The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 24 of 39

We do no work in bringing magnet a into position, for there are no forces against which work can be done. After the operation of placing a in position, the potential of the field is f!a. The operation of bringing magnet a from infinity has of course been simply that of moving a field of force of potential fla from infinity, where this same field of force had previously existed.

On bringing up magnet b, the work done is that of placing magnet b in a field of force of potential I2a. The work done is accordingly Qa (b).

446-448] Energy of a Magnetic Field 397

The work done in bringing up magnet c is that of placing magnet c in a field of force of potential £la + n6. It is therefore fla (c) + n6 (c).

Continuing this process we find that the total work done, W, is given by

w= na(b)

• na (C) + n6 (C)

• aa (d) + nb (d) + nc (d) + etc.

If, however, the magnets had been brought up in the reverse order, we should have had

W= nb (a) + Hc (a) +Qd(a) + ... + nn (a)

• nc{b) + nd(b) + ...+nn(b) +£id(c) + ...+nn(c)

• etc. so that by addition of these two values for W, we have

2W= nb(a) + nc(a)+£ld(a) + ... + nn(a)

• aa(b) + nc (b) + nd(b) + ... +nn (b)

• na(c) + n6(c) +nd(c) + ... + n„(c)

• na(d) + nb (d) + nc(d) +... + nn (d)

• etc.

The first line is equal to Q (a) except for the absence of the term Q,a (a), and so on for the other lines. Thus we have

2W= ft(a)-nB(a)

• H (6) - nb (b) + etc. = 2f2(a)-2na(a) (378).

The quantity Qa(a), the potential energy of the magnet a in its own field of force, is purely a constant of the magnet a, being entirely independent of the properties or positions of the other magnets b, c, d, .... Thus in equation (378), we may regard the term XHa(a) as a constant, and may replace the equation by

W-^tCl (a) + constant (379).

448. If we take the magnets a, b, c, ... n to be the ultimate magnetic particles, the values of Oa(a), H6(6), ... etc. all vanish, and their sum also vanishes. Thus equation (379) assumes the form

W=%£l{a) (380),

where the standard configuration from which W is measured is one in which the ultimate particles are scattered at infinity. The value of H (a) for a single particle is (cf. § 420)

an an an>

/7ai2 an en\

398 Permanent Magnetism [ch. xi

On replacing /u, by Idxdydz, we find for the energy of a system of magnetised bodies

-£+J,f+a£)** <381>-

the integration being taken throughout all magnetised matter.

449. An alternative proof can be given of equations (380) and (381), following the method of § 106, in which we obtained the energy of a system of electric charges.

Out of the magnetic materials scattered at infinity, it will be possible to construct n systems, each exactly similar as regards arrangement in space to the final system, but of only one-nth the strength of the final system. If n is made very great, it is easily seen that the work done in constructing a

single system vanishes to the order of — , so that, in the limit when n is very

great, the work done in constructing the series of n systems is infinitesimal. Thus the energy of the final system may be regarded as the work done in superposing this series of n systems.

Let us suppose so many of the component systems to have been super- posed, that the system in position is k times its final strength, where k is a positive quantity less than unity. The potential of the field at any point will be /eft. On bringing up a new system let us suppose that k is increased to k + die, so that the strength of the new system is d/c times that of the final system. In bringing up the new system, we place a magnet of d/c times the strength of a in a field of force of potential /eft, and so on with the other magnets. Thus the work done is

die . «ft (a) + die . kCI (b) + ..., and on integration of the work performed, we obtain

W=f /td* {fl (a) + ft (&) + ...}

Jo

= iSft (a), agreeing with equation (380), and leading as before to equation (381).

450. If the magnetic matter consists solely of normally magnetised shells, we may replace equation (381) by

where ds denotes thickness and dS an element of area of a shell. Replacing Ids by <j>, so that </> is the strength of a shell, we have

w=^lhd£ds-

448-451] Energy in the Medium 399

For uniform shells, </> may be taken outside the sign of integration, and the equation becomes

(cf. § 423), where n is the number of lines of induction which cross the shell.

This calculation measures the energy from a standard configuration in which the magnetic materials are all scattered at infinity. To calculate the energy measured from a standard configuration in which the shells have already been constructed and are scattered at infinity as complete shells, we use equation (378), namely

W=±X{n(a)-na(a)},

from which we obtain TT = iS

//£<■

where — — denotes the values - — at the surface of any shell if the shell itself on dn

is supposed annihilated.

If all the shells are uniform, this may again be written

W=-%S<f>n' (382),

where n' is the number of tubes of force from the remaining shells, which cross the shell of strength <£. An example of this has already occurred in

§424.

Energy in the Medium.

451. We  have  seen  that  the  energy  of  a  magnetic  field  is  given  by 
     

(cf. equation (381))

"1I + S + 0)* (383>-

the integration being taken over all magnetic matter. As a preliminary to transforming this into an integral taken through all space, we shall prove that

flf(aGL + b/3 + cy)dxdydz = 0 (384),

the integration being through all space.

The integral on the left can be written as

and this, by Green's Theorem, may be transformed into

1 1 1 H hp + =- + ^-\ dxdydz - j I O (la + mb + nc) dS,

400 Permanent Magnetism [ch. xi

the latter integral being taken over a sphere at infinity. Now at infinity O

is of the order of — (cf. § 67), while la + mb + nc vanishes, and dS is of

the order of r2, so that the surface integral vanishes on passing to the limit r = oo . Also the volume integral vanishes since

da db do _ dx dy dz '

and hence the theorem is proved.

Replacing a, b, c by their values, as given by equations (359), we find that equation (384) becomes

ff [ (a2 + /32 + 72) dxdydz + 4tt j (Act + B/3 + Cy) dxdydz = 0 . . .(385).

Both integrals are taken through all space, but since A—B = G=0 except in magnetic matter, we can regard the latter integral as being taken only over the space occupied by magnetic matter. This integral is therefore equal, by equation (383), to — 2 W, so that equation (385) becomes

W = -^jf!(a? + /32 + ^)dxdydz (386),

the integral being taken through all space.

This expression is exactly analogous to that which has been obtained for the energy of an electrostatic system, namely,

W=^rjfj(X2+Y2 + Z2)dxdydz.

And, as in the case of an electrostatic system, equation (386) may be interpreted as meaning that the energy may be regarded as spread through

the medium at a rate 5— (a2 + /32 + y2) per unit volume.

07T

Terrestrial Magnetism.

452. The  magnetism  of  the  earth  is  very  irregularly  distributed  and  is 
     

constantly changing. The simplest and roughest approximation of all to the

state of the earth's magnetism is obtained by regarding it as a bar magnet,

possessing two poles near to its surface, the position of these in 1906 being

as follows :

North Pole 70°30'N., 97°40'W.

South Pole* 73°39'S., 146° 15' E.

Another approximation, which is better in many ways although still very rough, is obtained by regarding the earth as a uniformly magnetised sphere.

• Sir E. Shackleton gives the position of the South Pole in 1909 as 72° 25' S., 155° 10' E.

451-454] Terrestrial Magnetism 401

With the help of a compass-needle, it will be possible to find the direction of the lines of force of the earth's field at any point. It will also be possible to measure the intensity of this field, by comparing it with known magnetic fields, or by measuring the force with which it acts on a magnet of known strength.

453. At any point on the earth, let us suppose that the angle between the line of magnetic force and the horizontal is 0, this being reckoned positive if the line of force points down into the earth, and let the horizontal projection of the line of force make an angle 8 with the geographical meridian through the point, this being reckoned positive if this line points west of north. The angle fr is called the dip at the point, the angle 8 is called the declination.

Let H be the horizontal component of force, then the total force may be regarded as made up of three components :

X = H cos 8, towards the north, Y = H sin 8, towards the west, Z = H tan 6, vertically downwards. If fl is the potential due to the earth's field at a point ot latitude I, longitude X, and at distance r from the centre, we have (cf. equations (331))

y-_I» r=- l a", Z = f (387).

r oi r cos I dX or 7

Analysis of Potential of Earth's field.

454. Since O is the potential of a magnetic system, the value of H in regions in which there is no magnetisation must (by § 408) be a solution of Laplace's equation, and must therefore (by § 233) be capable of expansion in the form

12= (^1 + ^ +...) + (£/ + &V + £,V + ...) (388),

in which Slt S2, ... S0', $/> &/> ••• are surface harmonics, of degrees indicated by the subscripts.

At the earth's surface, the first term is the part of the potential which arises from magnetism inside the earth, while the second term arises from magnetism outside.

The surface harmonic Sn can, as in § 275, be expanded in the form

m=n

Sn= S P% (sin I) (AniTn cos m\ + Bn<m sin m),

m = 0

so that XI can be put in the form

« = oo m = n (JPm (sin Z)

n= S 2 ] n )l+1 (Anim cos m\ + Bn<m sin mX)

• rnP% (sin I) (A'n>m cos m\ + B'nt7nsin m)[ .

26

402 Permanent Magnetism [ch. xi

Hence from equations (387) we obtain the values of X, Y, Z at any point in terms of the longitude and latitude of the point and the constants such

as xi. T^Tn} -Dji.wij -£«• n,m> ■& n,m,'

By observing the values of X, Y, Z at a great number of points, we obtain a system of equations between the constants An<m, etc., and on solving these we obtain the actual values of the constants, and therefore a knowledge of the potential as expressed by equation (388).

If the magnetic field arose entirely from magnetism inside the earth, we should of course expect to find #/ = 82' = • • . = 0, while if the magnetic field arose from magnetism entirely outside the earth, we should find Sl = S2=... = 0.

455. The results actually obtained are of extreme interest. The mag- netic field of the earth, as we have said, is constantly changing. In addition to a slow, irregular, and so-called "secular" change, it is found that there are periodic changes of which the periods are, in general, recognisable as the periods of astronomical phenomena. For instance there is a daily period, a yearly period, a period equal to the lunar month, a period of about 26 g days (the period of rotation of the inner core of the sun*), a period of about 11 years (the period of sun-spot variations), a period of 19 years (the period of the motion of the lunar nodes), and so on. Thus the potential can be divided up into a number of periodic parts and a residual constant, or slowly and irregularly changing, part. All the periodic parts are extremely small in comparison with the latter. It is found, on analysing the potentials of these different parts of the field, that the constant field arises from magnetisation inside the earth, while the daily variation arises mainly from magnetisation outside the earth. The former result might have been anticipated, but the latter could not have been predicted with any confidence. For the variation might have represented nothing more than a change in the permanent magnetism of the earth due to the cooling and heating of the earth's mass, or to the tides in the solid matter of the earth produced by the sun's attraction.

This daily variation is not such as could be explained by the magnetism of the sun itself; Chreef has found that it cannot be explained by the cooling and heating either of the earth's mass, or of the atmosphere as suggested by Faraday. Balfour Stewart J put forward the hypothesis that the daily variation was due mainly to electric currents circulating in the upper atmosphere as a result of the electromotive forces induced by the connective

• The outer surface of the sun is not rigid, and rotates at different rates in different latitudes. Thus it is impossible to discover the actual rate of rotation of the inner core except by such indirect methods as that of observing periods of magnetic variation.

f Roy. Soc. Phil. Trans., 202, p. 335.

J Art. "Terrestrial Magnetism," in the 9th Edn. of the Encyc. Brit. (1882).

454-456] Terrestrial Magnetism 403

motion of the atmosphere across the earth's magnetic field. This hypothesis was examined and developed by Schuster*, who examined the daily varia- tions by the method of harmonic analysis, already explained. Schuster found the origin of the magnetic field to be mainly external ; he suggested also that the convection currents indicated by the diurnal barometric changes were ulti- mately responsible for the phenomenon, and further found that a small part of the field must be attributed to origins inside the earth : these it was suggested might be a system of currents induced in the earth by the atmo- spheric currents above.

Chapmanf has recently reexamined the question, and obtains results in substantial agreement with Schuster's theory. He finds that the contribution from inside the earth is about 28 per cent, of the total diurnal variation. It is supposed that the conducting layer in the upper atmosphere in which the induced currents flow is that of which we already have evidence in the pheno- menon of the bending of electromagnetic waves round the earth; this layer is also the seat of the aurora borealis. Chapman finds that the internal magnetic field of induced currents would be explained by assuming that, beneath an upper non-conductive layer of 150 or 200 miles depth, the earth has a specific resistance of about 4 x 10~n C.G.s. units.

Besides the variation just considered, there is found to be a lunar diurnal variation, of period equal to the apparent period of motion of the moon. This appears to be the result of a semi-diurnal tidal oscillation of the atmosphere the mechanism being otherwise similar to that already explained.

456. The non-periodic part of the earth's field is found to arise entirely from magnetism inside the earth, having a potential of the form

«9

This method of analysing the earth's field is due to Gauss, who calculated the coefficients, with such accuracy as was then possible, for the year 1830. The most complete analysis of the field which now exists has been calculated by Neumayer for the year 1885, using observations of the field at 1800 points on the earth's surface.

The first few coefficients obtained by Neumayer are as follows :

(Ahl= -0248,

41|D--3157 {Bhi = _.060S}

A2<0 = -0079 -I"2'1" ".™? t2>2

[42fl = --0498, ^l2i2 = --0057, [B2A = -0130, J52>2=--0126, = --0<H4 {^,i = -039G, ^3,2 = --0279, ^3>3= --0033,

U?3,i = -0074, B3>2 = - -0004, £3i3 = --0055, A --0344 j^4,i= - -0306, ^4>2 = -:01983 Aii3 = -0068, ^l4>4 = --0008,

4,0 l#4,i =--0119, J94>2= -0071, £4)3 = -0051, Bi>4 = -0010.

• Phil. Trans. A, 180 (1889), p. 467, aud A, 203 (1907), p. 163. t Phil. Trans. A, 213 (1913), p. 279, and A, 218 (1919), p. 1.

26—2

404 Permanent Magnetism [ch. xi

457. The simplest approximation is of course obtained by ignoring all harmonics beyond the first. This gives as the magnetic potential

XI = — -UMPi (sin I) + iY (sin I) (Ahl cos X + 2?M sin X)[ =- 1*3157 sin I + cos I (-0248 cos \ - -0603 sin X)l .

The expression in brackets is necessarily a biaxial harmonic of order unity (cf. § 276) ; it is easily found to be equal to *3224 cos 7, where 7 is the angular distance of the point (I, X) from the point

lat. 78° 20' N., long. 67° 17' W (389).

The potential is now £1 = -3224 52i2 ,

which is the potential of a uniformly magnetised sphere, having as direction of magnetisation the radius through the point (§ 415). Or again, it is the potential of a single magnetic particle at the centre of the earth, pointing in this same direction. It is naturally impossible to distinguish between these two possibilities by a survey of the field outside the earth. Green's theorem has already shewn that we cannot locate the sources of a field inside a closed surface by a study of the field outside the surface.

EXAMPLES.

1. Two small magnets float horizontally on the surface of water, one along the direction of the straight line joining their centres, and the other at right angles to it. Prove that the action of each magnet on the other reduces to a single force at right angles to the straight line joining the centres, and meeting that line at one-third of its length from the longitudinal magnet.

2. A small magnet A CB, free to turn about its centre C, is acted on by a small fixed magnet PQ. Prove that in equilibrium the axis ACB lies in the plane PQC, and that tan# = — i tan 6', where 6, & are the angles which the two magnets make with the line joining them.

3. Three small magnets having their centres at the angular points of an equilateral triangle ABC, and being free to move about their centres, can rest in equilibrium with the magnet at A parallel to BC, and those at B and C respectively at right angles to AB and AC. Prove that the magnetic moments are in the ratios

V3 : 4 : 4.

4. The axis of a small magnet makes an angle <£ with the normal to a plane. Prove that the line from the magnet to the point in the plane where the number of lines of force crossing it per unit area is a maximum makes an angle 6 with the axis of the magnet, such that

2 tan <9 = 3 tan 2 ($-0).

Examples 405

5. Two small magnets lie in the same plane, and make angles 6, ff with the line joining their centres. Shew that the line of action of the resultant force between them divides the line of centres in the ratio

tan ff + 2 tan 6 : tan 8 + 2 tan ff

6. Two small magnets have their centres at distance r apart, make angles 8, ff with the line joining them, and an angle e with each other. Shew that the force on the first magnet in its own direction is

3mm' ,„ „ .

— — (5 cos2 6 cos 8 - cos ff - 2 cos e cos 8).

Shew that the couple about the line joining them which the magnets exert on one another is

mm'

o?sini where d is the shortest distance between their axes produced.

7. Two magnetic needles of moments M. M' are soldered together so that their directions include an angle a. Shew that when they are suspended so as to swing freely in a uniform horizontal magnetic field, their directions will make angles 8, & with the lines of force, given by

sin 6 _ sin 6' _ sin a

M' M (i/2 + M>2 + 2MM' cos a)* '

8. Prove that if there are two magnetic molecules, of moments M and M', with their centres fixed at A and B, where AB=r, and one of the molecules swings freely, while the other is acted on by a given couple, so that when the system is in equilibrium this molecule makes an angle 8 with AB, then the moment of the couple is

f MM' sin 28 jr3 (3 cos2 5 + 1)*, where there is no external field.

9. Two small equal magnets have their centres fixed, and can turn about them in a magnetic field of uniform intensity H, whose direction is perpendicular to the line r joining the centres. Shew that the position in which the magnets both point in the direction of the lines of force of the uniform field is stable only if

H > 3M/r*.

10. Two magnetic particles of equal moment are fixed with their axes parallel to the axis of 2, and in the same direction, and with their centres at the points + a, 0, 0. Shew that if another magnetic molecule is free to turn about its centre, which is fixed at the point (0, y, z), its axis will rest in the plane x = 0, and will make with the axis of z the angle

, Svz tan-1— = — ■— 5.

2zi-az — yi

Examine which of the two positions of equilibrium is stable.

11. Prove that there are four positions in which a given bar magnet may be placed so as to destroy the earth's control of a compass-needle, so that the needle can point indifferently in all directions. If the bar is short compared with its distance from the needle, shew that one pair of these positions are about 1^ times more distant than the other pair.

I

406 Permanent Magnetism [ch. xi

12. Three small magnets, each of magnetic moment ju, are fixed at the angular points of an equilateral triangle ABC, so that their north poles lie in the directions AC, AB, BC respectively. Another small magnet, moment //, is placed at the centre of the triangle, and is free to move about its centre. Prove that the period of a small oscillation is the same as that of a pendulum of length Ibzgj\l^blp.yL, where b is the length of a side of the triangle, and / the moment of inertia of the movable magnet about its centre.

13. Three magnetic particles of equal moments are placed at the corners of an equilateral triangle, and can turn about those points so as to point in any direction in the plane of the triangle. Prove that there are four and only four positions of equilibrium such that the angles, measured in the same sense of rotation, between the axes of the magnets and the bisectors of the corresponding angles of the triangle are equal. Also prove that the two symmetrical positions are unstable.

14. Four small equal magnets are placed at the corners of a square, and oscillate under the actions they exert on each other. Prove that the times of vibration of the principal oscillations are

Mk2cP ) 2 |m23(2 + l/2v/2)J

f Mk*-d? } h (m2 (3 -1/2^/2)/ '

. (Mk2d32 v/21 *

277 \— 3^— r •

where m is the magnetic moment, and M&2 the moment of inertia, of a magnet, and d is a side of the square.

15. A system of magnets lies entirely in one plane and it is found that when the axis of a small needle travels round a contour in the plane that contains no magnetic poles, the needle turns completely round. Prove that the contour contains at least one equilibrium point.

16. Prove that the potential of a body uniformly magnetised with intensity I is, at any external point, the same as that due to a complex magnetic shell coinciding with the surface of the body and of strength Ix, where x is a coordinate measured parallel to the direction of masrnetisation.

17. A sphere of hard steel is magnetised uniformly in a constant direction and a magnetic particle is held at an external point with the axis of the particle parallel to the direction of magnetisation of the sphere. Find the couples acting on the sphere and on the particle.

18. A spherical magnetic shell of radius a is normally magnetised so that its strength at any point is Si, where Si is a spherical surface harmonic of positive order i. Shew that the potential at a distance r from the centre is

-47rfirK«y when>-<«,

2i + l

i + 1

Si(-) when r>a.

19. If  a  small  spherical  cavity  be  made  within  a  magnetised  body,  prove  that  the 
    

components of magnetic force within the cavity are

a+$A, (3 + ±B, y+£C/.

Examples 407

20. If the earth were a uniformly magnetised sphere, shew that the tangent of the dip at any point would be equal to twice the tangent at the magnetic latitude.

21. Prove that if the horizontal component, in the direction of the meridian, of the earth's magnetic force were known all over its surface, all the other elements of its magnetic force might be theoretically deduced.

22. From the principle that the line integral of the magnetic force round any circuit ordinarily vanishes, shew that the two horizontal components of the magnetic force at any station may be deduced approximately from the known values for three other stations which lie around it. Shew that these sis known elements are not independent, but must satisfy one equation of condition.

23. If the earth were a sphere, and its magnetism due to two small straight bar magnets of the same strength situated at the poles, with their axes in the same direction along the earth's axis, prove that the dip 8 in latitude X would be given by

/\ x\ ,X „, X „, cx <

8 cot I 8 + - j = cot - - 6 tan - - 3 tan2 - .

24. Assuming  that  the  earth  is  a  sphere  of  radius  a,  and  that  the  magnetic  potential 
    

Q is represented by

-©+©'+* ©'+*©'■

shew that 12 is completely determined by observations of horizontal intensity, declination and dip at four stations, and of dip at four more.

25. Assuming that in the expansion of the earth's magnetic potential the fifth and higher harmonics may be neglected, shew that observations of the resultant magnetic force at eight points are sufficient to determine the potential everywhere.

26. Assuming that the earth's magnetism is entirely due to internal causes, and that in latitude X the northerly component of the horizontal force is A cos X + B cos3 X, prove that in this latitude the vertical component reckoned downwards is

2(,4+fS)sinX-§flsin3X,

CHAPTER XII

INDUCED MAGNETISM

Physical Phenomena.

458. Reference has already been made to the well-known fact that a magnet will attract small pieces of iron or steel which are not themselves magnets. Here we have a phenomenon which at first sight does not seem to be explained by the law of the attractions and repulsions of magnetic poles. It is found, however, that the phenomenon is due to a magnetic " induction " of a kind almost exactly similar to the electrostatic induction already discussed. It can be shewn that a piece of iron or steel, placed in the presence of a magnet, will itself become magnetised. Temporarily, this piece of iron or steel will be possessed of magnetic poles of its own, and the system of attractions and repulsions between these and the poles of the original permanent magnet will account for the forces which are observed to act on the metal.

It has, however, been seen that pairs of corresponding positive and negative poles cannot be separated by more than molecular distances, so that we are led to suppose that each particle of the body in which magnetism is induced must become magnetised, the adjacent poles neutralising one another as in a permanent magnet.

Taking this view, it will be seen that the attraction of a magnet for an unmagnetised body is analogous to the attraction of an electrified body for a piece of dielectric (§ 197), rather than to its attraction for an uncharged conductor. The attraction of a charged body for a fragment of a dielectric has been seen to depend upon a molecular phenomenon taking place in the dielectric. Each molecule becomes itself electrified on its opposite faces, with charges of opposite sign, these charges being equal and opposite so that the total charge on any molecule is nil. In the same way, when magnetism is induced in any substance, each molecule of the substance must be supposed to become a magnetic particle, the total charge of magnetism on each particle being nil. It follows that the attraction of a magnet for a non-magnetic body is merely the aggregate of the attractive forces acting on the different individual particles of the body.

459. Confirmation of this view is found in the fact that the intensity of the attraction exerted by a magnet on a non-magnetised body depends on

458-460] Induced Magnetism 409

the material of the latter. The significance of this fact will, perhaps, best be realised by comparing it with the corresponding fact of electrostatics. When an uncharged conductor is attracted by a charged body, the phenomena in the former body which lead to this attraction are mass-phenomena : currents of electricity flow through the mass of the body until its surface becomes an equipotential. Thus the attraction depends solely upon the shape of the body and not upon its structure. On the other hand, the phenomena which lead to the attraction of a fragment of dielectric are, as we have seen, molecular phenomena. They are conditioned by the shape and arrangement of the molecules, with the result that the total force depends on the nature of the dielectric material.

All magnetic phenomena occurring in material bodies must be molecular, as a consequence of the fact that corresponding positive and negative poles cannot be separated by more than molecular distances. Hence we should naturally expect to find, as we do find, that all magnetic phenomena in material bodies, and in particular the attraction of unmagnetised matter by a magnet, would depend on the nature of the matter. There would be a real difficulty if the attraction were found to depend only on the shape of the bodies.

460. The amount of the action due to magnetic induction varies enormously more with the nature of the matter than is the case with the corresponding electric action. Among common substances the phenomenon of magnetic induction is not at all well-marked except in iron and steel. These substances shew the phenomenon to a degree which appears very surprising when compared with the corresponding electrostatic phenomenon. After these substances, the next best for shewing the phenomena of induction are nickel and cobalt, although these are very inferior to iron and steel. It is worth noticing that the atomic weights of iron, nickel and cobalt are very close together*, and that the three elements hold corresponding positions in the table of elements arranged according to the periodic law.

It has recently been found that certain rare metals shew magnetic induction to an extent comparable with iron, and that alloys can be formed to shew great powers of induction although the elements of which these alloys are formed are almost entirely non-magneticf.

It appears probable that all substances possess some power of magnetic induction, although this is generally extremely feeble in comparison with that of the substances already mentioned. In some substances, the effect is of the opposite sign from that in iron, so that a fragment of such matter is repelled from a magnetic pole. Substances in which the effect is of the

• Iron = 55-5, nickel = 58-3, cobalt = 58-56.

t For an account of the composition and properties of Heusler's alloys, see a paper by J. C. McLennan, Phijs. Review, Vol. 24, p. 449.

1

410 Induced Magnetism [ch. xii

same kind as in iron are called 'paramagnetic, while substances in which the effect is of the opposite kind are called diamagnetic.

The phenomenon of magnetic induction is much more marked in para- magnetic, than in diamagnetic, substances. The most diamagnetic substance known is bismuth, and its coefficient of susceptibility (§ 461, below) is only

about 77T- of that of the most paramagnetic samples of iron. 109

Coefficients of Susceptibility and Permeability.

461. When a body which possesses no permanent magnetism of its own is placed in a magnetic field, each element of its volume will, for the time it remains under the influence of the magnetic field, be a magnetic particle. If the body is non-crystalline the direction of the induced magnetisation at any point will be that of the magnetic force at the point. Thus if H denote the magnetic force at any poinf, we can suppose that the induced magnetism, of an intensity /, has its direction the same as that of H.

Thus if a, /?, <y are the components of magnetic force, and A, B, C the components of induced magnetisation, we shall have equations of the form

A =/ca \

B = kJ3\ (390),

C=*7J

the quantity k being the same in each equation because the directions of I and H are the same.

The quantity k is called the magnetic susceptibility.

If the body has no permanent magnetisation, the whole components of magnetisation are the quantities A, B, G given by equations (390), and the components of induction are given (cf. equations (359)) by

a = a + 4nrA = a (1 + 4>7tk),

& = /3 + 47t£ = /3(1 + 47™),

c = 7 + 4nrC = 7(1+ 47J7C).

If we put ^ = 1 + 4™ (391),

we have a = fiot '

b = fx/3 > (392),

c = fx<y , and fi is called the magnetic permeability.

462. The quantities k and fi are by no means constant for a given substance. Their value depends largely upon the physical conditions, particularly the temperature, of the substance, upon the strength of the magnetic field in which the substance is placed, and upon the previous magnetic experiences of the substance m question.

400-463]

Physical Phenomena

411

We pass to the consideration of the way in which the magnetic coefficients vary with some of these circumstances. As k and /u, are connected by a simple relation (equation (391)), it will be sufficient to discuss the variations of one of these quantities only, and the quantity /m will be the most convenient for this purpose. Moreover, as the phenomenon of induced magnetisation is almost insignificant in all substances except iron and steel, it will be sufficient to consider the magnetic phenomena of these substances only.

463. Dependence of p on H. The way in which the value of //, depends on H is, in its main features, the same for all kinds of iron. For small forces, fi is a constant, for larger forces fx increases, finally it reaches a maximum, and after this decreases in such a way that ultimately fiH approximates to a constant value, known as the "saturation" value. This is represented graphically in a typical case in fig. 113, which represents the results obtained by Ewing from experiments on a piece of iron wire.

mH = 1500.0

/iH = 10000

MH=5000

M=3000

ix = 2000

^ = 1000

H =