A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 5 of 27

In empty space the coefficient of induction is unity. In bodies capable of induced magnetization the coefficient of induction is 1 + 4 TT K = /x, where K is the quantity already defined as the co efficient of induced magnetization.

429.] Let p, [k be the values of p on opposite sides of a surface

E

52 INDUCED MAGNETIZATION. [4^9-

separating two media, then if F, V are the potentials in the two media, the magnetic forces towards the surface in the two media

dV , dV' are -7- and -3-7- • Av dv

The quantities of magnetic induction through the element of

dV dV

surface dS are u-^-dS and u? -^-j-dS in the two media respect- r dv dv

ively reckoned towards dS.

Since the total flux towards dS is zero, dV ,dV

But by the theory of the potential near a surface of density o-,

dV dV

•      4.47r(r:r=  o. 
  

dv dv

Hence -7- (l — —A + 4 TT or = 0.

c?i> V ju, /

If K! is the ratio of the superficial magnetization to the normal force in the first medium whose coefficient is jot, we have

4 77 KI =

Hence K± will be positive or negative according as /ut is greater or less than //. If we put ju = 4 TT /c + 1 and p' '= 4 77 / + 1 ,

"47T/+1

In this expression K and K' are the coefficients of induced mag netization of the first and second medium deduced from experiments made in air, and KX is the coefficient of induced magnetization of the first medium when surrounded by the second medium.

If K is greater than K, then /q is negative, or the apparent magnetization of the first medium is in the opposite direction from the magnetizing force.

Thus, if a vessel containing a weak aqueous solution of a para magnetic salt of iron is suspended in a stronger solution of the same salt, and acted on by a magnet, the vessel moves as if it were magnetized in the opposite direction from that in which a magnet would set itself if suspended in the same place.

This may be explained by the hypothesis that the solution in the vessel is really magnetized in the same direction as the mag netic force, but that the solution which surrounds the vessel is magnetized more strongly in the same direction. Hence the vessel is like a weak magnet placed between two strong ones all mag-

43°-] POISSON'S THEORY OP MAGNETIC INDUCTION. 53

netized in the same direction, so that opposite poles are in contact. The north pole of the weak magnet points in the same direction as those of the strong- ones, but since it is in contact with the south pole of a stronger magnet, there is an excess of south magnetism in the neighbourhood of its north pole, which causes the small magnet to appear oppositely magnetized.

In some substances, however, the apparent magnetization is negative even when they are suspended in what is called a vacuum.

If we assume K = 0 for a vacuum, it will be negative for these substances. No substance, however, has been discovered for which

K has a negative value numerically greater than — — , and therefore for all known substances /x is positive.

Substances for which K is negative, and therefore p less than unity, are called Diamagnetic substances. Those for which K is positive, and ^ greater than unity, are called Paramagnetic, Ferro magnetic, or simply magnetic, substances.

We shall consider the physical theory of the diamagnetic and paramagnetic properties when we come to electromagnetism, Arts. 831-845.

430.] The mathematical theory of magnetic induction was first given by Poisson *. The physical hypothesis on which he founded his theory was that of two magnetic fluids, an hypothesis which has the same mathematical advantages and physical difficulties as the theory of two electric fluids. In order, however, to explain the fact that, though a piece of soft iron can be magnetized by induction, it cannot be charged with unequal quantities of the two kinds of magnetism, he supposes that the substance in general is a non-conductor of these fluids, and that only certain small portions of the substance contain the fluids under circumstances in which they are free to obey the forces which act on them. These small magnetic elements of the substance contain each pre cisely equal quantities of the two fluids, and within each element the fluids move with perfect freedom, but the fluids can never pass from one magnetic element to another.

The problem therefore is of the same kind as that relating to a number of small conductors of electricity disseminated through a dielectric insulating medium. The conductors may be of any form provided they are small and do not touch each other.

If they are elongated bodies all turned in the same general

• Memoires de I'lnstitut, 1824.

54 INDUCED MAGNETIZATION. [43O.

direction, or if they are crowded more in one direction than another, the medium, as Poisson himself shews, will not be isotropic. Poisson therefore, to avoid useless intricacy, examines the case in which each magnetic element is spherical, and the elements are dissem inated without regard to axes. He supposes that the whole volume of all the magnetic elements in unit of volume of the substance is k.

We have already considered in Art. 314 the electric conductivity of a medium in which small spheres of another medium are dis tributed.

If the conductivity of the medium is ^ , and that of the spheres ju2, we have found that the conductivity of the composite system is

P = f*l-j

Putting fa = 1 and /ot2 = oc, this becomes

_ 1 + 2/fc

This quantity ju is the electric conductivity of a medium con sisting of perfectly conducting spheres disseminated through a medium of conductivity unity, the aggregate volume of the spheres in unit of volume being k.

The symbol ^ also represents the coefficient of magnetic induction of a medium, consisting of spheres for which the permeability is infinite, disseminated through a medium for which it is unity.

The symbol k, which we shall call Poisson's Magnetic Coefficient, represents the ratio of the volume of the magnetic elements to the whole volume of the substance.

The symbol K is known as Neumann's Coefficient of Magnet ization by Induction. It is more convenient than Poisson's.

The symbol ^ we shall call the Coefficient of Magnetic Induction. Its advantage is that it facilitates the transformation of magnetic problems into problems relating to electricity and heat.

The relations of these three symbols are as follows :

47TK

3 * =

3*

477

If we put K = 32, the value given by Thalen's* experiments on

• Recherches sur les Proprietes Magnetiques dufer, Nova Ada, Upsal, 1863.

430.] POISSON'S THEORY OF MAGNETIC INDUCTION. 55

soft iron, we find k = |f|-. This, according to Poisson's theory, is the ratio of the volume of the magnetic molecules to the whole volume of the iron. It is impossible to pack a space with equal spheres so that the ratio of their volume to the whole space shall be so nearly unity, and it is exceedingly improbable that so large a proportion of the volume of iron is occupied by solid molecules whatever be their form. This is one reason why we must abandon Poisson's hypothesis. Others will be stated in Chapter VI. Of course the value of Poisson's mathematical investigations remains unimpaired, as they do not rest on his hypothesis, but on the experimental fact of induced magnetization.

CHAPTER V.

PARTICULAR PROBLEMS IN MAGNETIC INDUCTION.

A Hollow Spherical Shell.

431.] THE first example of the complete solution of a problem in magnetic induction was that given by Poisson for the case of a hollow spherical shell acted on by any magnetic forces whatever.

For simplicity we shall suppose the origin of the magnetic forces to be in the space outside the shell.

If V denotes the potential due to the external magnetic system, we may expand V in a series of solid harmonics of the form

7= CQ80 + C1S1r + to. + CiSiiA, (1)

where r is the distance from the centre of the shell, #< is a surface harmonic of order i, and Ci is a coefficient.

This series will be convergent provided r is less than the distance of the nearest magnet of the system which produces this potential. Hence, for the hollow spherical shell and the space within it, this expansion is convergent.

Let the external radius of the shell be a2 and the inner radius alf and let the potential due to its induced magnetism be H. The form of the function H will in general be different in the hollow space, in the substance of the shell, and in the space beyond. If we expand these functions in harmonic series, then, confining our attention to those terms which involve the surface harmonic Si9 we shall find that if Q^ is that which corresponds to the hollow space within the shell, the expansion of Q^ must be in positive har monics of the form Al St r*, because the potential must not become infinite within the sphere whose radius is a^.

In the substance of the shell, where r± lies between aL and a2, the series may contain both positive and negative powers of /*, of the form

Outside the shell, where r is greater than a2, since the series

HOLLOW SPHERICAL SHELL. 57

must be convergent however great r may be, we must have only negative powers of /, of the form

The conditions which must be satisfied by the function 12, are : It must be (1) finite, and (2) continuous, and (3) must vanish at an infinite distance, and it must (4) everywhere satisfy Laplace's equation.

On account of (1) Bl = 0.

On account of (2) when r = a^

(4-4,H2i+1-52=0, (2)

and when r = «2,

(^2-J3)^2i+1 + ^2-^3 = 0. (3)

On account of (3) Az = 0, and the condition (4) is satisfied everywhere, since the functions are harmonic.

But, besides these, there are other conditions to be satisfied at the inner and outer surface in virtue of equation (10), Art. 427.

At the inner surface where r = alt

, d£l9 d&, dV ,..

<1+4*«>V-ifr+4"'* = <)'

and at the outer surface where r = a2,

d dV

,KN 0.

From these conditions we obtain the equations

iCia12i+l = <), (6)

«22»+1-(^+l)^2)+(^+l)^3+47r^^22i+1=0^ (7) and if we put

we find

/ /, 2» + l\

4 = -(4™)^ + l)(l-Q) }NtClt (9)

[I a 2t+l^-j

2^+l+477K(^+l)(l-(^) )J^Ci, (10)

(11) «12i+1)^Ci. (12)

These quantities being substituted in the harmonic expansions give the part of the potential due to the magnetization of the shell. The quantity Ni is always positive, since 1 -f 4 ir K can never be negative. Hence A1 is always negative, or in other words, the

58 MAGNETIC PEOBLEMS. [432.

action of the magnetized shell on a point within it is always op posed to that of the external magnetic force whether the shell he paramagnetic or diamagnetic. The actual value of the resultant potential within the shell is

or (l + 4wjc)(2i+ l^NiCtS.r. (13)

432.] When K is a large number, as it is in the case of soft iron, then, unless the shell is very thin, the magnetic force within it is hut a small fraction of the external force.

In this way Sir W. Thomson has rendered his marine galvano meter independent of external magnetic force hy enclosing it in a tube of soft iron.

433.] The case of greatest practical importance is that in which i = 1. In this case

(14)

9(l+47TK)+2(477K)2(l-0')

= -477*13+ 8w«(l — (^) )UViQ, !> (15)

L X dr> —I

£3= 4 7TK(3 + 8 7TK)(#23 — «13)^V1 Ci.

The magnetic force within the hollow shell is in this case uniform and equal to

9(1+477*)

If we wish to determine K by measuring the magnetic force within a hollow shell and comparing it with the external magnetic force, the best value of the thickness of the shell may be found from the equation

1_

2 (4 TT K)2

The magnetic forc"e inside the shell is then half of its value outside. Since, in the case of iron, K is a number between 20 and 30, the thickness of the shell ought to be about the hundredth part of its radius. This method is applicable only when the value of K is large. When it is very small the value of A^ becomes insensible, since it depends on the square of K.

434-1 SPHERICAL SHELL. 59

For a nearly solid sphere with a very small spherical hollow,

. 2(4ir«)«

1J

4 77 K

The whole of this investigation might have been deduced directly from that of conduction through a spherical shell, as given in Art. 312, by putting ^ = (1 -f 47TK)/£2 in the expressions there given, remembering that A^ and A2 in the problem of conduction are equi valent to C1 + A1 and C1 + A2 in the problem of magnetic induction.

434.] The corresponding solution in two dimensions is graphically represented in Fig. XV, at the end of this volume. The lines of induction, which at a distance from the centre of the figure are nearly horizontal, are represented as disturbed by a cylindric rod magnetized transversely and placed in its position of stable equi librium. The lines which cut this system at right angles represent the equipotential surfaces, one of which is a cylinder. The large dotted circle represents the section of a cylinder of a paramagnetic substance, and the dotted horizontal straight lines within it, which are continuous with the external lines of induction, represent the lines of induction within the substance. The dotted vertical lines represent the internal equipotential surfaces, and are continuous with the external system. It will be observed that the lines of induction are drawn nearer together within the substance, and the equipotential surfaces are separated farther apart by the paramag netic cylinder, which, in the language of Faraday, conducts the lines of induction better than the surrounding medium.

If we consider the system of vertical lines as lines of induction, and the horizontal system as equipotential surfaces, we have, in the first place, the case of a cylinder magnetized transversely and placed in the position of unstable equilibrium among the lines of force, which it causes to diverge. In the second place, considering the large dotted circle as the section of a diamagnetic cylinder, the dotted straight lines within it, together with the lines external to it, represent the effect of a diamagnetic substance in separating the lines of induction and drawing together the equipotential surfaces, such a substance being a worse conductor of magnetic induction than the surrounding medium.

60 MAGNETIC PROBLEMS. [435-

Case of a Sphere in which the Coefficients of Magnetization are Different in Different Directions.

435.] Let a, (B, y be the components of magnetic force, and A, £, C those of the magnetization at any point, then the most general linear relation between these quantities is given by the equations A = ^0+^3/3+ q2y, \

£ = q9a+r2p+fly, { (1)

C = p2a+q1h2 + 7-3 y, )

where the coefficients r,jo, q are the nine coefficients of magnet ization.

Let us now suppose that these are the conditions of magnet ization within a sphere of radius a, and that the magnetization at every point of the substance is uniform and in the same direction, having the components A, 13, C.

Let us also suppose that the external magnetizing force is also uniform and parallel to one direction, and has for its components X, Y, Z.

The value of V is therefore

and that of &' the potential of the magnetization outside the sphere is

(3)

The value of H, the potential of the magnetization within the sphere, is 4-n-

(4)

o

The actual potential within the sphere is V-- £1, so that we shall have for the components of the magnetic force within the sphere a = X — ^TtA, \ 0 = 7-J.ir-B, (5)

y =Z-

Hence

+i*r1)^+ twftjjB + iir&

C = &J+ r2Y+frZ, (6)

+(1 +

Solving these equations, we find A = r/^+K

'' (7)

43^.] CRYSTALLINE SPHERE. 61

where I/ /•/ = r± + ^ TT ( rB rl — p2 q2 4 r-± r2 —

;-A^i)>

&c.,

where D is the determinant of the coefficients on the right side of equations (6), and D' that of the coefficients on the left.

The new system of coefficients p' ', /, / will be symmetrical only when the system p, q, r is symmetrical, that is, when the co efficients of the form p are equal to the corresponding ones of the form q.

436.] The moment of the couple tending to turn the sphere about the axis of x from y towards z is

f. n Y\ (Q\

— Jr2 ))* \ /

If we make

X = 0, Y = Fcos 0, Y = Fsin 0,

this corresponds to a magnetic force F in the plane of yz, and inclined to y at an angle 0. If we now turn the sphere while this force remains constant the work done in turning the sphere will

T27T

be / LdQ in each complete revolution. But this is equal to

0

Hence, in order that the revolving sphere may not become an inexhaustible source of energy, j»1/= fa', and similarly j»./= q2 and

These conditions shew that in the original equations the coeffi cient of B in the third equation is equal to that of C in the second, and so on. Hence, the system of equations is symmetrical, and the equations become when referred to the principal axes of mag netization, TI

A = rr*"i '

C =

(11)

The moment of the couple tending to turn the sphere round the axis of x is

62 MAGNETIC PROBLEMS. [437-

In most cases the differences between the coefficients of magnet ization in different directions are very small, so that we may put

This is the force tending to turn a crystalline sphere about the axis of oo from y towards z. It always tends to place the axis of greatest magnetic coefficient (or least diamagnetic coefficient) parallel to the line of magnetic force.

The corresponding case in two dimensions is represented in Fig. XVI.

If we suppose the upper side of the figure to be towards the north, the figure represents the lines of force and equipotential surfaces as disturbed by a transversely magnetized cylinder placed with the north side eastwards. The resultant force tends to turn the cylinder from east to north. The large dotted circle represents a section of a cylinder of a crystalline substance which has a larger coefficient of induction along an axis from north-east to south-west than along an axis from north-west to south-east. The dotted lines within the circle represent the lines of induction and the equipotential surfaces, which in this case are not at right angles to each other. The resultant force on the cylinder is evidently to turn it from east to north.

437.] The case of an ellipsoid placed in a field of uniform and parallel magnetic force has been solved in a very ingenious manner by Poisson.

If V is the potential at the point (as, y, z\ due to the gravitation

dV of a body of any form of uniform density p, then — -=- is the

potential of the magnetism of the same body if uniformly mag netized in the direction of x with the intensity I = p.

For the value of -- =— 8# at any point is the excess of the value clx

of V3 the potential of the body, above V, the value of the potential when the body is moved — §x in the direction of x.

If we supposed the body shifted through the distance — 8#, and its density changed from p to — p (that is to say, made of repulsive

dV instead of attractive matter,) then — y-8# would be the potential

due to the two bodies.

Now consider any elementary portion of the body containing a volume b v. Its quantity is pbv, and corresponding to it there is

437-] ELLIPSOID. 63

an element of the shifted body whose quantity is — pbv at a distance — 8#. The effect of these two elements is equivalent to that of a magnet of strength pbr and length 8#. The intensity of magnetization is found hy dividing the magnetic moment of an element by its volume. The result is p 8#.

dV Hence -=- 8# is the magnetic potential of the body magnetized

rl V

with the intensity p bx in the direction of x, and — is that of

ax

the body magnetized with intensity p.

This potential may be also considered in another light. The body was shifted through the distance — 8# and made of density —p. Throughout that part of space common to the body in its two positions the density is zero, for, as far as attraction is con cerned, the two equal and opposite densities annihilate each other. There remains therefore a shell of positive matter on one side and of negative matter on the other, and we may regard the resultant potential as due to these. The thickness of the shell at a point where the normal drawn outwards makes an angle e with the axis of a? is 8 a? cos e and its density is p. The surface-density is therefore

dV p bx cos 6, and, in the case in which the potential is — , the

surface-density is p cos e.

In this way we can find the magnetic potential of any body uniformly magnetized parallel to a given direction. Now if this uniform magnetization is due to magnetic induction, the mag netizing force at all points within the body must also be uniform and parallel.

This force consists of two parts, one due to external causes, and the other due to the magnetization of the body. If therefore the external magnetic force is uniform and parallel, the magnetic force due to the magnetization must also be uniform and parallel for all points within the body.

Hence, in order that this method may lead to a solution of the

clV

problem of magnetic induction, -=- must be a linear function of

doc

the coordinates x, y> z within the body, and therefore V must be a quadratic function of the coordinates.

Now the only cases with which we are acquainted in which V is a quadratic function of the coordinates within the body are those in which the body is bounded by a complete surface of the second degree, and the only case in which such a body is of finite dimen-

64 MAGNETIC PROBLEMS. [437-

sions is when it is an ellipsoid. We shall therefore apply the method to the case of an ellipsoid.

be the equation of the ellipsoid, and let 4>0 denote the definite integral

f

'0

Then if we make

dfr

the value of the potential within the ellipsoid will be

70 = - (L x2 + My* + Nz*} + const. (4)

2

If the ellipsoid is magnetized with uniform intensity / in a direction making angles whose cosines are I, m, n with the axes of #, y, z, so that the components of magnetization are

A = II, B = Im, C = In, the potential due to this magnetization within the ellipsoid will be

a = —I(Llx + Mmy + Nnz). (5)

If the external magnetizing force is Ǥ, and if its components are a, ft, y, its potential will be

r=Xx + Yy + Zz. (6)

The components of the actual magnetizing force at any point within the body are therefore

X-AL, Y-BM, Z-CN. (7)

The most general relations between the magnetization and the magnetizing force are given by three linear equations, involving nine coefficients. It is necessary, however, in order to fulfil the condition of the conservation of energy, that in the case of magnetic induction three of these should be equal respectively to other three, so that we should have

A = K,(X-AL) + Kfs(Y-BM) + K'2(Z-CN}, B = K\ (X-AL) + K2i(Y-BM) + K(Z-CN], (8)

C = K'2(X-AL) + K(Y-BM) + Kz(Z-CN}. From these equations we may determine J, B and C in terms of X, Y} Z, and this will give the most general solution of the problem.

The potential outside the ellipsoid will then be that due to the

• See Thomson and Tait's Natural Philosophy, § 522.

438.] ELLIPSOID. 65

magnetization of the ellipsoid together with that due to the external magnetic force.

438.] The only case of practical importance is that in which

K\ = K2 = K3 = 0. (9)

We have then

If the ellipsoid flattened form,

A —

"i

X 1

(10)

and is of the planetary or : (ID

7?

K2

T ' JJ

V

C = has two b= c

1+K2M~

K3 g

l+K3N

axes equal, a

(12) l-e

M = N = 2 , (±-^ sin-'- ™) . \ e e2 ' J

If the ellipsoid is of the ovary or elongated form

a — b = A/1 — e*c; (13)

In the case of a sphere, when e = 0,

— .«. -^- ^ j

In the case of a very flattened planetoid L becomes in the limit equal to 4 TT, and M and JV become 7r2 - •

In the case of a very elongated ovoid L and M approximate to the value 2 TT, while N approximates to the form

a2,, 2c ,

and vanishes when e = 1 .

It appears from these results that —

(1) When K, the coefficient of magnetization, is very small, whether positive or negative, the induced magnetization is nearly equal to the magnetizing force multiplied by K, and is almost independent of the form of the body.

VOL. II. F

66 MAGNETIC PROBLEMS.

(2) When K is a large positive quantity, the magnetization depends principally on the form of the body,, and is almost independent of the precise value of /c, except in the case of a longitudinal force acting on an ovoid so elongated that NK is a small' quantity though K is large.

(3) If the value of K could be negative and equal to — we

should have an infinite value of the magnetization in the case of a magnetizing force acting normally to a flat plate or disk. The absurdity of this result confirms what we said in Art. 428.

Hence, experiments to determine the value of K may be made on bodies of any form provided K is very small, as it is in the case of all diamagnetic bodies, and all magnetic bodies except iron, nickel, and cobalt.

If, however, as in the case of iron, K is a large number, experi ments made on spheres or flattened figures are not suitable to determine K ; for instance, in the case of a sphere the ratio of the magnetization to the magnetizing force is as 1 to 4.22 if K = 30, as it is in some kinds of iron, and if K were infinite the ratio would be as 1 to 4.19, so that a very small error in the determination of the magnetization would introduce a very large one in the value of K.

But if we make use of a piece of iron in the form of a very elongated ovoid, then, as long as NK is of moderate value com pared with unity, we may deduce the value of K from a determination of the magnetization, and the smaller the value of JV the more accurate will be the value of K.

In fact, if NK be made small enough, a small error in the value of N itself will not introduce much error, so that we may use any elongated body, such as a wire or long rod, instead of an ovoid.

We must remember, however, that it is only when the product JV~/c is small compared with unity that this substitution is allowable. In fact the distribution of magnetism on a long cylinder with flat ends does not resemble that on a long ovoid, for the free mag netism is very much concentrated towards the ends of the cylinder, whereas it varies directly as the distance from the equator in the case of the ovoid.

The distributi6n of electricity on a cylinder, however, is really comparable with that on an ovoid, as we have already seen, Art. 152.

439-] CYLINDER. 67

These results also enable us to understand why the magnetic moment of a permanent magnet can be made so much greater when the magnet has an elongated form. If we were to magnetize a disk with intensity / in a direction normal to its surface, and then leave it to itself, the interior particles would experience a constant demagnetizing force equal to 4 TT I, and this, if not sufficient of itself to destroy part of the magnetization, would soon do so if aided by vibrations or changes of temperature.

If we were to magnetize a cylinder transversely the demagnet izing force would be only 2 TT I.

If the magnet were a sphere the demagnetizing force would be £*/.

In a disk magnetized transversely the demagnetizing force is

a

7T2 - 1) and in an elongated ovoid magnetized longitudinally it

0

a2 2c

is least of all, being 4 TT -^ 7 log --- G a

Hence an elongated magnet is less likely to lose its magnetism than a short thick one.

The moment of the force acting on an ellipsoid having different magnetic coefficients for the three axes which tends to turn it about the axis of #, is

Hence, if *2 and K3 are small, this force will depend principally on the crystalline quality of the body and not on its shape, pro vided its dimensions are not very unequal, but if K2 and *3 are considerable, as in the case of iron, the force will depend principally on the shape of the body, and it will turn so as to set its longer axis parallel to the lines of force.

If a sufficiently strong, yet uniform, field of magnetic force could be obtained, an elongated isotropic diamagnetic body would also set itself with its longest dimension parallel to the lines of magnetic force.

439.] The question of the distribution of the magnetization of an ellipsoid of revolution under the action of any magnetic forces has been investigated by J. Neumann*. Kirchhofff has extended the method to the case of a cylinder of infinite length acted on by any force.

• Crelle, bd. xxxvii (1848). t Crelle, bd. xlviii (1854).

F 2

68 MAGNETIC PROBLEMS. [439-

Green, in the 17th section of his Essay, has given an invest igation of the distribution of magnetism in a cylinder of finite length acted on by a uniform external force parallel to its axis. Though some of the steps of this investigation are not very rigorous, it is probable that the result represents roughly the actual magnetization in this most important case. It certainly expresses very fairly the transition from the case of a cylinder for which K is a large number to that in which it is very small, but it fails entirely in the case in which K is negative, as in diamagnetic substances.

Green finds that the linear density of free magnetism at a distance x from the middle of a cylinder whose radius is a and whose length is 2 I, is

px

ea +e

where p is a numerical quantity to be found from the equation

0.231863 — 2 \ogep + 2p = - -— The following are a few of the corresponding values of p and K.

K K

oo 0

336.4 0.01

62.02 0.02

48.416 0.03

29.475 0.04

20.185 0.05

14.794 0.06

11.802 0.07

9.137 0.08

7.517 0.09

6.319 0.10

0.1427 1.00

0.0002 10.00

0.0000 oo

negative imaginary.

When the length of the cylinder is great compared with its radius, the whole quantity of free magnetism on either side of the middle of the cylinder is, as it ought to be,

M= v2aKX.

Of this \p M is on the flat end of the cylinder, and the distance of the centre of gravity of the whole quantity M from the end

a

of the cylinder is - P

When K is very small p is large, and nearly the whole free magnetism is on the ends of the cylinder. As K increases p diminishes, and the free magnetism is spread over a greater distance

44O-] FORCE ON PARA- AND DIA-MAGNETIC BODIES. 69

from the ends. When K is infinite the free magnetism at any point of the cylinder is simply proportional to its distance from the middle point, the distribution being similar to that of free electricity on a conductor in a field of uniform force.

440.] In all substances except iron, nickel, and cobalt, the co efficient of magnetization is so small that the induced magnetization of the body produces only a very slight alteration of the forces in the magnetic field. We may therefore assume, as a first approx imation, that the actual magnetic force within the body is the same as if the body had not been there. The superficial magnetization

dV dV

of the body is therefore, as a first approximation, K -j- , where -=-

is the rate of increase of the magnetic potential due to the external magnet along a normal to the surface drawn inwards. If we now calculate the potential due to this superficial distribution, we may use it in proceeding to a second approximation.

To find the mechanical energy due to the distribution of mag netism on this first approximation we must find the surface-integral

taken over the whole surface of the body. Now we have shewn in Art. 100 that this is equal to the volume-integral

///* ^rT7 ^ j 77" 2

taken through the whole space occupied by the body, or, if R is the resultant magnetic force,

E = -

Now since the work done by the magnetic force on the body during a displacement 8# is Xbos where X is the mechanical force in the direction of SB, and since

/

= constant,

which shews that the force acting on the body is as if every part of it tended to move from places where R2 is less to places where it is greater with a force which on every unit of volume is

rf.JP K dx '

70 MAGNETIC PEOBLEMS.

If K is negative, as in diamagnetic bodies, this force is, as Faraday first shewed, from stronger to weaker parts of the magnetic field. Most of the actions observed in the case of diamagnetic bodies depend on this property.

Skip's Magnetism.

441.] Almost every part of magnetic science finds its use in navigation. The directive action of the earth's magnetism on the compass needle is the only method of ascertaining the ship's course when the sun and stars are hid. The declination of the needle from the true meridian seemed at first to be a hindrance to the appli cation of the compass to navigation, but after this difficulty had been overcome by the construction of magnetic charts it appeared likely that the declination itsylf would assist the mariner in de termining his ship's place.

The greatest difficulty in navigation had always been to ascertain the longitude ; but since the declination is different at different points on the same parallel of latitude, an observation of the de clination together with a knowledge of the latitude would enable the mariner to find his position on the magnetic chart.

But in recent times iron is so largely used in the construction of ships that it has become impossible to use the compass at all without taking into account the action of the ship, as a magnetic body, on the needle.

To determine the distribution of magnetism in a mass of iron of any form under the influence of the earth's magnetic force, even though not subjected to mechanical strain or other disturb ances, is, as we have seen, a very difficult problem.

In this case, however, the problem is simplified by the following considerations.