The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 23 of 39

Substituting these values for O in the formulae just obtained, we have as the mutual potential energy of the two magnets,

32 (V

dsds' \r,

1 1+ iA/V— + '— + '— V-

dx dy bzj \ dx' dy' dz'J \rj

This is symmetrical with respect to the two magnets, as of course it ought to be — it is immaterial whether we bring the first magnet into the field of the second, or the second into the field of the first.

If we now put

r {(x-xy + (y-yj + (z-z'y}^ we obtain on differentiation,

3 /1\ _ x — x' _x—x

dx' \r) ~ {(a. _ xy + (y _ yy + (Z _ zyfi ~ r3

82 /1\ 1 S(x-x'Y

so that

dx'dx' \rj r° r

^ /l^_ S(x-x')(y-y')

r5

dydx' \r Hence we obtain as the value of W,

W= ^{U' + mm' + nn')

, etc.

fy*d

2>fAIJL

[l (x - x') + m (y - y') + n{z- z')} [V (x - x') +m'(y-y')+n'(z-z')}.

Let us now denote the angle between the axes of the two magnets by e, and the angles between the line joining the two magnets and the axes of the first and second magnets respectively by 8 and 6'. Then cos e —ll'+ mm! + nn',

cos 6 = - {I (x ~x')+m(y — y') + n(z- z')},

cosd' = ^{l'(x-x') + m'(y-y') + n'(z-z')}

420-422] Potential Energy 379

so that W can be expressed in the form

F=^(cose-3cos0cos6>') (354).

If we take the line drawn from the first magnet to the second as pole in spherical polar coordinates, and denote the azimuths of the axes of the two magnets by yfr, yjr', then the polar coordinates of the directions of the axes of the two magnets will be 0, y{r and 0', yfr' respectively, and we shall have

cos e = cos 0 cos & + sin 0 sin 0' cos (yjr — yjr'). On substituting this value for cos e in equation (354), we obtain

W= &£ {sin 0 sin ff cos (f -f)-2 cos 0 cos ff) (355).

422. Knowing the mutual potential energy W, we can derive a know- ledge of all the mechanical forces by differentiation. For instance the repulsion between the two magnets, i.e. the force tending to increase r, is

-wor

dW , or

r4

{sin 0 sin & cos (yjr — yjrf) — 2 cos 0 cos ff).

Thus, whatever the position of the magnets, the force between them varies as the inverse fourth power of the distance.

If the magnets are parallel to one another, 0 = 0' and yjr = yjr', so that the repulsion

r*

(sin2 0-2 cos2 0).

Thus when # = 0, i.e. when the magnets lie along the line joining them, the force is an attractive force -~- . When 0 = k, so that the magnets are

q '

at right angles to the line joining them, the force is a repulsive force — — .

In passing from the one position to the other the force changes from one of attraction to one of repulsion when sin2 0 — 2 cos2 0 = 0, i.e. when 0 = tan-1 *J2.

The couples can be found in the same way. If ^ is any angle, the couple tending to increase the angle % is — -~ — , or

-^-k- {sin 0 sin & cos (^ - •«//) - 2 cos 0 cos 0% so that all the couples vary inversely as the cube of the distance.

380 Permanent Magnetism [ch. xi

For instance, taking x *° De the same as i/r, we find that the couple tending to rotate the first magnet about the line joining it to the second, in the direction of ty increasing

so that this couple vanishes if either of the magnets is along the line joining them, or if they are in the same plane, results which are obvious enough geometrically.

Potential Energy of a Shell in a Field of Force.

423. Consider a shell of which the strength at any point is cj), placed in a field of potential O. The element dS of the shell is a magnetic particle of strength <j>dS, so that its potential energy in the field of force will, by formula (353), be

*<•

where »- denotes differentiation along the normal to the shell. Thus the potential energy of the whole shell will be

W^JJ^dS (356).

If the shell is of uniform strength, this may be replaced by

^-♦//i <857>-

Since the normal component of force at a point just outside the shell

and on its positive face is — ^— , it is clear that Ij-^-dS is equal to minus

the surface integral of normal force taken over the positive face of the shell, and this again is equal to minus the number of unit tubes of force which emerge from the shell on its positive face. Denoting this number of unit tubes by n, equation (357) may be expressed in the form

W=-<f>n (358).

Here it must be noticed that we are concerned only with the original field before the shell is supposed placed in position. Or, in other terms, the number n is the number of tubes which would cross the space occupied by the shell, if the shell were annihilated. Since the tubes are counted on the positive face of the shell, we see that n may be regarded as the number of unit tubes of the external field which cross the shell in the direction of its magnetisation.

1 dxdydz u -^ — -m-~- + n ^-)

422-426] Force inside a Magnetised Body 381

424. Consider a field consisting only of two shells, each of unit strength. Let n^ be the number of tubes from shell 1 which cross the area occupied by 2, and let n% be the number of tubes from shell 2 which cross the area occupied by 1. The potential energy of the field may be regarded as being either the energy of shell 1 in the field set up by 2, or as the energy of shell 2 in the field set up by 1. Regarded in the first manner, the energy of the field is found to be — n2 ; regarded in the second manner, the energy is found to be — Jij. Hence we see that ni = n2. This result, which is of great importance, will be obtained again later (§ 446) by a purely geometrical method.

Potential Energy of any Magnetised Body in a Magnetic Field of Force.

425. Let / be the intensity of magnetisation and I, m, n the direction- cosines of the direction of magnetisation at any point x, y, z of a magnetised body, and let 12 be the potential, at this point, of an external field of magnetic force. The element dxdydz of the magnetised body is a magnetic particle of strength I dxdydz, of which the axis is in the direction I, m, n. Thus its potential energy in the field of force is, by formula (353),

dy and by integration the potential of the whole magnet is

Force inside a Magnetised Body.

426 So far the magnetic force has been defined and discussed only in regions not occupied by magnetised matter : it is now necessary to consider the more difficult question of the measurement of force at points inside a magnetised body.

At the outset we are confronted with a difficulty of the same kind as that encountered in discussing the measurement of electric force inside a dielectric, on the molecular hypothesis explained in § 143. We found that the molecules of a dielectric could be regarded as each possessing two equal and opposite charges of electricity on two opposite faces. If we replace " electricity " by " magnetism " the state is very similar to what we believe to be the state of the ultimate magnetic particles. In the electric problem a difficulty arose from the fact that the electric force inside matter varied rapidly as we passed from one molecule to another, because the intensity of the field set up by the charges on the molecules nearest to any point was

382 Permanent Magnetism [ch. xi

comparable with the whole field. A similar difficulty arises in the magnetic problem, but will be handled in a way slightly different from that previously adopted. There are two reasons for this difference of treatment — in the first place, we are not willing to identify the ultimate magnetic particles with the molecules of the matter, and in the second place, we are not willing to assume that the magnetism of an ultimate particle may be localised in the form of charges on the two opposite faces. We shall follow a method which rests on no assumptions as to the connection between molecular structure and magnetic properties, beyond the well-established fact that on cutting a magnet new magnetic poles appear on the surfaces created by cutting.

427. One way of measuring the force at a point Q inside a magnet will be to imagine a cavity scooped out of the magnetic matter so as to enclose the point Q, and then to imagine the force measured on a pole of unit strength placed at Q. This method of measurement will only determine a definite force at Q if it can be shewn that the force is independent of the position, shape and size of the cavity, and this, as will be obvious from what follows, is not generally the case.

428. Let us suppose that, in order to form a cavity in which to place the imaginary unit pole, we remove a small cylinder of magnetic matter, the axis of this cylinder being in the direction of magnetisation at the point. Let this cylinder be of length I and cross-section S, and let the intensity of magnetisation at the point be /. Let the size of the cylinder be supposed to be very great in comparison with the scale of molecular structure, although very small in comparison with the scale of variation in the magnetisation of the body.

In steel or iron there are roughly 1023 molecules to the cubic centimetre, so that a length of 1 millimetre may be regarded as large when measured by the molecular scale, although in most magnets the magnetisation may be treated as constant within a length of a millimetre.

At a point near the centre of this cavity we are at a distance from the nearest magnetic particles, which is, by hypothesis, great compared with molecular dimensions. Hence, by § 416, we may regard the potential at points near the centre of the cavity as being that due to the following distributions of imaginary magnetic matter. —

I. A distribution of surface-density IA + mB + nC, spread over the surface of every magnet.

II. A distribution of volume-density

fdA dB do

dB dC\ dy + dzj'

\dx dy

spread throughout the whole space which is occupied by magnetic matter after the cavity has been scooped out.

426-430] Force inside a Magnetised Body 383

III. A distribution of surface-density IA + mB + nO, spread over the walls of the cavity.

From the way in which the cavity has been chosen, it follows that IA + mB + nC vanishes over the side-walls, and is equal to + 1 on the two ends.

The force acting on an imaginary unit pole placed at or near the centre of the cavity may be regarded as the force arising from these three distributions.

429. The force from distribution III can be made to vanish by taking the length of the cavity to be very great in comparison with the linear dimensions of its ends. For the ends of the cavity may then be treated as points, and the force exerted by either end upon a unit pole placed at the centre of the cavity will be

SI

and this will vanish if S is small compared with I2. The resultant force will therefore arise solely from distributions I and II.

The force arising from distribution II may be regarded as the force arising from a distribution of volume-density

fd_A dB dC V dx dy dz ,

spread throughout the whole of the magnetised matter, regardless of the existence of the cavity, together with a distribution of volume-density

\dx dy dz

spread through the space occupied by the cavity. The force from this latter distribution vanishes in the limit when the size of the cavity is infinitesimal, so that the force from distribution II may be regarded as that from a volume-density

fd_A d_B d_G \ dx dy dz

spread through all the original magnetised matter.

We have now arrived at a force which is independent of the shape, size and position of the cavity, provided only that these satisfy the conditions which have already been laid down. This force we define to be the magnetic force, at the point under discussion, inside the magnetised body.

430. In the notation of § 416, the force which has just been defined is due to a distribution of surface-density a, and a distribution of volume-density

384 Permanent Magnetism [ch. xi

p throughout the whole magnetised matter. The potential of these distribu- tions is

jj'dS + jjfP-dxdych,

or ClQ if we regard this as defined by equation (348). Thus, with this meaning assigned to £Iq, the components of force at a point Q inside a magnetic body will be

_d^Q _BJIq dnQ

dx ' dy * dz

At the same time it must be remembered that Qq has not been shewn to be the true value of the potential except when the point Q is outside the magnetic matter. The true potential inside magnetised matter will vary rapidly as we pass from one magnetic particle to another.

431. Let us next suppose that the length I of the cylindrical cavity is very small compared with the linear dimensions of an end. The force, as before, is that due to the distributions I, II and III of § 428. The force from distribution III, however, will no longer vanish, for this distribution con- sists of distributions + / over the ends of the cavity,

Fid 1 OR

and the force from these is not now negligible. From analogy with the distribution of electricity on a parallel plate condenser, it is clear that the force arising from distribution III is a force 4>7rl in the direction of magnetisation. The forces from distributions I and II are easily seen to be the same as in the former case. Thus the force on a unit pole placed at a point Q inside a cavity of the kind we are now considering is the resultant of

(i) the magnetic force at Q, as defined in § 429,

(ii) a force 4nrl in the direction of the intensity of magnetisation at Q.

The resultant of these forces is called the magnetic induction at Q.

432. The magnetic force will be denoted by H, and its components by a, /3, 7.

The induction will be denoted by B, and its components by a, b, c.

We have seen that the force B is the resultant of a force H and a force 47r/. The components of this latter force are \irA, 4nrB, 4nrG. Hence we have the equations

a= a + 4>7tA'

6 = /3+ 4tt5J- (359).

c = <y + 4<7rG

430-434] Force inside a Magnetised Body

385

Fig. 109.

433. Let  us  next  consider  the  force  on  a  unit  pole  inside  a  cylindrical 
     

cavity when the cavity is disc-shaped, as in § 431, but its axis is not in the direction of magnetisation. The force can, as in § 428, be regarded as arising from three distributions.

Distributions I and II are the same as before, but distribution III will now consist of charges both on the end and on the side- walls of the cylinder. By making the length of the cylinder small in comparison with the linear dimensions of its cross-section, the force from the distri- bution in the side-walls can be made to vanish. And if 0 is the angle between the axis of the cavity and the direction of magnetisation, the distribution on the ends is one of density + I cos 0. Thus the force arising from distribution III is a force 4nrl cos 0 in the direction of the axis of the cavity.

Thus the force on a pole placed inside this cavity may be regarded as compounded of the force H (arising from distributions I and II), and a force 4<7rl cos 0 in the direction of magnetisation, arising from distribution III.

Let e be the angle between the direction of the force H and the axis of the cavity, then the component force in the direction of the axis of the cavity

= H cos e + 4s7rl cos 0.

If I, m, n are the direction-cosines of this last direction,

H cos e = la. + m/3 + ny, 4-77-7 cos 0 = 4nr(lA+ mB + nO), so that, by equations (395),

H cos e + 477- 1 cos 0 = la + mb + nc.

Thus the component of the force in the direction of the axis of the cavity is the same as the component, in the same direction, of the magnetic induc- tion, namely la + mb + nc.

434. We are now in a position to understand the importance of the vector which has been called the induction. This arises entirely from the property of the induction which is expressed in the following theorem :

Theorem. The surface-integral of the normal component of induction, taken over any surface whatever, vanishes,

or in other words (cf. § 177),

field.

The induction is a soleuoidal vector throughout the whole of the magnetic

25

386

Permanent Magnetism

[CH. XI

To prove this let us take any closed surface S in the field, this surface cutting any number of magnetised bodies. Along those parts of the surface which are inside magnetic bodies, let us remove a layer of matter, so that the surface no longer actually passes through any magnetic matter.

Fig. 110. Then by Gauss' Theorem (§ 409),

ms=o

.(360),

where JSf is the component of force in the direction of the outward normal to S, acting on a unit pole placed at any point of the surface S. This force, however, is exactly identical with that considered in § 433, and its normal component has been seen to be identical with the normal component of the induction. Thus iV, in equation (360), will be the normal component of induction, so that this equation proves the theorem.

Analytically, the theorem may be stated in the form

lf(la + mb + nc)dS = 0 (361),

and this, by Green's Theorem (§ 179), is identical with

(362).

II

da db dc dx By dz

0

435. Definition. By a line of induction is meant a curve in the magnetic field such that the tangent at every point is in the direction of the magnetic induction at that point.

Definition. A tube of induction is a tubular surface of small cross- section, which is bounded entirely by lines of induction.

By a proof exactly similar to that of § 409, it can be shewn that the product of the induction and cross-section of a tube retains a constant value along the tube. This constant value is called the strength of the tube.

434-437] Force inside a Magnetised Body 387

In free space the lines and tubes of induction become identical with the lines and tubes of force, and the foregoing definition of the strength of a tube of induction is such as to make the strengths of the tubes also become identical.

436. At any point of a surface let B be the induction, and let e be the angle between the direction of the induction and the normal to the surface. The aggregate cross-section of all the tubes which pass through an element dS of this surface is dS cos e, so that the aggregate strength of all these tubes is B cos edS. Since Bcose = N, where N is the normal induction, this may be written in the form N dS. Thus the aggregate strength of the tubes of induction which cross any area is equal to

NdS.

This, we may say, is the number of unit-tubes of induction which cross this area.

The theorem that fjNdS=0,

where the integration extends over a closed surface, may now be stated in the form that the number of tubes which enter any closed surface is equal to the number which leave it. This is true no matter where the surface is situated, so that we see that tubes of induction can have no beginning or ending.

437. Let  us  take  any  closed  circuit  s  in  space,  and  let  n  be  the  number 
     

of tubes of induction which pass through this circuit in a specified direction.

Then n will also be the number <3f tubes which cut any area whatever which is bounded by the circuit s. If S is any such area, this number is

known to be lllfdS, where the integration is taken over the area S, so that

"*-//

NdS.

The number n, however, depends only on the position of the curve s by which the area S is bounded, so that it must be possible to express n in a form which depends only on the position of the curve s, and not on the area S.

In other words, it must be possible to replace 1 1 N dS by an expression which

depends only on the boundary of the area s. This we are enabled to do by a theorem due to Stokes.

25—2

388

Permanent Magnetism

[CH. XI

Stokes' Theorem. 438. Theorem. If X, Y, Z are continuous functions of position in space,

then

[(X^+Y^ + Zplds

J \ as as as/

'fdZ 3F\

dX dz

D+- £-£)** •••<**

where the line integral is taken round any closed curve in space, and the surface integral is taken over any area (or shell) bounded by the contour.

Here I, m, n are the direction-cosines of the normal to the surface. A rule is needed to fix the direction in which the normal is to be drawn. The following is perhaps the simplest. Imagine the shell turned about in space so that the tangent plane at any point P is parallel to the plane of xy, and so that the direction in which the line integral is taken round the contour is the same as that of turning from the axis of cc to the axis of y. Then the normal at P must be supposed drawn in the direction of the positive axis of z.

439. To prove the theorem, let us select any two points A, B on the contour, and let us introduce a quantity / defined by

IX/O U/o too /

/ =

B

the path from A to B being the same as that followed in the integral of equation (363). Let us also introduce a quantity J equal to the same

Fig. ill.

integral taken from A to B, but along the opposite edge of the shell. Th the whole integral on the left of equation (363) is equal to I — J.

en

438, 439] Stokes' Theorem 389

It will be possible to connect A and B by a series of non-intersecting lines drawn in the shell in such a way as to divide the whole shell into narrow strips. Let us denote these lines by the letters a, b, ... n, the lines being taken in order across the shell, starting with the line nearest to that along which we integrate in calculating /. Let us denote the value of

B f„dx „dy „dz\ T 1 X-j' + Y-f + Z-j-ids as ds dsj

/;(

taken along the line a by Ia.

Then the left-hand member of equation (363) = I-J = (I-Ia)+(Tm-Ib) + (Ib-Ie) + . .. + (/„- J>

Let us consider the value of any term of this series, say Ia — Ib.

Let us take each point on the line a and cause it to undergo a slight displacement, so that the coordinates of any point x, y, z are changed to x + 8x, y + 8y, z + 8z. If 8x, 8y, 8z are continuous functions of x, y, z the result will be to displace the line a into some adjacent position, and by a suitable choice of the values of 8x, 8y, 8z this displaced position of line a can be made to coincide with line b. If this is done, it is clear that the value of I a, after replacing x, y, z by x+8x,y + 8y, z+ 8z, will be Ib. Hence if we denote this new value of Ia by Ia + 81, we shall have

Ia+8I = Ib,

so that Ia — Ib = — 81

J a \ ds ds ds)

and the value of this quantity can be obtained by the ordinary rules of the calculus of variations.

"We have

rB fjT rB (jr rB rj

8 X^ds= 8XCpds+ X~{8x)ds JA ds J A ds JA dsx

(B/dx ax dx s \ dx , ,

=]AteSx+dtBy+te8Vds-ds+

X8x

X 8x

B rB jx — I —r—8xds,

A J A ds

B

may be omitted,

and since 8x vanishes both at A and B, the term and the whole expression put equal to

[B(dAs —8 dX-8zy\ — -(—— — ^+— — )Sx\ds

J a \dx dy y dz J ds \dx ds dy ds dz ds) j

390 Permanent Magnetism

or again, on simplifying, to

[CH. XI

/.

A

M (S,j > - sM - d fa * - S, p)\ els.

\dy \ J ds dsj dz \ ds dsj)

This may De written in the form

\jr~ (tydx "~ Bxdy) — -x— (&ccfo — &3cfcc)L

/;

.(364).

Fig. 112.

Now in fig. 112, let P, Q, P' be the points x, y, z; x + dx, y + dy, z + dz ; and x + Sx, y + By, z+ Bz. Let dS denote the area of the parallelogram PQQ'P', and let I, m, n be the direction-cosines of the normal to its plane. Then the projection of the parallelogram on the plane of xy will be of area ndS, while the coordinates of three of its angular points will be x, y\ x + dx, y + dy; and x + 8x, y + By. Using the usual formula for the area, we obtain

ndS = (Bydx — Bxdy),

and using this relation in expression (364), we obtain

8JBX^ds = j(^ndS-^mdS) (365),

Jy

the integral denoting summation over all those elements of area of the shell which lie between lines a and b. type of (365), we obtain

"B dx

By summation of three equations of the

[B dx Ia — Ib = — 8 I X -Y~ds — B J a ds

ds i

• hr

A dX

dz

B d?

ZT-ds ds

~)^dS +

dx

d4)ndS

where the integration has the same meaning as before. If we add a system of equations of this type, one for each strip, the left-hand, as already seen, becomes I — J, which is equal to the left-hand member of equation (363), while the right-hand member of the new equation is also the right-hand member of equation (363). This proves the theorem.

439-441] Stokes7 Theorem 391

440. Stokes' Theorem can be readily expressed in a vector notation. If X, Y, Z are the components of any vector F, it is usual to denote by curl P the vector of which the components are

dZ_dY dX_dZ d_Y_dX

dy dz ' dz d%' da dy '

Hence Stokes' Theorem assumes the form

J'

(component of F along ds) ds

= /(components of curl P along normal to dS) dS.

The theorem enables us to transform any line integral taken round a closed circuit into a surface integral taken over any area by which the circuit can be filled up. The converse operation of changing a surface integral into a line integral may or may not be possible.

441. Theorem.     It  will  be  possible  to  transform  the  surface  integral 
     

(lu + mv + niv)dS (366)

//<

into a line integral taken round the contour of the area S if, and only if,

du dv div

ai + ^ + ^ = 0 (367)

at every point of the area S.

It is easy to see that this condition is a necessary one. Let S' denote any area having the same boundary as S, and being adjacent to it, but not coinciding with it. Then if I is the line integral into which the surface integral can be transformed, we must have

I=jj(lu + mv + nw)dS (368),

and also I = (((I'u + m'v + n'w) dS' (369).

On equating these two values for i" we obtain an equation which may be expressed in the form

ff(lu + mv + niv)dS = 0 (370),

where the integration is over a closed surface bounded by S and S', and I, m, n are the direction-cosines of the outward normal to the surface at any point. From equation (370), the necessity of condition (367) follows at once.

Condition (367) is most easily proved to be sufficient by exhibiting an actual solution of the problem when this condition is satisfied. We have to

392

Permanent Magnetism

[CH. XI

shew that, subject to condition (367) being satisfied, there are functions

X, Y, Z such that

d_Z_d7= \

dy dz

dX_d_Z dz dx

dY_d_X

dx dy j

= v V

= w

•(371),

for if this is so, the required line integral is (IX + mY+ nZ) dS. By inspection a solution of equations (371) is seen to be

X=fvdz, Y=-fudz, Z = 0 (372),

du dv\ 7 fdw 7

7T- ) dz = -^dz = w,

dx dy! J dz

for it is obvious that the first two equations are satisfied, and on substituting in the third, we obtain

d_Y_d_X_ dx dy

shewing that the proposed solution satisfies all the conditions.

442. The absence of symmetry from solution (372) suggests that this solution is not the most general solution. The most general solution can, however, be easily found. If we assume it to be

X=jvdz+X', Y=-judz + Y', Z=Z'

then we find, on substitution in equations (371), that we must have

ar_aF az; = az' dY' _dxr

• dz '

.(373),

.(374),

dy dz ' dz dx ' dx dy and if we introduce a new variable % defined by %= jX'dx, we find at once that

dx' dy' dz'

so that the most general solution of equations (371) is

*~h+& Y=~h^y zJi <375>-

Substituting these values, the line integral is found to be

dy

ds+f^ds,

![{hz)-£-(hz)ds_

and the condition that this shall be equal to the surface integral is that or that x shall be single-valued.

441-444] Vector-Potential 393

Thus if x is any single-valued function, equations (375) represent a solu- tion, and the most general solution, of equations (371).

Vector-Potential. 443. The discussion as to the transformation from surface to line inte- grals arose in connection with the integral jjJ^dS or I J (la + mb + nc) dS, in

which a, b, c are the components of magnetic induction. Since the condition

da db dc _ „

dx dy dz

is satisfied throughout all space, it must always be possible (cf. § 441) to transform the surface integral into a line integral by a relation of the form

Jfoa + mb + nc) dS=f(F^ + G^ + H^jds.

The vector of which the components are F, G, H is known as the magnetic vector- -potential.

From what has been said in § 442, it is clear that the vector-potential is

not fully determined when the magnetic field is given. On the other hand,

if the vector-potential is given the magnetic field is fully determined", being

given by the equations

= d_H_dG\

dy dz

dF dH

dz dx

_dG_dF

dx dy

(376).

We shall calculate some possible values of the components of vector- potential in a few simple cases. It must be remembered that the values obtained, although solutions of equations (376), will not be the most general solutions.

Magnetic Particle.

444. Let us first suppose that the field is produced by a single magnetic particle at the point x', y', z in free space, parallel to the axis of z. Then,

7. /I \ by equation (338), Q, = fx ^-> ( - ] , so that at any point x, y, z,

dn a2 /i\ aj /i

a = a = — and similarly

dx dxdz'WJ dxdz\7'

b = ^dyTz[-r)' C = ^aT2lr

394

Permanent Magnetism

[OH. XI

The equations to be solved (equations (376)) are

dH_dG = J!_/1N dy dz dxdz \r.

dF_d_H= 8* /1>

dz dx dydz \r,

dG_ dF_ &_ (V

dx dy dz2 \rj

and the simplest solution, similar to that given by equations (372), is

F=fi

G = -

f*

dx

H = 0.

"' dy [r.

The components of vector-potential for a magnet parallel to the axes of x or y can be written down from symmetry. In terms of the coordinates x', y, z' of the magnetic particle, this solution may be expressed as

F=-

dy

4)-

G = /i

dx'

H = 0.

445. Let us superpose the fields of a magnetic particle of strength l\i parallel to the axis of x, one of strength mp parallel to the axis of y, and one of strength n/j, parallel to the axis of z. Then we obtain the vector- potential at x, y, z due to a magnetic particle of strength fi and axis (I, m, n) at x , y\ z' in the forms

1 _ JL^ 1 « . f — -n —

dz dy) r \ dz' dy'

d_ d

dx

F= — fx(m

dz ) r

1/9 d\

H=

H*

7 d d\ i /, a a

dy dx] r \ oy

dx

1 \

r

i>

r

1

r I

....(377).

The number of lines of induction which cross the circuit from a magnetic particle is (§ 437)

/(

F^ + G^ + H^ds,

ds

which may be written in the form

dx ds'

I,

ds

dy ds3

m,

d_

dx

\r)' dy\r)'

ds)

dz ds

n dz \r

ds,

the integral being taken round the circuit in the direction determined by the rule given in § 438 (p. 388).

444-446] Vector- Potential 395

Uniform Magnetic Shell.

446. Next let us suppose that the lines of force proceed from a uniform magnetic shell, supposed for simplicity to be of unit strength. Let V, m, n' be the direction-cosines of the normal to any element dS' of this shell. Then the element dS' will be a magnetic particle of moment dS' and of direction-cosines I', m', n. The element accordingly contributes to F a term which, by equations (377), is seen to be

K-4)(£K

dy'.

where x, y', z are the coordinates of the element dS'. Thus the whole value of F is

*=//K4-4)(>'-

This surface integral satisfies the condition of § 441, so that it must be possible to transform it into a line integral of the form

The equations giving /, g, h are

Clearly a solution is

dh dy''

dz' U'

df dz''

dh _ d n\

dx dz \r) '

dx'

dy' dy' \r

f=-r, #=0, h = 0,

so that on substitution the value of F is

F= finds'.

J r ds

Similarly G = / - -^ ds',

J r ds

Thus the number of tubes of induction crossing the circuit s from a magnetic shell of unit strength bounded by the circuit s', is given by

~J('l+»3f+5)

or

— ([(— ^x' 4. dy dy' dz dz'\ 1 t j / JJ\ds ds' ds ds ds ds J r

396 Permanent Magnetism [ch. xi

If e is the angle between the two elements ds, ds', the direction of these elements being taken to be that in which the integration takes place, then

dx dx dy dy' dz dz _

ds ds ds ds' ds ds' '

f f COS 6

so that n = 1 1 dsds'.

From the rule as to directions given on p. 388, it will be clear that if the integration is taken in the same direction round both circuits, then the direction in which the n lines cross the circuit will be that of the direction of magnetisation of the shell.

Clearly n is symmetrical ao regards the two circuits s and s', so that we have the important result :

The number of tubes of induction crossing the circuit s from a shell of unit strength bounded by the circuit s' is equal to the number of tubes of induction crossing the circuit s' from a shell of unit strength bounded by the circuit s.

Here we have arrived at a purely geometrical proof of the theorem already obtained from dynamical principles in § 424.

Energy of a Magnetic Field.

447. Let a, b, c, . . . n be a system of magnetised bodies, the magnetisation of each being permanent, and let us suppose that the total magnetic field arises solely from these bodies. Let us suppose that the potential fl at any point is regarded as the sum of the potentials due to the separate magnets. Denoting these by I2a, H&, ... fln, we shall have

Let us denote the potential energy of magnet a, when placed in the field of force of potential H, by £2 (a) ; if placed in the field of force arising from magnet b alone, by H& (a), etc.

Let us imagine that we construct the magnetic field by bringing up the magnets a, b, c, ... n in this order, from infinity to their final positions.