The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 25 of 39

The abscissae represent values of H, the ordinate of the thick curve the value of fxH, and the ordinate of the thin curve the value of //,. The corre- sponding numerical values are as follows :

H

fiH

M

H

/jlH

f-

0-32

40

120

5-17

12680

2450

0-84

170

200

6-20

13640

2200

1-37

420

310

7-94

14510

1830

2-14

1170

550

9-79

14980

1530

2-67

3710

1390

11-57

15230

1320

3-24

7300

2250

15-06

15570

1030

3-89

9970

2560

19-76

15780

800

4*50

11640

2590

21-70

15870

730

412

Induced Magnetism,

[en. xii

464. Retentiveness and Hysteresis. It is found that after the magnetising force is removed from a sample of iron, the iron still retains some of its mag- netism. Here we have a phenomenon similar to the electrostatic phenomenon of residual charge already described in § 397.

Fig. 114 is taken from a paper by Prof. Ewing (Phil. Trans. Roy. Soc. 1885). The abscissae represent values of H, and ordinates values of B, the induction. The magnetic field was increased from H = 0 to H=22, and as H increased the value of B increased in the manner shewn by the curve OP of the graph. On again diminishing H from H = 22 to H=0, the graph for B was found to be that given by the curve PE. Thus during this operation there was always more magnetisation than at the corresponding stage of the original operation, and finally when the inducing field was entirely removed, there was magnetisation left, of intensity represented by OE. The field was then further decreased from H = 0 to H = — 20, and then increased again from H = — 20 to H = 22. The changes in B are shewn in the graph.

H

465. Dependence of fi on temperature. As has already been said, the value of /j, depends to a large extent on the temperature of the metal. In general, the value of fju continually increases as the temperature is raised, this increase being slow at first but afterwards more rapid, until a temperature known as the " temperature of recalescence " is reached. This temperature has values ranging from 600° to 700° for steel and from 700° to 800° for iron. This temperature takes its name from the circumstance that a piece of metal cooling through this temperature will sink to a dull glow before reaching it, and will then become brighter again on passing through it.

After passing the temperature of recalescence, the value of fi falls with extreme rapidity, and at a temperature only a few degrees above this temperature, iron appears to be almost completely non-magnetic.

'464-467] Mathematical Theory 413

For paramagnetic substances, it appears to be a general law that the susceptibility k varies inversely as the absolute temperature (Curie's Law).

Mathematical Theory.

466. If  £1  is  the  magnetic  potential,  supposed  to  be  defined  at  points 
     

inside magnetic matter by equation (348), we have, as in equations (341)

(cf. § 430), a = — -=- etc., so that

an , ao an

The quantities a, b, c, as we have seen (§ 434), satisfy

da db dc _ /«««*

S + *+»-° <393>

at every point, and

r(la + mb + nc)dS = 0 (394),

//<

where the integration is taken over any closed surface. In terms of the potential, equation (393) becomes

^^J+tyl^J+9l^fcJ = 0 (39o)'

while equation (394) becomes

//■

nd^dS = 0 (396).

If fx, is constant throughout any volume, equation (395) becomes

V2D = 0.

Thus inside a mass of homogeneous non- magnetised matter, the magnetic potential satisfies Laplace's Equation.

467. At a surface at which the value of /j, changes abruptly we may take a closed surface formed of two areas fitting closely about an element dS of the boundary, these two areas being on opposite sides of the boundary. On applying equation (396), we obtain

fi1~- + fi2~- = 0 (397),

where a1} u2 are the permeabilities on the two sides, and =— . =- denote

OVx ov2

differentiations with respect to normals to the surface drawn into the two

media respectively.

Equations (397) and (395) (or (396)), combined with the condition that fl must be continuous, suffice to determine 12 uniquely. The equations

414 Induced Magnetism [ch. xii

satisfied by O, the magnetic potential, are exactly the same as those which would be satisfied by V, the electrostatic potential, if ja were the Inductive Capacity of a dielectric. Thus the law of refraction of lines of magnetic induction is exactly identical with the law of refraction of lines of electric force investigated in § 138, and figures (43) and (78) may equally well be taken to represent lines of magnetic induction passing from one medium to a second medium of different permeability.

468. At any external point Q, the magnetic potential of the magnetisation induced in a body in which fj, and k have constant values is, by equation (342),

Transforming by Green's Theorem,

y JJJ r JJ\ Ox cy dzj r

-.«//©> (399)'

shewing that the potential is the same if there were a layer of magnetic matter of surface density — k -=- spread over the surface of the body. This is Poisson's expression for the potential due to induced magnetism. We can also transform equation (398) into

—/K©*9 (400)-

shewing that the potential at any external point Q of the induced magnetism is the same as if there were a magnetic shell of strength — fcfl coinciding with the surface of the body.

Body in which permanent and induced magnetism coexist.

469. If a permanent magnet has a permeability different from unity, we shall have a magnetisation arising partly from permanent and partly from induced magnetism. If k is the susceptibility and / the intensity of the permanent magnetisation at any point, the components of the total magnet- isation at any point will be

A = II + ko, etc (401),

467-471] Energy of a Magnetic Field 415

and the components of induction are

a = a + 4<7rA = 4>ttII + fia, etc. (402).

For such a substance, it is clear that equations (395) and (396) will not in general be satisfied.

Energy of a Magnetic Field.

470. To obtain the energy of a magnetic field in which both permanent and induced magnetism may be present, we return to the general equation obtained in § 451,

I (aa + b/3 + cy) dxdydz = 0 (403).

On substituting for a, b, c from equations (402), this becomes

4tt \\I (la + m& + ny) dxdydz + 1 1 J /x (a2 + /32 + y*) dxdydz = 0 . . .(404).

Whether or not induced magnetism is present, it is proved, in § 448, that the energy of the field is

w=kISSt <?i§+ m^+ n!r) dxdydz>

where the integral is taken through all space. This is equal to — 3— times the first term in equation (404). Thus

W=^rjff p (tf + /32+r) dxdydz (405).

This could have been foreseen from analogy with the formula

W= i- f I f K(X2+ Y* + Z2) dxdydz,

which gives the energy of an electrostatic field.

From formula (405) we see that the energy of a magnetic field may be

LlH2

supposed spread throughout the medium, at a rate ^ — per unit volume.

Mechanical Forces in the Field.

471. The mechanical forces acting on a piece of matter in a magnetic field can be regarded as the superposition of two systems — first, the forces acting on the matter in virtue of its permanent magnetism (if any), and, secondly, the forces acting on the matter in virtue of its induced magnetism (if any).

The problem of finding expressions for the mechanical forces in a magnetic field is mathematically identical with that of finding the forces in an electro- static field. This is the problem of which the solution has already been

416 Induced Magnetism [ch. xii

given in § 196. The result of the analysis there given may at once be applied to the magnetic problem.

In equation (117), p. 175, we found the value of 2, the ^-component of the mechanical force per unit volume, in the form

dV RdK d {R dK\

dx 87r dec dec \87r 8t / *

To translate this result to the magnetic problem, we must regard p as specifying the density of magnetic poles, R must be replaced by H, the magnetic intensity, and K by /*, the magnetic permeability. Also the electrostatic potential V must be replaced by the magnetic potential X2. We then have, as the value of 5 in a magnetic field,

a = -PTx-^r^+dx\MTd-r) (406)'

Clearly the first term in the value of H is that arising from the per- manent magnetism of the body, while the second and third terms arise from the induced magnetism. The first term can be transformed in the manner already explained in the last chapter. It is with the remaining terms that we are at present concerned. These will represent the forces when no per- manent magnetism is present. Denoting the components of this force by 2', H', Z\ we have

. ~=-^£+dx{^T£) <407>

472. This general formula assumes a special form in a case which is of great importance, namely when the magnetic medium is a fluid.

All liquid magnetic media in which the susceptibility is at all marked consist of solutions of salts of iron, and the magnetic properties of the liquid arise from the presence of the salts in solution. According to Quincke, the solution having the greatest susceptibility is a solution of chloride of iron in methyl alcohol, and for this the value of fi — 1 is about toW*. In such a liquid, the field arising from the induced magnetism will be small compared with that arising from the original field, so that the magnetisation of any single particle of the salt in the solution may be regarded as produced entirely by the original field. Hence we have conditions similar to those which obtain electrostatically in a gas. The induced field may be regarded simply as the aggregate of the fields arising from the different particles of the magnetic medium, and is therefore jointly proportional to the density of these particles and to the strength of the inducing field. The latter fact shews that, for a given density of the medium, fi ought to be independent of H, a result to which we shall return later. The former fact shews that, as

• Cf. G. T. Walker, " Aberration " (Cambridge Univ. Press, 1900), p. 7G.

471-474]

Magnetostriction

417

the density t changes, /j, — 1 ought to be proportional to t — a result analogous to the result that K — 1 is proportional to the density in a gas. It has been found experimentally by Quincke* that p — 1 is approximately proportional to T.

In gases we have conditions precisely similar to those which obtain when a gas is placed in an electrostatic field. Hence /j, — 1 must, for a gas, be proportional to r, for exactly the same reason for which K — 1 is proportional to t This result also has been verified by Quincke f.

Thus we may say that for fluid media, whether liquid or gaseous, /a — 1 is, in general, proportional to r, where t is the density of the magnetic liquid, in the case of a liquid in solution, or of the gas itself, in the case of a gas.

473. If  we  assume  the  relation 
     

fi~l = cr (408),

where c is a constant, we find that expression (407) may be put in the simpler form

h-, a— l d

a =~ — r-

87r dx

m,

shewing that the whole mechanical force is the same as would be set up by a

hydrostatic pressure at every point of the medium of amount

8

H*

7T

If H varies from point to point of the field, the effect of this pressure will clearly be to urge the medium to congregate in the more intense parts of the field. This has been observed by MatteucciJ for a medium consisting of drops of chloride of iron dissolved in alcohol placed in a medium of olive oil. The drops of solution were observed to move towards the strongest parts of the field.

Magnetostriction.

474. If a liquid is placed in a magnetic field, it yields under the influence of the mechanical forces acting upon it, so that we have a phenomenon of magnetostriction, analogous to the phenomenon of electro- striction already explained (§ 203). Clearly the liquid will expand until the

pressure is decreased by an amount ^ — H2 at each point, the new pressure

and the mechanical forces resulting from the magnetic field now producing equilibrium in the fluid. By measuring the expansion of a liquid placed in a magnetic field Quincke has been able to verify the agreement between theory and experiment.

• Wied. Ann. 24, p. 347.

X Comptes Rendus, 36, p. 917.

• Wied. Ann. 34, p. 401.

J.

27

418 Induced Magnetism [ch. xii

Molecular Theories.

Poisson s Molecular Theory of Induced Magnetism.

475. In Chapter V it was found possible to account for all the electro- static properties of a dielectric by supposing it to consist of a number of perfectly conducting molecules. Poisson attempted to apply a similar explanation to the phenomenon of magnetic induction.

Poisson's theory can, however, be disproved at once, by a consideration of the numerical values obtained for the permeability /a. This quantity is analogous to the quantity K of Chapter V, so that its value may be estimated in terms of the molecular structure of the magnetic matter. The fact with respect to which Poisson's theory breaks down is the existence of substances (namely, different kinds of soft iron) for which the value of fi is very large. To understand the significance of the existence of such substances, let us consider the field produced when a uniform infinite slab of such a substance is placed in a uniform field of magnetic force, so that the face of the slab is at right angles to the lines of force. If the value of /* is very large, the fall of potential in crossing the slab is very small. Throughout the supposed perfectly-conducting magnetic molecules the potential would, on Poisson's theory, be constant, so that the fall of potential could occur only in the interstices between the molecules. In these interstices (cf. fig. 46), the fall of potential per unit length would be comparable with that outside the slab. Hence a very large value of /a could be accounted for only by supposing the molecules to be packed together so closely as to leave hardly any interstices. Samples of iron can be obtained for which ^ is as large as 4000 ; it is known, from other evidence, that the molecules of iron are not so close together that such a value of fi could be accounted for in the manner proposed by Poisson.

It is worth noticing, too, that Poisson's theory does not seem able, without modification, to give any reasonable account of the phenomena of saturation, hysteresis, etc.

Weber's Molecular Theory of Induced Magnetism.

476. A theory put forward by Weber shews much more ability than the theory of Poisson to explain the facts of induced magnetism.

Weber supposes that, even in a substance which shews no magnetisation, every molecule is a permanent magnet, but that the effects of these different magnets counteract one another, owing to their axes being scattered at random in all directions. When the matter is placed in a magnetic field each molecule tends, under the influence of the field, to set itself so that its axis is along the lines of force, just as a compass-needle tends to set itself along the lines of force of the earth's magnetic field. The axes of the

475-477] Molecular Theories 419

molecules no longer point in all directions indifferently, so that the magnetic fields of the different molecules no longer destroy one another, and the body as a whole shews magnetisation. This, on Weber's theory, is the magnetisa- tion induced by the external field of force.

Weber supposes that each molecule, in its normal state, is in a position of equilibrium under the influence of the forces from all the neighbouring molecules, and that when it is moved out of this position by the action of an external magnetic field, the forces from the other molecules tend to restore it to its old position. It is, therefore, clear that so long as the external field is small, the angle through which each axis is turned by the action of the field will be exactly proportional to the intensity of the field, so that the magnetisation induced in the body will be just proportional to the strength of the inducing field. In other words, for small values of H, [X must be independent of H.

There is, however, a natural limit imposed upon the intensity of the induced magnetisation. Under the influence of a very intense field all the molecules will set themselves so that their axes are along the lines of force. The magnetisation induced in the body is now of a quite definite intensity, and no increase of the inducing field can increase the intensity of the induced magnetisation beyond this limit. Thus Weber's theory accounts quite satisfactorily for the phenomenon of saturation, a phenomenon which Poisson's theory was unable to explain.

477. In connection with this aspect of Weber's theory, some experi- ments of Beetz are of great importance. A narrow line was scratched in a coat of varnish covering a silver wire The wire was placed in a solution of a salt of iron, arranged so that iron could be deposited electrolytically on the wire at the points at which the varnish had been scratched away. The effect was of course to deposit a long thin filament of iron along the scratch. If, however, the experiment was performed in a magnetic field whose lines of force were in the direction of the scratch, it was found not only that the filament of iron deposited on the wire was magnetised, but that its magnetisation was very intense. Moreover, on causing a powerful magnetising force to act in the same direction as the original field, it was found that the increase in the intensity of the induced magnetisation was very small, shewing that the magnetisation had previously been nearly at the point of saturation.

Now if, as Weber supposed, the molecules of iron were already magnets before being deposited on the silver wire, then any magnetic force sufficient to arrange them in order on the wire ought to have produced a filament in a state of magnetic saturation, while if, as Poisson supposed, the magnetism in the molecules was merely induced by the external magnetic field, then the magnetisation of the filament ought to have been proportional to the

420

Induced Magnetism

[CH. XII

original field, and ought to have disappeared when the field was destroyed. Thus, as between these two hypotheses, the experiments decide conclusively for the former.

478. Weber's  theory  is  illustrated  by  the  following  analysis. 
     

Consider a molecule which, in the normal state of the matter, has its axis in the direction OP, and let the field of force from the neigh- bouring molecules be a field of in- tensity D, the direction of the lines of force being of course parallel to OP Now let an external field of intensity H be applied, its direction being a direction OA making an angle a with OP. The total field

acting on the molecule is now com- es

pounded of D along OP and H along OA.

Fig. 115.

In fig. 115, let SO, OP represent H and D in magnitude and direction, then SP will represent the resultant field, so that the new direction of the axis of the molecule will be SP. Suppose that there are n molecules per unit volume, each of moment m. Originally, when the axes of the molecules were scattered indifferently in all directions, the number for which the angle a had a value between a and a 4- da was hn sin ada. These molecules now have their axes pointing in the direction SP, and therefore making an angle PSA (= 6, say) with the direction of the external magnetic field. The aggregate moment of all these molecules resolved in the direction of OA is accordingly

hmn sin a cos 6 da,

and on integration the aggregate moment of all the molecules per unit volume, which is the same as the intensity of the induced magnetisation I, is given by

ro=7r 1=1 ^mn sin a cos dda (409).

J <x=0

If R is the value of SP, measured on the same scale on which SO and OP represent H and D respectively, then

B?=H* + Dn--2HD cos a.

so that, on changing the variable from a to R, we must have the relation, obtained by differentiation of the above equation,

RdR = HD sin ada.

477-479]

We also have cos 6 =

so that equation (409) becomes

I = ^mn I

Molecular Theories

421

2RH

R* + H*- Z)2 2H*D

dR.

In fig. 115 the limits of integration for R are R = D + H and R = D — H. If, however, H > I), then the point 8 falls outside the circle APR and the limits for R are R = D + H and R = H-D.

On integrating,

we

find

as the values of /,

len X < D,

l = %mn^t

„ X = D,

I = %mn,

„ X>D,

I = ran (l - j{j ,

„ X = qo ,

I = mn.

Fig. 116.

In fig. 116, the abscissae represent values of H, the ordinates of the thick curve the values of I, and the ordinates of the dotted curve the values of B or pH, drawn on one-tenth of the vertical scale of the graph for I

Maxwell's Molecular Theory of Induced Magnetism.

479. It will be seen that Weber's theory fails to account for the increase in the value of /j, before i" reaches its maximum, and also that it gives no account of the phenomenon of retentiveness. Maxwell has shewn how the theory may be modified so as to take account of these two phenomena. He supposes that, so long as the forces acting on the molecules are small, the molecules experience small deflexions as imagined by Weber, but that as soon as these deflexions exceed a certain amount, the molecules are wrenched away entirely from their original positions of equilibrium, and take up positions relative to some new position of

422

Induced Magnetism

[CH. XII

equilibrium. It might be, for instance, that originally the molecule had two

possible positions of equilibrium, OP and OQ in fig. 117. Suppose the

molecule to be in position OP and to be

acted upon by a gradually increasing force

in some direction OA. At first the molecule

will turn from the position OP towards OA.

But it may be that, as soon as the molecule

passes some position OR, it suddenly swings

round and takes up a position in which it FlG- 117#

must be regarded as being deflected from the position of equilibrium OQ and

not from OP. Let its new position be OS, then the deflexion produced is

the angle SOP instead of the angle ROP which would be given by Weber's theory.

In Maxwell's original discussion, no distinction was made between the position OR, at which the magnet broke away from its old position of equi- librium, and OS, the new position of equilibrium. Maxwell accordingly had to assume that in some unknown way, the force of restitution broke down as soon as the magnet reached the position OR.

The improvement of distinguishing between the position OR, the limit of stability under the old position of equilibrium, and OS, the new position of equilibrium, was introduced by Ewing. In Ewing's form of the theory, no forces are needed beyond those provided by the mutual action of the magnets upon one another.

On either form of the theory, it is clear that the ratio of / to H will remain approximately constant until the molecules begin to break away from their original positions of equilibrium. As soon as this happens, the induced magnetism will increase more rapidly than the inducing force — i.e. /u, will increase with H, in agreement with observation.

If the magnetising force is now removed, the molecule in the position OS will not return to its original position OP, but to the position OQ. It will therefore still have a deflexion QOP, called by Maxwell its "permanent set," and this will account for the " retentiveness " of the substance.

No molecular theory of this kind can, however, be regarded as at all complete. We shall return to the discussion of molecular theories of magne- tism in Chapter xvi.

EXAMPLES.

1. A small magnet is placed at the centre of a spherical shell of radii a and b. Determine the magnetic force at any point outside the shell.

2. A system of permanent magnets is such that the distribution in all planes parallel to a certain plane is the same. Prove that if a right circular solid cylinder be placed in the field with its axis perpendicular to these planes, the strength of the field at any point inside the cylinder is thereby altered in a constant ratio.

Examples 423

3. A magnetic particle of moment m lies at a distance a in front of an infinite block of soft iron bounded by a plane face, to which the axis of the particle is perpendicular. Find the force acting on the magnet, and shew that the potential energy of the system is

-m2(^-l)/8a3(/x + l).

4. The whole of the space on the negative side of the yz plane is filled with soft iron, and a magnetic particle of moment m at the point (a, 0, 0) points in the direction (cos a, 0, sin a). Prove that the magnetic potential at the point x, y, z inside the iron is

2m z sin a - (a — x) cos a

1+^ {(a-xf+y^ + z^i •

5. A small magnet of moment M is held in the presence of a very large fixed mass of soft iron of permeability p with a very large plane face : the magnet is at a distance a from the plane face and makes an angle 6 with the shortest distance from it to the plane. Shew that a certain force, and a couple

0 - 1 ) J/2 sin 6 cos (9/8 (/* + 1) a\

are required to keep the magnet in position.

6. A small sphere of radius b is placed near a circuit which, when carrying unit current, would produce a field of strength IT at the point where the centre of the sphere is placed. Shew that if < is the coefficient of magnetic induction for the sphere, the presence of the sphere increases the self-induction of the wire by, approximately,

87r63K(3 + 27r*)7Z2 (3 + 4tt/c)2

7. If the magnetic field within a body of permeability /x be uniform, shew that any spherical portion can be removed and the cavity filled up with a concentric spherical nucleus of permeability /xx and a concentric shell of permeability /x2 without affecting the external field, provided fi lies between /xx and n2, and the ratio of the volume of the nucleus to that of the shell is properly chosen. Prove also that the field inside the nucleus is uniform, and that its intensity is greater or less than that outside according as p. is greater or less than pv

8 A sphere of radius a has at any point (x, y, z) components of permanent magneti- sation (Px, Qy, 0), the origin of coordinates being at its centre. It is surrounded by a spherical shell of uniform permeability p, the bounding radii being a and b. Determine the vector potential at an outside point.

9. A sphere of soft iron of radius a is placed in a field of uniform magnetic force parallel to the axis of z. Shew that the lines of force external to the sphere lie on surfaces of revolution, the equation of which is of the form

{*+2-£#(")>^>=—.

r being the distance from the centre of the sphere.

10. A sphere of soft iron of permeability p. is introduced into a field of force in which the potential is a homogeneous polynomial of degree n in x, y, z. Shew that the potential inside the sphere is reduced to its original value multiplied by

2» + l rip + n + 1 "

11. If a shell of radii a, b is introduced in place of the sphere in the last question, shew that the force inside the cavity is altered in the ratio

(2»+l)2/x : (%/*+ra+l)(w/x+ra+/i)-»(» + J)(/i-l)2(r)

424 Induced Magnetism [ch. xii

12. An infinitely long hollow iron cylinder of permeability fx, the cross-section being concentric circles of radii a, b, is placed in a uniform field of magnetic force the direction of which is perpendicular to the generators of the cylinder. Shew that the number of lines of induction through the space occupied by the cylinder is changed by inserting the cylinder in the field, in the ratio

6»0»+l)«-o^0*-l)«:2/i{6«G»+l)-a^0*-l)}.

13. A cylinder of iron of permeability fi has for cross-section the curve

r = a(l+ecos2<9),

where e2 may be neglected. Find the distribution of potential when the cylinder is placed in a field of force of which the potential before the introduction of the cylinder was

Q. = Axy

14. An infinite elliptic cylinder of soft iron is placed in a uniform field of potential — (Xx+Yy), the equation of the cylinder being — 2 + T2 = l. Shew that the potential of the induced magnetism at any internal point is

• (u - 1) f-r^— Xx + — ?—r Yy

15. A solid elliptic cylinder whose equation is | = a given by

x + ty = c cosh (£ + irj) is placed in a field of magnetic force whose potential is A (x2 — y2). Shew that in the space external to the cylinder the potential of the induced magnetism is

-^c2cosech2(a+i3)sin4ae2(a~^~f)cos27?, where coth 2/3 is the permeability.

16. A solid ellipsoid of soft iron, semi-axes a, b, c and permeability p, is placed in a uniform field of force X parallel to the axis of x, which is the major axis. Verify that the internal and external potentials of the induced magnetisation are

Q1 = PA1x, Q0 = PAox,

where

! r <** a = T-

1 Jo (at + ylrfiW + yl,)* (<? + &)¥ ° J *(a2

d^r

(a2 + ^ (b2 + f)i (c2 + ^' J ^(a2 + yl,)§(b2+yl,)h(c2 + ,},)$'

P=(^-l)X/{(F-l)A1 + 2(abc)-^ ana X is the parameter of the confocal through the point considered.

17. A unit magnetic pole is placed on the axis of z at a distance / from the centre of a sphere of soft iron of radius a. Shew that the potential of the induced magnetism at anv external point is

n -i-

f+xdtd6

1 fx-l a?

TT^+1/2

OJ

Z + luZ cos 6 —

dH^ 2 '

where z, w are the cylindrical coordinates of the point. Find also the potential at an internal point.

18. A  magnetic  pole  of  strength  m  is  placed  in  front  of  an  iron  plate  of  permeability 
    

/* and thickness c. If this pole be the origin of rectangular coordinates x, y, and if x be

perpendicular and y parallel to the plate, shew that the potential behind the plate is

given by

a-1

O = m(l-p2)jo l_°p^Jtct , where p =

fi+1'

CHAPTEE XIII

THE MAGNETIC FIELD PRODUCED BY ELECTRIC CURRENTS

(1)

Experimental Basis.

480. So far the subjects of electricity and magnetism have been developed as entirely separate groups of physical phenomena. Although the mathe- matical treatment in the two cases has been on parallel lines, we have not had occasion to deal with any physical links connecting the two series of phenomena.

The first definite link of the kind was discovered by Oersted in 1820. Oersted's discovery was the fact that a current of electricity produced a magnetic field in its neighbourhood.

The nature of this field can be investigated in a simple manner. We

first double back on itself a wire in which ^

a current is flowing (fig. 118, 1). It is found that no magnetic field is produced.

Next we open the end into a small plane loop PQRS (fig. 118, 2). It is found that at distances from the loop which are great compared with its linear dimensions, such a loop exercises the same magnetic forces as a magnetic particle of which the axis is perpendicular to the plane PQRS,

and the moment is jointly proportional to the strength of the current and to the area PQRS The single current flowing in the circuit OPQRST is obviously equivalent to two currents of equal strength, the one flowing in the circuit OP ST obtained by joining the points P and S, and the other flowing in the closed circuit PQRSP. The former current is shewn, by the preliminary experiment, to have no magnetic effects, so that the whole magnetic field may be ascribed to the small closed circuit PQRS.

Q

i R

(2)

Fig. 118.

Fig. 119.

426 The Magnetic Field produced by Electric Currents [en. xm

481. Instead of regarding this field as due to a particle of moment jointly proportional to the area PQRS and to the current-strength, we may regard it as due to a small magnetic shell, coinciding with the area PQRS, and of strength simply proportional to the current flowing in PQRS.

482. Next, let us consider the current flowing in a closed circuit of any shape we please, and not necessarily in one plane. Let us cover in the closed circuit by an area of any kind having the circuit for its boundary, and let us cut up this area into infinitely small meshes by two systems of lines. A current of strength i flowing round the boundary circuit, is exactly equivalent to a current of strength i flowing round each mesh in the same direction as the current in the boundary. For, if we imagine this latter system of currents in existence, any line such as AB in the interior will have two currents flowing through it, one from each of the two meshes which it separates, and these currents will be equal but in opposite directions. Thus all the currents in the lines which have been introduced in the interior of the circuit annihilate one another as regards total effect, while the currents in those parts of the meshes which coincide with the original circuit just combine to reproduce the original current flowing in this circuit.

Thus the original circuit is equivalent, as regards magnetic effect, to a system of currents, one in each mesh. By taking the meshes sufficiently small, we may regard each mesh as plane, so that the magnetic effect of a current circulating in it is known : the magnetic effect of the current in a single mesh is that of a magnetic shell of strength proportional to the current and coinciding in position with the mesh. Thus, by addition, we find that the whole system of currents produces the same magnetic effects as a single magnetic shell coinciding with the surface of which the original current- circuit is the boundary, and of strength proportional to the current. This shell, then, produces the same magnetic effect as the original single current. The magnetic shell is spoken of as the " equivalent magnetic shell."

Thus we have obtained the following result :

" A current floiving in any closed circuit produces the same magnetic field as a certain magnetic shell, known as the ' equivalent magnetic shell.' This shell mag be taken to be any shell having the circuit for its boundary, its strength being uniform and proportional to that of the current"

481-484] Experimental Basis 427