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retentivity, of the metal.1 Steel, which is well suited for the construe- Demagrietizirtg Factors. tion of permanent magnets, is said to possess great “ coercive force.” To this term, which had long been used in a loose and indefinite manner, ]. Hopkinson supplied a precise meaning (Phil. Trans. clxxvi. 460). The coercive force, or coercizfity, of a material is that reversed magnetic force which, while it is acting, just suffices to reduce the residual induction to nothing after the material has been temporarily submitted to any great magnetizing force. A metal which has great retentiveness may at the same time have small coercive force, and it is the latter quality which is of chief importance in permanent magnets.

Demagnetizirzg Force.-It has already been mentioned that when a ferroma netic body is placed in a magnetic field, the resultant magnetic iorce H, at a point within the body, is compounded of the force HD, due to the external held, and of another force, Hi, arising from the induced magnetization of the body. Since Hi generally tends to oppose the external force, thus making H less than Ho, it may be called the demagnetizirtg force. Except in the few special cases when a uniform external field produces uniform magnetization, the value of the demagnetizing force cannot be calculated, and an exact determination of the actual magnetic force within the body is therefore impossible. An important instance in which the calculation can be made is that of an elongated ellipsoid of revolution placed in a uniform field HD, with its axis of revolution parallel to the lines of force. The magnetization at any point inside the ellipsoid will then be


rm <'9>

where N=41r (i-1) (ilogg-1,

e being the eccentricity (see Maxwell's Treatise, § 438). Since I =xH, we have

KH +, <N1=.<H0, (30)

or H = H0 - N I,

NI being the demagnetizing force Hi. N may be called, after H. du Bois (Magnetic Circuit, p. 33), the demagrtetizirtg factor, and the ratio of the length of the ellipsoid 26 to its equatorial diameter 20 (=c/a), the dimensional ratio, denoted by the symbol m. az 1

Since e= 1-2a= 1-E

theiabove expression for N may be written N =, ,-$2 ~ .-L los l" ll' Q';. 1

2/m”- 1 m - 4/m2-1


=ITég%' Qi-.iilog m'l-w/ml-I>-Ii,

from which the value of N for a given dimensional ratio can be calculated. When the ellipsoid is so much elongated that 1 is negligible in relation to 1112, the expression approximates to the simpler form

N=% log 2m~1 . (31)

In the case of a sphere, e =O and N =§ 1r; therefore from (29) xvhenee H =3-Qian. = H., (33)

and B =;iH = Ho. (34)

Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of, a is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three tirnes that of the external force, and nearly independent of the permeability. The demagnetizing force inside a cylindrical rod placed longitudinally in a uniform field H0, is not uniform, being greatest at the ends and least in the middle part. Denoting its mean value by Ht, and that of the demagnetizing factor by N, we have H=H0-H;=H°-NI. (35)

Du Bois has shown that When the dimensional ratio m (=length/ diameter) exceeds 100, NIlI2=COHSt3Ht=45, and hence for long thin rods

N =45/mf. (36)

From an analysis of a number of experiments made with rods of different dimensions H. du Bois has deduced the corresponding mean demagnetizing factors. These, together with values of m2N for cylindrical rods, and of N and m2N for ellipsoids of revolution, are given in the following useful table (loc. cit. p. 41):- I

the “ residual induction ” (=41rI).

Hopkinson specified the retentiveness by the numerical value of Cylinder. Ellipsoid.

m. -

N. WN. N. m2N.

0 12-5664 0 12-5664 0

0-5 - — 6-5864-1

- - 4-1888-5

- - 0-7015-IO

0-2160 21-6 0-2549 25-5

I5 0-1206 27 1 0-1350 30-5

20 0-0775 31 0 0-0848 34-0

25 0'0533 33'4 0'0579 35'2

30 @0393 35'4 0'0432 33'3

40 0-0238 38-7 0'O266 42-5

50 0-0162 40-5 0-0181 45-3

60 0-0118 42-4 0-0132 47-5

70 0-0089 43-7 0-0101 49-5

80 0-0069 -44-4 0-0080 51-2

Q0 0-0055 44-8 0-0065 52-5

100 0-0045 45-0 0-0054 54-0

150 0-0020 45-0 0-0026 58-3

200 0-0011 45-0 0.0016 64-0

300 0-00050 45-0 0-00075 67-5

400 0-00028 45 0 0-00045 72-0

500 0-00018 45 0 0-00030 75-0

1000 0-00005 45 0 0-00008 80-0

In the middle part of a rod which has a length of 400 or 500 diameters the effect of the ends is insensible; but for many experiments the condition of endlessness may be best secured by giving the metal the shape of a ring of uniform section, the magnetic field being produced by an electric current through a coil of wire evenly wound round the ring. In such cases Hi =o and H =H0. The residual magnetization I, retained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing force NI, , which is greater the smaller the dimensional ratio. Hence the difficulty of imparting any considerable permanent magnetization to a short thick bar not possessed of great coercive force. The magnetization retained by a long thin rod, even when its coercive force is small, is sometimes little less than that which was produced by the direct action of the field. Demagrtetizaliori by Rei/ersals.-In the course of an experiment it is often desired to eliminate the effects of previous magnetization, and, as far as possible, wipe out the, magnetic history of a specimen. In order to attain this result it was formerly the ractice to raise the metal to a bright red heat, and allow it to cool whilie carefully guarded from magnetic influence. This operation, besides being very troublesome, was open to the objection that it was almost sure to produce a material but uncertain change in the physical constitution of the metal, so that, in fact, the results of experiments made before and after the treatment were not comparable. Ewing introduced the method (Phil. Trans. clxxvi. 539) of demagnetizing a specimen by subjecting it to a succession' of magnetic forces which alternated in direction and gradually diminished in strength from a high value to zero. By means of a simple arrangement, which will be described farther on, this process can be carried out in a few seconds, and the metal can be' brought as often as desired to a definite condition, which, if not quite identical with the virgin state, at least closely approximates to it.

Forces acting on a Small Body in the Magnetic Field.-If a small magnet of length ds and pole-strength rn is brought into a magnetic field such that the values of the magnetic potential at the negative and positive poles respectively are V1 and V2, the work done upon the magnet, and therefore its potential energy, will be W =m(V2 -V1) =mdV,

which may be written

dV dV

W=md.1E;- =Mm= - MH0 = - 1/IH0,

where M is the moment of the magnet, v the volume, I the magnetization, and H0 the magnetic force along ds. The small magnet may be a sphere rigidly magnetized in the direction of Ho; if this is replaced by an isotropic sphere inductively magnetized by the field, then, for a displacement so small that the magnetization of the sphere may be regarded as unchanged, we shall have dW = - vIdH0 = -vé-1;-¢H0dH@;



The mechanical force acting on the sphere in the direction of displacement x IS

7l ..l . .

W 21-I-§ 1rxH2°

dW x dH”

F= - dx:UI-l-$11-ic dxo' (38)