Page:EB1911 - Volume 17.djvu/342

retentivity, of the metal.^[1] Steel, which is well suited for the construction of permanent magnets, is said to possess great “coercive force.” To this term, which had long been used in a loose and indefinite manner, J. Hopkinson supplied a precise meaning (Phil. Trans. clxxvi. 460). The coercive force, or coercivity, 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.

Demagnetizing Force.—It has already been mentioned that when a ferromagnetic body is placed in a magnetic field, the resultant magnetic force H, at a point within the body, is compounded of the force H₀, due to the external field, and of another force, H_i, arising from the induced magnetization of the body. Since H_i generally tends to oppose the external force, thus making H less than H₀, it may be called the demagnetizing 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 H₀, with its axis of revolution parallel to the lines of force. The magnetization at any point inside the ellipsoid will then be

I =	κH0
	1 + κN

(29)

where

N = 4π ${\Big (}$	1	− 1 ${\Big )}{\Big (}$	1	log	1 + e	− 1 ${\Big )}$,
	e2		2e		1 − e

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

κH + κNI = κH₀,

(30)

or

H = H₀ − NI,

NI being the demagnetizing force H_i. N may be called, after H. du Bois (Magnetic Circuit, p. 33), the demagnetizing factor, and the ratio of the length of the ellipsoid 2c to its equatorial diameter 2a (= c/a), the dimensional ratio, denoted by the symbol m.

Since

e = √( 1 −	a2	) = √( 1 −	1	),
	c2		m2

the above expression for N may be written

N =	4π	${\Big (}$	m	log	m + √(m2 − 1)	− 1 ${\Big )}$
	m2 − 1		2√(m2 − 1)		m − √(m2 − 1)

=	4π	{	m	log ( m+ √(m2 − 1) ) − 1 },
	m2 − 1		√(m2 − 1)

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 m², the expression approximates to the simpler form

N =	4π	${\big (}$ log 2m − 1 ${\big )}$
	m2

(31)

In the case of a sphere, e = O and N = 4/3π; therefore from (29)

I = κH =	κH0	=	3κ	H0,
	1 + 4/3πκ		3 + 4πκ

(32)

Whence

H =	3	H0 =	3	H0,
	3 + 4πκ		μ + 2

(33)

and

B = μH =	3μ	H0.
	μ + 2

(34)

Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of μ 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 times 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 H_i, and that of the demagnetizing factor by N, we have

H = H₀ − H_i = H₀ − NI.

(35)

Du Bois has shown that when the dimensional ratio m (= length/diameter) exceeds 100, Nm² = constant = 45, and hence for long thin rods

N = 45 / m².

(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):—

Demagnetizing Factors.

m.	Cylinder.		Ellipsoid.
	N.	m2N.	N.	m2N.
0	12.5664	0	12.5664	0
0.5	—	—	6.5864	—
1	—	—	4.1888	—
5	—	—	0.7015	—
10	0.2160	21.6	0.2549	25.5
15	0.1206	27.1	0.1350	30.5
20	0.0775	31.0	0.0848	34.0
25	0.0533	33.4	0.0579	36.2
30	0.0393	35.4	0.0432	38.8
40	0.0238	38.7	0.0266	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
90	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 H_i = 0 and H = H₀.

The residual magnetization I_r retained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing force NI_r, 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.

Demagnetization by Reversals.—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 practice to raise the metal to a bright red heat, and allow it to cool while 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 m 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

W = mds	dV	= M	dV	= −MH0 = −vIH0,
	ds		ds

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 H₀; 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 = −vI dH0 = −v	κ	H0 dH0;
	1 + 4/3πκ

whence

W = −	v		κ	H20.
	2		1 + 4/3πκ

(37)

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

F = −	dW	= v	κ		dH20	.
	dx		1 + 4/3πκ		dx

(38)

1. Hopkinson specified the retentiveness by the numerical value of the “residual induction” (= 4πI).