or of B; in the same specimen its va ue may, under different conditions, vary from less than 2 to upwards of 5000. The magnetic susceptibility x expresses the numerical relation of the magnetization to the magnetizing force. From the equation x= (u-1)/41r, it follows that the magnetic susceptibility of a vacuum (where it = I) is 0, that of a diamagnetic substance (where p < 1) has a negative value, while the susceptibility of paramagnetic and ferromagnetic substances (for which p>1) is positive. No substance has yet been discovered having a. negative susceptibility sufficiently great to render the permeability (= 1 +4-/rx) negative.

Magnetic Circuit.-The circulation of magnetic induction or fiux through magnetic and non-magnetic substances, such as iron and air, is in many respects analogous to that of an electric current through good and bad conductors. lust as the lines of flow of an electric current all pass inclosingd curves through the battery or other generator, so do all the lines of induction pass inclosingd curves through the magnet or magnetizing coil. The total magnetic induction or fiux corresponds to the current of electricity (practically measured in amperes); the induction or flux density B to the density 'of the current (number of amperes to the square centimetre of section); the magnetic permeability to the specific electric conductivity; and the line integral of the magnetic force, sometimes called the magneto motive force, to the electro-motive force in the circuit. The principal points of difference are that (I) the magnetic permeability, unlike the electric conductivity, which is independent of the strength of the current, is not in general constant; (2) there is no perfect insulator for magnetic induction, which will pass more or less freely through all known substances. Nevertheless, many important problems relating to the distribution of magnetic induction may be solved by methods similar to those employed for the solution of analogous problems in electricity. For the elementary theory of the magnetic circuit see ELECTRO-MAGNETISM.

Hysteresis, Coercive Force, Retentiveness.-It is found that when a piece of ferromagnetic metal, such as iron, is subjected toa magnetic field of changing intensity, the changes which take place in the induced magnetization of the iron exhibit a tendency to lag behind those which occur in the intensity of the field-a phenomenon to which ]. A. Ewing (Phil. Trans. clxxvi. 524) has given the name of hysteresis (Gr. bo'-repéw, to lag behind). Thus it happens that there is no definite relation between the magnetization of a piece of metal which has been previously magnetized and the strength of the field irf which it is pliaced. Much depends upon its antecedent magnetic condition, and indeed upon its whole magnetic history. A well-known example of hysteresis is presented by the case of permanent magnets. If a bar of hard steel is placed in a strong magnetic field, a certain intensity of magnetization is induced in the bar; but when the strength of the field is afterwards reduced to zero, the magnetization does not entirely disappear. That portion which is permanently retained, and which may amount to considerably more than one-half, is called the residual magnetization. The ratio of the residual magnetization to its previous maximum value measures the retentizfeness, or