The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 22 of 39

4. Prove that the product of the resistance to leakage per unit length between two practically infinitely long parallel wires insulated by a uniform dielectric and at different potentials, and the capacity per unit length, is EpjAir, where K is the inductive capacity and p the specific resistance of the dielectric. Prove also that the time that elapses before the potential difference sinks to a given fraction of its original value is independent of the sectional dimensions and relative positions of the wires.

5. If the right sections of the wires in the last question are semicircles described on opposite sides of a square as diameters, and outside the square, while the cylindrical space whose section is the semicircles similarly described on the other two sides of the square is filled up with a dielectric of infinite specific resistance, and all the neighbouring space is filled up with a dielectric of resistance p, prove that the leakage per unit length in unit time is 2 Vjp, where V is the potential difference.

6. If <p + fy=f{x--iy), and the curves for which $ = cons. bo closed curves, shew that the insulation resistance between lengths I of the surfaces <p = <po, <p = <pi, is

p (<ftx ~ (ftp)

where [^J is the increment of \j/ on passing once round a 0-curve, and p is the specific resistance of the dielectric.

• Wied. Ann. xxxv. (1888), p. 291. f Wied. Ann. xl. (1890), p. 328.

Examples 363

7. Current enters and leaves a uniform circular disc through two circular wires of small radius e whose central lines pass through the edge of the disc at the extremities of a chord of length d. Shew that the total resistance of the sheet is

(2tr/»r) log (d/e).

8. Using the transformation

log (x+iy) = i + ir],

prove that the resistance of an infinite strip of uniform breadth it between two electrodes distant 2a apart, situated on the middle line of the strip and having equal radii 8, is

|log(|tanh«)-

9. Shew  that  the  transformation 
   

x' + itj = cosh it (x + iy)\ 'a enables us to obtain the potential due to any distribution of electrodes upon a thin conductor in the form of the semi-infinite strip bounded by y=0, y = a, and x=0.

If the margin be uninsulated, find the potential and flow due to a source at the point x=c, y = 5 . Shew that if the flows across the three edges are equal, then 7rc=acosh-1 2.

10. Equal  and  opposite  electrodes  are  placed  at  the  extremities  of  the  base  of  an 
    

isosceles triangular lamina, the length of one of the equal sides being a, and the vertical

angle — Shew that the lines of flow and equal potential are given by

a

smh^ - + 1=^/3 - ,

2 ^ 1 - en u

where 35r Q) ua = T C£\ T Q V ze « - a V

and the modulus of en u is sin 75°, the origin being at the vertex.

11. A circular sheet of copper, of specific resistance o-j per unit area, is inserted in a very large sheet of tinfoil (<r0), and currents flow in the composite sheet, entering and leaving at electrodes. Prove that the current-function in the tinfoil corresponding to an electrode at which a current e enters the tinfoil is the coefficient of * in the imaginary part of

2tt

"log (*-c)+^ log « 1

where a is the radius of the copper sheet, 2 is a complex variable with its origin at the centre of the sheet, and c is the distance of the electrode from the origin, the real axis passing through the electrode.

Generalise the expression for any position of the electrode in the copper or in the tinfoil, and investigate the corresponding expressions determining the lines of flow in the copper.

12. A  uniform  conducting  sheet  has  the  form  of  the  catenary  of  revolution 
    

w2 + z2 = c2 cosh2 - . c

Prove that the potential at any point due to an electrode at Xq, yo, Zq, introducing a current C, is

constant - ^ log ( cosh * -*° - ^0+ZZ°

^ °\ o nV+^)W+^)

f>

CHAPTER XI

PERMANENT MAGNETISM

Physical Phenomena.

398. It is found that certain bodies, known as magnets, will attract or repel one another, while a magnet will also exert forces on pieces of iron or steel which are not themselves magnets, these forces being invariably attractive. The most familiar fact of magnetism, namely the tendency of a magnetic needle to point north and south, is simply a particular instance of the first of the sets of phenomena just mentioned, it being found that the earth itself may be regarded as a vast aggregation of magnets.

The simplest piece of apparatus used for the experimental study of magnetism is that known as a bar-magnet. This consists of a bar of steel which shews the property of attracting to itself small pieces of steel or iron. Usually it is found that the magnetic properties of a bar-magnet reside largely or entirely at its two ends. For instance, if the whole bar is dipped into a collection of iron filings, it is found that the filings are attracted in great numbers to its two ends, while there is hardly any attraction to the middle parts, so that on lifting the bar out from the collection of filings, we shall find that filings continue to cluster round the ends of the bar, while the middle regions will be comparatively free.

Poles of a Magnet.

399. The two ends of a magnet — or, more strictly, the two regions in which the magnetic properties are concentrated — are spoken of as the "poles" of the magnet. If the magnet is freely suspended, it will turn so that the line joining the two poles points approximately north and south. The pole which places itself so as to point towards the north is called the " north-seeking pole," while the other pole, pointing to the south, is called the " south-seeking pole."

By experimenting with two or more magnets, it is found to be a general law that similar poles repel one another, while dissimilar poles attract one another.

398-401] Permanent Magnetism 365

The earth may roughly be regarded as a single magnet of which the two magnetic poles are at points near to the geographical north and south poles. Since the northern magnetic pole of the earth attracts the north-seeking pole of a suspended bar-magnet, it is clear that this northern magnetic pole must be a south-seeking pole ; and similarly the southern pole of the earth must be a north-seeking pole. Lord Kelvin speaks of a south-seeking pole as a " true north " pole — i.e. a pole of which the magnetism is of the kind found in the northerly regions of the earth. But for purposes of mathematical theory it will be most convenient to distinguish the two kinds of pole by the entirely neutral terms, positive and negative. And, as a matter of convention, we agree to call the north-seeking pole positive. Thus we have the following pairs of terms :

North-seeking = True South = Positive,

South-seeking = True North = Negative.

Law of Force betiveen Magnetic Poles.

400. By experiments with his torsion-balance, Coulomb established that the force between two magnetic poles varies inversely as the square of the distance between them. It was found also to be proportional to the product of two quantities spoken of as the " strengths " of the poles. Thus if F is the repulsion between two poles of strengths m, w! at a distance r apart, we have

„ cmm'

F=— (328).

It is found that c depends on the medium in which the poles are placed, but is otherwise constant. Clearly if we agree that the strength of positive poles is to be reckoned as positive, while that of negative poles is reckoned negative, then c will be a positive quantity

The Unit Magnetic Pole.

401. Just as Coulomb's electrostatic law of force supplied a convenient way of measuring the strength of an electric charge, so the law expressed by equation (328) provides a convenient way of measuring the strength of a magnetic pole, and so gives a system of magnetic units. A system of units, analogous to the electrostatic system (§§ 17, 18) is obtained by defining the unit pole to be such as to make c = 1 in equation (328). This system is called the Magnetic (or, more generally, Electromagnetic) system of units. We define a unit pole, in this system, to be a pole of strength such that when placed at unit distance from a pole of equal strength the repulsion between the two poles is one of unit force.

366 Permanent Magnetism [ch. xi

Thus the force F between two poles of strengths m, m', measured in the Electromagnetic system of units, is given by

F=— (329).

r2

The physical dimensions of the magnetic unit can be discussed in just the same way in which the physical dimensions of the electrostatic unit have already been discussed in § 18.

Moment of a Line-Magnet.

402. It is found that every positive pole has associated with it a negative pole of exactly equal strength, and that these two poles are always in the same piece of matter.

Thus not only are positive and negative magnetism necessarily brought into existence together and in equal quantities, as is the case with positive and negative electricity, but, further, it is impossible to separate the positive and negative magnetism after they have been brought into existence, and in this respect magnetism is unlike electricity.

It follows that it is impossible to have a body " charged with magnetism " in the way in which we can have a body charged with electricity. A mag- netised body may possess any number of poles, and at each pole there is, in a sense, a charge of magnetism ; but the total charge of magnetism in the body will always be zero.

Hence it follows that the simplest and most fundamental piece of matter we can imagine which is of interest for the theory of magnetism, is not a small body carrying a charge of magnetism, but a small body carrying (so to speak) two equal and opposite charges at a certain distance apart.

This leads us to introduce the conception of a line-magnet. A line- magnet is an ideal bar-magnet of which the width is infinitesimal, the length finite, and the poles at the two extreme ends. Thus geometrically the ideal line-magnet is a line, while its poles are points.

. The strengths of the two poles of a line-magnet are necessarily equal and opposite. The product of the numerical strength of either pole and the distance between the poles is called the " moment " of the line-magnet.

Magnetic Particle.

403. If we imagine the distance between the two poles of a line-magnet to shrink until it is infinitesimal, the magnet becomes what is spoken of as a magnetic particle. If ± m are the strengths of its poles and ds is the distance between the two poles, the moment of the magnetic particle is mds.

401-403] Physical Phenomena 367

It is easily shewn that, as regards all phenomena occurring at a finite distance away, two magnetic particles have the same effect if their moments are equal ; their length and the strengths of their poles separately are of no importance. To see this we need only consider the case of two magnetic particles, each having poles + m, and length ds, and therefore moment mds. Clearly these will produce the same effect at finite distances whether they are placed end to end or side by side. In the latter case, we have a magnet of length ds, poles + 2m, while in the former case the two contiguous poles, being of opposite sign, neutralise one another, and the arrangement is in effect a magnet of length 2ds and poles + m. Thus in each case the moment is the same, namely 2mds, while the strengths of the poles and their distances apart are different.

If we place a large number n of similar magnetic particles end to end, all the poles will neutralise one another except those at the extreme ends, so that the arrangement produces the same effect as a line-magnet of length

nds. By taking n = -r , where I is a finite length, we see that the effect of

a line-magnet of length I can be produced exactly by n magnetic particles of length ds.

The two arrangements will be indistinguishable by their magnetic effects at all external points. There is, however, a way by which it would be easy to distinguish them. If the arrangement were simply two poles + m, at the ends of a wire of length I, then on cutting the wire into two pieces, we should have one pole remaining in each piece. If, however, the arrangement were

± —

• -+     -+     ~+     -+     ~±      — +  ,..,.    +     — f     — b  4-  -+  -+  --f-    - 
  

——————— —^—^——— (in — — — — — — — — ■ —

Fig. 104.

that of a series of magnetic particles, we should be able to divide the series between two particles, and should in this way obtain two complete magnets. The pair of poles on the two sides of the point of division which have so far been neutralising one another now figure as independent poles.

As a matter of experiment, it is not only found to be possible to produce two complete magnets by cutting a single magnet between its poles, but it is found that two new magnets are produced, no matter at what point the cutting takes place. The inference is not only that a natural magnet must be supposed to consist of magnetic particles, but also that these particles are so small that when the magnet is cut in two, there is no possibility of

368 Permanent Magnetism [oh. xi

cutting a magnetic particle in two, so that one pole is left on each side of the division. In other words, we must suppose the magnetic particles either to be identical with the molecules of which the matter is composed or else to be even smaller than these molecules. At the same time, it will not be necessary to limit the magnetic particle of mathematical analysis by assigning this definite meaning to it: any collection of molecules, so small that the whole space occupied by it may be regarded as infinitesimal, will be spoken of as a magnetic particle.

404. Axis of a magnetic particle. The axis of a magnetic particle is defined to be the direction of a line drawn from the negative to the positive pole of the particle.

It will be clear, from what has already been said, that the effect of a magnetic particle at all external points is known when we know its position, axis and moment.

Intensity of Magnetisation.

405. In considering a bar-magnet, which must be supposed to have breadth as well as length, we have to consider the magnetic particles as being stacked side by side as well as placed end to end. For clearness, let us suppose that the magnet is a rectangular parallelepiped, its length being parallel to the axis of x, while its height and breadth are parallel to the two other axes. The poles of this bar-magnet may be supposed to consist of a uniform distribution of infinitesimal magnetic poles over each of the two faces parallel to the plane of yz, let us say a distribution of poles of aggregate strength I per unit area at the positive pole, and — / per unit area at the negative pole, so that if A is the area of each of these faces, the poles of the magnet are of strengths + I A.

As a first step, we may regard the magnet as made up of an infinite number of line-magnets placed side by side, each line-magnet being a rectangular prism parallel to the length of the magnet, and of very small cross-section. Thus a prism of cross-section dydz may be regarded as a line- magnet having poles + 1 dydz. This again may be regarded as made up of a number of magnetic particles. As a type, let us consider a particle of length dx, so that the volume of the magnet occupied by this particle is dxdydz. The poles of this particle are of strength ±Idydz, so that the moment of the particle is

I dxdydz.

If we take any small cluster of these particles, occupying a small volume dv, the sum of their moments is clearly Idv, and these produce the same magnetic effects at external points as a single particle of moment

Idv.

403-407] The Magnetic Field of Force 3G9

The quantity / is called the " intensity of magnetisation " of the magnet. The magnetisation has direction as well as magnitude. In the present instance the direction is that of the axis of x.

406. In general, we define the intensity and direction of magnetisation as follows :

The intensity of magnetisation at any point of a magnetised body is defined to be the ratio of the magnetic moment of any small particle at this point to the volume of the particle.

The direction of magnetisation at any point of a magnetised body is defined to be the direction of the magnetic axis of a small particle of magnetic matter at the point.

Instead of specifying the magnetisation of a body in terms of its poles, it is both more convenient from the mathematical point of view, and more in accordance with truth from the physical point of view, to specify the intensity at every point in magnitude and direction. Thus the bar-magnet which has been under consideration would be specified by the statement that its intensity of magnetisation at every point is / parallel to the axis of x. A body such that the intensity is the same at every point, both in magnitude and direction, is said to be uniformly magnetised.

The Magnetic Field of Force.

407. The field of force produced by a collection of magnets is in many respects similar to an electrostatic field of force, so that the various conceptions which were found of use in electrostatic theory will again be employed.

The first of these conceptions was that of electric intensity at a point. In electrostatic theory, the intensity at any point was defined to be the force per unit charge which would act on a small charged particle placed at the point. It was necessary to suppose the charge to be of infinitesimal amount, in order that the charges on the conductors in the field might not be disturbed by induction.

There is, as we shall see later, a phenomenon of magnetic induction, which is in many respects similar to that of electrostatic induction, so that in defining magnetic intensity we have again to introduce a condition to exclude effects of induction.

Also, to avoid confusion between the magnetic intensity and the intensity of magnetisation defined in § 406, it will be convenient to speak of magnetic force at a point, rather than of magnetic intensity. We accordingly have the following definition, analogous to that given in § 30.

j. '24

370 Permanent Magnetism [ch. xi

The magnetic force at any point is given, in magnitude and direction, by the force per unit strength of pole, which would act on a magnetic pole situated at this point, the strength of the pole being supposed so small that the magnetism of the field is not affected by its presence.

408. The other quantities and conceptions follow in order, as in Chapter II. Thus we have the following definitions:

A line of force is a curve in the magnetic field, such that the tangent at every point is in the direction of the magnetic force at that point (cf. § 31).

The potential at any point in the field is the work per unit strength of pole which has to be done on a magnetic pole to bring it to that point from infinity, the strength of the pole being supposed so small that the magnetism of the field is not affected by its presence (cf. § 33).

Let O denote the magnetic potential and a, /3, y the components of magnetic force at any point x, y, z, then we have from this definition (cf. equation (6)),

a = - r'V'\adx + ^dy + r/dz) (330),

and the relations (cf. equations (9)),

a = -^' ^ = -dy" 7 = "^ ( }-

A surface in the magnetic field such that at every point on it the potential has the same value, is called an Equipotential Surface (cf. § 35).

From this definition, as in § 35, follows the theorem :

Equipotential Surfaces cut lines of force at right angles.

The law of force being the same as in electrostatics, we have as the value of the potential (cf. equation (10)),

V = 2™ (332),

where m is the strength of any typical pole, and r is the distance from it to the point at which the potential is being evaluated.

As in § 42, we have Gauss' Theorem :

fJ~dS=-4>7r^m (333),

where the integration is over any closed surface, and %m is the sum of the strengths of all the poles inside this surface. If the surface is drawn so as not to cut through any magnetised matter, Sm will be the aggregate strength of the poles of complete magnetic particles, and therefore equal to zero. Thus for a surface drawn in this way

d^dS = 0 (334).

//;

407-410] The Magnetic Field of Force 371

If the position of the surface S is determined by geometrical conditions — if, for instance, it is the boundary of a small rectangular element dxdydz — then we cannot suppose it to contain only complete magnetic particles, and equation (334) will not in general be true.

If there is no magnetic matter present in a certain region, equation (334) is true for any surface in this region, and on applying it to the surface of the small rectangular element dxdydz, we obtain, as in § 50,

82o a2n d2n _ /oorN

^+a^+^ = ° ^'

the differential equation satisfied by the magnetic potential at every point of a region in which there is no magnetic matter present.

Tubes of Force.

409. A tubular surface bounded by lines of force is, as in electrostatics, called a tube of force. Let wl, tw2 be the areas of any two normal cross- sections of a thin tube of force, and let Hu H2 be the values of the intensities at these points. By applying Gauss' Theorem to the closed surface formed by the two cross-sections and the portion of the tube which lies between them, we obtain, as in § 56,

H1 &>i — i72 o)2 = 0,

provided there is no magnetic matter inside this closed surface.

Thus in free space the product Hco remains constant. The value of this product is called the strength of the tube.

In electrostatics, it was found convenient to define a unit tube to be one which ended on a unit charge, so that the product of intensity and cross-section was not equal to unity but to 4n-

Potential of a Magnetic Particle.

410. Let a magnetic particle consist of a pole of strength — m, at 0, and a pole of strength +mx at P, the distance OP being infinitesimal.

The potential at any point Q will be

nQ = PQ~ot (336)'

If we put OQ = r, and denote the angle P0Q by 0, -»«; +ml this becomes Flo- 1Q5.

m1 (0Q-PQ)<nh OP cos 0 _fi cos 0 Uq~~ PQ.OQ ~~ PQ.OQ ~ 1^ {6°n'

where /x = ml. OP, the moment of the particle.

24—2

372

Permanent Magnetism

[CH. XI

The analysis here given and the result reached are exactly similar to those already given for an electric doublet in § 64. The same result can also be put in a different form.

Let us put OP = ds, and let ^- denote differentiation in the direction of

OP, the axis of the particle. Then equation (336) admits of expression in the form

°.-".*5©-*i(?) (338)-

Let I, m, n be the direction-cosines of the axis of the particle, then formula (338) can also be written

8m- 3/1N- dlM .(339),

nQ = fj,

dx \r

3 /1\

• m„- - + oy \rj

3

where, in differentiation, x, y, z are supposed to be the coordinates of the particle, and not of the point Q.

411. Resolution of a magnetic particle. Equation (339) shew* that the potential of the single particle we have been considering is the same as the potential of three separate particles, of strengths liI, fim and fin, and axes in the directions Ox, Oy, Oz respectively Thus a magnetic particle may be resolved into components, and this resolution follows the usual vector law.

The same result can be seen geometrically.

Let us start from 0 and move a distance Ids parallel to the axis of x, then a distance mds parallel to the axis of y, and then a distance nds parallel to the axis of z. This series of movements brings us from 0 to P, a distance ds in the direction I, m, n. Let the path be OqrP in fig. 106. The magnetic particle under consideration has poles — mx at 0 and + m, at P. Without altering the field, we can super- pose two equal and opposite poles ± mx at q, and also two equal and opposite poles + ni^ at r.

The six poles now in the field can be taken in three pairs so as to constitute three doublets of strengths m^ . Oq, m^ . qr and m^ rP respec- tively along Oq, qr and rP. These, however, are doublets of strengths fil, fim and /xn parallel to the coordinate axes.

Potential of a Magnetised Body.

412. Let I be the intensity of magnetisation at any point of a mag- netised body, and let I, in, n be the direction-cosines of the direction of magnetisation at this point.

410-413]

The Magnetic Field of Force

373

The matter occupying any element of volume dxdydz at this point will be a magnetic particle of which the moment is I dxdydz and the axis is in direction I, m, n. By formula (339), the potential of this particle at anv external point is

1 dx \r) ' dy

^{l)+nFz{l)}dx^det

so that, by integration, we obtain as the potential of the whole body at any external point Q,

nQ =

dx \r) dy

4 ©♦"I©}*** <340>

in which r is the distance from Q to the element dxdydz, and the integration extends over the whole of the magnetised body.

If we introduce quantities A, B, G defined by

A=Il\ B = Im G=In then equation (340) can be put in the form

9 /1\ . „ 3 t\ „ d

.(341),

n«=

A

dx \r

dy \r) dz

-)> dxdydz.

.(342).

The quantities A, B, G are called the components of magnetisation at the point x, y, z. Equation (342) shews that the potential of the original magnet, of magnetisation I, is the same as the potential of three superposed magnets, of intensities A, B, G parallel to the three axes. This is also obvious from the fact that the particle of strength I dxdydz, which occupies the element of volume dxdydz, may be resolved into three particles parallel to the axes, of which the strengths will be Adxdydz, Bdxdydz and Gdxdydz, if J., B, G are given by equations (341).

Potential of a uniformly Magnetised Body.

413. If  the  magnetisation  of  any  body  is  uniform,  the  values  of  A,  B,  G 
     

are the same at all points of the body.

Let the coordinates of the point Q in equation (342) be x', y', z', so that i - [(* - x'f + (y - y')' + (s - *')'] " K

Then, clearly, | (I) =- 1, (J) , etc.

374 Permanent Magnetism [ch. xi

Replacing differentiation with respect to x, y, z by differentiation with respect to x, y', z in this way, we find that equation (342) assumes the form

n^-{Al+Bh+cM\dxd^ (343)-

7.   ri       7) 
   

the quantities A, B, G and the operators r— ; , ~—, , =p ■, being taken outside the

sign of integration, since they are not affected by changes in x, y, z.

If V denote the potential of a uniform distribution of electricity of volume density unity throughout the region occupied by the magnet, we have

VQ=jji^dxdydz (344),

so that equation (343) becomes

Q«=-^-#-<# (3«>.

or nQ = AX + BY+CZ,

where X, Y, Z are the components of electric intensity at Q produced by this distribution.

Or again if ^-, denotes differentiation with respect to the coordinates of Q

in a direction parallel to that of the magnetisation of the body, namely that of direction-cosines I, m, n, equation (345) becomes

"« = - ^ (346). ■

414. Yet another expression for the potential of a uniformly magnetised body is obtained on transforming equation (342) by Green's Theorem. If V, m, n' are the direction-cosines of the outward-drawn normal to the magnet at any element dS of its surface, the equation obtained after transformation is

nQ =

Jf(Al' + Bm'+Cri)ldS.

By equations (341),

Al' + Bm' + On' = I (W + mm' + nn')

= i" cos 8,

where 9 is the angle between the direction of magnetisation and the outward normal to the element dS of surface. The equation now becomes

'/cos 6

O,.//:

dS (347),

shewing that the potential at any external point is the same as that of a surface distribution of magnetic poles of density / cos 6 per unit area, spread over the surface of the magnet.

413-416] The Magnetic Field of Force 375

This distribution is of course simply the " Green's Equivalent Stratum " (§ 204) which is necessary to produce the observed external field.

The bar-magnet already considered in § 405, provides an obvious illustra- tion of these results.

415. Uniformly magnetised sphere. A second and interesting example of a uniformly magnetised body is a sphere, magnetised with uniform intensity I. This acquires its interest from the fact that the earth may, to a very rough approximation, be regarded as a uniformly magnetised sphere.

If we follow the method of § 313, we obtain for the value of Vq, defined by equation (344),

where a is the radius of the sphere. If we suppose the magnetisation to be in the direction of the axis of #, we have

Thus the potential at any external point is the same as that of a magnetic particle of moment %7ra3I at the centre of the sphere.

To treat the problem by the method of § 414, we have to calculate the potential of a surface density 7 cos 6 spread over the surface of the sphere. Regarding cos 6 as the first zonal harmonic Px (cos 6), the result follows at once from § 257

Poisson's imaginary Magnetic Matter.

416. if the magnetisation of the body is not uniform, the value of QQ given in equation (342) cannot be transformed into a surface integral, so that the potential of the magnet cannot be represented as being due to a surface charge of magnetic matter. If we apply Green's Theorem to the integral which occurs in equation (342), we obtain

• • lll\ (I + 1+ s) ***+//; <" + mB + ^ dS-

where I, m, n are the direction-cosines of the outward-drawn normal to the element dS of surface.

376 Permanent Magnetism [ch. xi

Thus nQ=jjj^dxdydz + jj^dS (348),

where p, a are given by

fdA dB dC\ ,_,ft.

r = -[te+^ + ^) <349)'

a= LA+mB + nG (350).

Thus the potential of the magnet at any external point Q is the same as if there were a distribution of magnetic charges throughout the interior, of volume-density p given by equation (349), together with a distribution over the surface, of surface-density a given by equation (350).

Potential of a Magnetic Shell.

417. A magnetised body which is so thin that its thickness at every point may be treated as infinitesimal, is called a " magnetic shell." Throughout the small thickness of a shell we shall suppose the magnetisation to remain constant in magnitude and direction, so that to specify the magnetisation of a shell we require to know the thickness of the shell and the intensity and direction of the magnetisation at every point.

Shells in which the magnetisation is in the direction of the normal to the surface of the shell are spoken of as "normally-magnetised shells." These form the only class of magnetic shells of any importance, so that we shall deal only with normally-magnetised shells, and it will be unnecessary to repeat in every case the statement that normal magnetisation is intended.

If I is the intensity of magnetisation at any point inside a shell of this kind, and if r is its thickness at this point, the product It is spoken of as the " strength " of the shell at this point. Any element dS of the shell will behave as a magnetic particle of moment IrdS, so that the strength of a shell is the magnetic moment per unit area, just as the intensity of magneti- sation of a body is the magnetic moment per unit volume.

Any element dS of a shell of strength (/> behaves like a magnetic particle of strength <fidS of which the axis is normal to dS.

The magnetisation of a magnetic shell may often be conveniently pictured as being due to the presence of layers of positive and negative poles on its two faces. Clearly if <f> is the strength and t the thickness of a shell at

any point, the surface-density of these poles must be taken to be + — ,

T

418. To obtain the potential of a shell at an external point, we regard any element dS of the shell as a magnetic particle of moment <f>dS and axis in the direction of the normal to the shell at this point, it being agreed that this normal must be drawn in the direction of magnetisation of the shell.

416-420] Potential Energy 377

The potential of the element dS Ox the shell at a point Q distant r from dS is then

*«4(;)-

so that the potential of the whole shell at Q is given by

-//♦^«

where 0 is the angle between the normal at dS and the line joining dS to P. Clearly dS cos 6 is the projection of the element dS on a plane perpendicular

to the line joining dS to P so that — is the solid angle subtended by

dS at Q. Denoting this by dco, we have the potential in the form

nQ

= ffij)dto (351).

419. Uniform shell. If the shell is of uniform strength, $ may be taken outside che sign of integration in equation (351), so that we obtain

£lQ = (j> (!dw = (})£l (352),

where 11 is the total solid angle subtended by the shell at Q.

Potential Energy of a Magnet in a Field of Force.

420. The potential energy of a magnet in an external field of force is equal to the work done in bringing up the magnet from infinity, the field of force being supposed to remain unaltered during the process.

Consider first the potential energy of a single particle, consisting of a pole of strength — 7nx at 0 and a pole of strength + mx at P. Let the potential of the field of force at 0 be H0 and at P be fLP. Then the amounts of work done on the two poles in bringing up this particle from infinity are respectively — m^o and mjfip, so that the potential energy of the particle when in the position OP

= raj (HP — fi0)

= m1 . OP -zr- , in the notation already used,

an /7an an am ,_s

378

Permanent Magnetism

[CH. XI

The potential energy of any magnetised body can be found by integration of expression (353), the body being regarded as an aggregation of magnetic particles.

421. Equation (353) assumes a special form if the magnetic field is due solely to the presence of a second magnetic particle. Let this be of moment fi, its axis having direction cosines V, m! , n', and its centre having coordinates x ', y , z'. Then we have as the value of X2, from § 410,

°-'»£M'»+<£+4)(?):