A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 3 of 27

as the value of the potential energy of the magnet with respect to the magnetic field in which it is placed.

The potential energy is here expressed in terms of the components of magnetization and of those of the magnetic force arising from external causes.

By integration by parts we may express it in terms of the distribution of magnetic matter and of magnetic potential

~ + -- + -dxdydzy (4)

where /, m, n are the direction-cosines of the normal at the element of surface dS. If we substitute in this equation the expressions for the surface- and volume-density of magnetic matter as given in Art. 386, the expression becomes

pdS. (5)

We may write equation (3) in the form

• Cy}dxdydz, (6)

where a, ft and y are the components of the external magnetic force.

On the Magnetic Moment and Axis of a Magnet.

390.] If throughout the whole space occupied by the magnet the external magnetic force is uniform in direction and magnitude, the components a, /3, y will be constant quantities, and if we write

IJJAdxdydz=lK, jjJBdxdydz=mK, [((cdxdydz = nKt (7)

the integrations being extended over the whole substance of the magnet, the value of ^may be written

y). (8)

16 ELEMENTAEY THEORY OF MAGNETISM.

In this expression I, m, n are the direction-cosines of the axis of

the magnet, and K is the magnetic moment of the magnet. If

e is the angle which the axis of the magnet makes with the

direction of the magnetic force «£), the value of W may be written

JF = -K$cos€. (9)

If the magnet is suspended so as to be free to turn about a vertical axis, as in the case of an ordinary compass needle, let the azimuth of the axis of the magnet be $, and let it be inclined 0 to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is 5 and dip £, then

a = «$p cos £ cos bj (3 = «£j cos £ sin 8, y = «£) sin f; (10)

I = cos 0 cos <£, m = cos 0 sin <£, n — sin 0 ; (11)

whence W— — KQ (cos £ cos 6 cos ($ — 8) + sin ( sin e). (12)

The moment of the force tending to increase $ by turning the magnet round a vertical axis is

_ ^L=_K cos Ccos<9 sin (<J>-5). (13)

On the Expansion of the Potential of a Magnet in Solid Harmonics.

391.] Let V be the potential due to a unit pole placed at the point (£, T?, f). The value of F" at the point #, y, z is

r= {(f-)2+(>/-,?o2 +(<r-)Ti (i)

This expression may be expanded in terms of spherical harmonics, with their centre at the origin. We have then

(2)

when FQ = - , r being the distance of (f, 77, f ) from the origin, (3)

(4)

_

2~ 2r5

fee.

To determine the value of the potential energy when the magnet is placed in the field of force expressed by this potential, we have to integrate the expression for W in equation (3) with respect to x, y and z, considering £, 77, (" and r as constants.

If we consider only the terms introduced by F~0, Ft and V2 the result will depend on the following volume-integrals,

392.] EXPANSION OF THE POTENTIAL DUE TO A MAGNET. 17 lK = jjJAdxdydz, mK = fjfsdxdydz, nK =JJJ Cdxdydz; (6)

L=jjJAxdxdydz> M = jjj Bydxdydz, N =jjJCzdxdydz', (7)

P = (B* + Cy)dxdydz, Q =

R = ^y + Bnyndydz- (8)

We thus find for the value of the potential energy of the magnet placed in presence of the unit pole at the point (^17, Q, _

r5

This expression may also be regarded as the potential energy of the unit pole in presence of the magnet, or more simply as the potential at the point £ , 17, f due to the magnet.

On ike Centre of a Magnet and its Primary and Secondary Axes.

392.] This expression may be simplified by altering the directions of the coordinates and the position of the origin. In the first place, we shall make the direction of the axis of x parallel to the axis of the magnet. This is equivalent to making

l—^ m = 0, n — 0. (10)

If we change the origin of coordinates to the point (#', y', /), the directions of the axes remaining unchanged, the volume-integrals IK, mK and nK will remain unchanged, but the others will be altered as follows :

L'=L-lKx', M'=M-mKy', Nf = N-nKz/-f (11)

P'=P—K(mz'+ny), Q'=Q- K(nx' + lz'\ R' — R— K(ly' + mx'}.

If we now make the direction of the axis of x parallel to the axis of the magnet, and put

, Zl-M-N , R , Q , .

x'= —^ > y = Tr> z = -^> (13)

2A A A

then for the new axes M and N have their values unchanged, and the value of 1! becomes \ (M+N). P remains unchanged, and Q and R vanish. We may therefore write the potential thus,

VOL. II.

18 ELEMENTARY THEOEY OF MAGNETISM. _392-

We have thus found a point, fixed with respect to the magnet, such that the second term of the potential assumes the most simple form when this point is taken as origin of coordinates. This point we therefore define as the centre of the magnet, and the axis drawn through it in the direction formerly defined as the direction of the magnetic axis may be defined as the principal axis of the magnet.

We may simplify the result still more by turning the axes of y

and z round that of x through half the angle whose tangent is

p -=£ — — . This will cause P to become zero, and the final form

of the potential may be written

Kt ttf-

3 2

This is the simplest form of the first two terms of the potential of a magnet. When the axes of y and z are thus placed they may be called the Secondary axes of the magnet.

We may also determine the centre of a magnet by finding the position of the origin of coordinates, for which the surface-integral of the square of the second term of the potential, extended over a sphere of unit radius, is a minimum.

The quantity which is to be made a minimum is, by Art. 141, 4 (Z2 + Mz + N*-MN-NL-LM] + 3 (P2 + Q2 +^2). (16)

The changes in the values of this quantity due to a change of position of the origin may be deduced from equations (11) and (12). Hence the conditions of a minimum are

21(2 L—M— N)+3nQ+3mR = 0, 2m(2M-N-L)+3lR+3nP = 0, (17)

2n (2N—Z—M)+3mP+3lQ = 0. If we assume I = I, m = 0, n = Q, these conditions become

2L-M—N=0, q = 0, R=0, (18)

which are the conditions made use of in the previous invest igation.

This investigation may be compared with that by which the potential of a system of gravitating matter is expanded. In the latter case, the most convenient point to assume as the origin is the centre of gravity of the system, and the most convenient axes are the principal axes of inertia through that point.

In the case of the magnet, the point corresponding to the centre of gravity is at an infinite distance in the direction of the axis,

394'] CONVENTION RESPECTING SIGNS. 19

and the point which we call the centre of the magnet is a point having- different properties from those of the centre of gravity. The quantities If, M, N correspond to the moments of inertia, and P, Q, R to the products of inertia of a material body, except that Z, M and N are not necessarily positive quantities.

When the centre of the magnet is taken as the origin, the spherical harmonic of the second order is of the sectorial form, having its axis coinciding with that of the magnet, and this is true of no other point.

When the magnet is symmetrical on all sides of this axis, as in the case of a figure of revolution, the term involving the harmonic of the second order disappears entirely.

393.] At all parts of the earth's surface, except some parts of the Polar regions, one end of a magnet points towards the north, or at least in a northerly direction, and the other in a southerly direction. In speaking of the ends of a magnet we shall adopt the popular method of calling the end which points to the north the north end of the magnet. When, however, we speak in the language of the theory of magnetic fluids we shall use the words Boreal and Austral. Boreal magnetism is an imaginary kind of matter supposed to be most abundant in the northern, parts of the earth, and Austral magnetism is the imaginary magnetic matter which prevails in the southern regions of the earth. The magnetism of the north end of a magnet is Austral, and that of the south end is Boreal. When therefore we speak of the north and south ends of a magnet we do not compare the magnet with the earth as the great magnet, but merely express the position which the magnet endeavours to take up when free to move. When, on the other hand, we wish to compare the distribution of ima ginary magnetic fluid in the magnet with that in the earth we shall use the more grandiloquent words Boreal and Austral magnetism.

394.] In speaking of a field of magnetic force we shall use the phrase Magnetic North to indicate the direction in which the north end of a compass needle would point if placed in the field of force.

In speaking of a line of magnetic force we shall always suppose it to be traced from magnetic south to magnetic north, and shall call this direction positive. In the same way the direction of magnetization of a magnet is indicated by a line drawn from the south end of the magnet towards the north end, and the end of the magnet which points north is reckoned the positive end.

20 ELEMENTARY THEORY OF MAGNETISM. _394--

We shall consider Austral magnetism, that is, the magnetism of that end of a magnet which points north, as positive. If we denote its numerical value by m> then the magnetic potential

and the positive direction of a line of force is that in which V diminishes.

CHAPTER II.

MAGNETIC FORCE AND MAGNETIC INDUCTION.

395.] WE have already (Art. 386) determined the magnetic potential at a given point due to a magnet, the magnetization of which is given at every point of its substance, and we have shewn that the mathematical result may be expressed either in terms of the actual magnetization of every element of the magnet, or in terms of an imaginary distribution of ' magnetic matter,' partly condensed on the surface of the magnet and partly diffused through out its substance.

The magnetic potential, as thus denned, is found by the same mathematical process, whether the given point is outside the magnet or within it. The force exerted on a unit magnetic pole placed at any point outside the magnet is deduced from the potential by the same process of differentiation as in the corresponding electrical problem. If the components of this force are a, /3, y,

dV dV dV m

a= — > /3 = j-j y— j-- (1)

dx dy dz

To determine by experiment the magnetic force at a point within the magnet we must begin by removing part of the magnetized substance, so as to form a cavity within which we are to place the magnetic pole. The force acting on the pole will depend, in general, in the form of this cavity, and on the inclination of the walls of the cavity to the direction of magnetization. Hence it is necessary, in order to avoid ambiguity in speaking of the magnetic force within a magnet, to specify the form and position of the cavity within which the force is to be measured. It is manifest that when the form and position of the cavity is specified, the point within it at which the magnetic pole is placed must be regarded as

22 MAGNETIC FORCE AND MAGNETIC INDUCTION. [396.

no longer within the substance of the magnet, and therefore the ordinary methods of determining the force become at once applicable.

396.] Let us now consider a portion of a magnet in which the direction and intensity of the magnetization are uniform. Within this portion let a cavity be hollowed out in the form of a cylinder, the axis of which is parallel to the direction of magnetization, and let a magnetic pole of unit strength be placed at the middle point of the axis.

Since the generating lines of this cylinder are in the direction of magnetization, there will be no superficial distribution of mag netism on the curved surface, and since the circular ends of the cylinder are perpendicular to the direction of magnetization, there will be a uniform superficial distribution, of which the surface- density is /for the negative end, and —/for the positive end.

Let the length of the axis of the cylinder be 2 b, and its radius a. Then the force arising from this superficial distribution on a magnetic pole placed at the middle point of the axis is that due to the attraction of the disk on the positive side, and the repulsion of the disk on the negative side. These two forces are equal and in the same direction, and their sum is

---!=. (2)

From this expression it appears that the force depends, not on the absolute dimensions of the cavity, but on the ratio of the length to the diameter of the cylinder. Hence, however small we make the cavity, the force arising from the surface distribution on its walls will remain, in general, finite.

397.] We have hitherto supposed the magnetization to be uniform and in the same direction throughout the whole of the portion of the magnet from which the cylinder is hollowed out. Wlien the magnetization is not thus restricted, there will in general be a distribution of imaginary magnetic matter through the substance of the magnet. The cutting out of the cylinder will remove part of this distribution, but since in similar solid figures the forces at corresponding points are proportional to the linear dimensions of the figures, the alteration of the force on the magnetic pole due to the volume-density of magnetic matter will diminish indefinitely as the size of the cavity is diminished, while the effect due to the surface-density on the walls of the cavity remains, in general, finite.

If, therefore, we assume the dimensions of the cylinder so small

399-1 MAGNETIC FORCE IN A CAVITY. 23

that the magnetization of the part removed may be regarded as everywhere parallel to the axis of the cylinder, and of constant magnitude I, the force on a magnetic pole placed at the middle point of the axis of the cylindrical hollow will be compounded of two forces. The first of these is that due to the distribution of magnetic matter on the outer surface of the magnet, and throughout its interior, exclusive of the portion hollowed out. The components of this force are a, /3 and y, derived from the potential by equations (1). The second is the force 72, acting along the axis of the cylinder in the direction of magnetization. The value of this force depends on the ratio of the length to the diameter of the cylindric cavity.

398.] Case I. Let this ratio be very great, or let the diameter of the cylinder be small compared with its length. Expanding the

expression for R in terms of j- , it becomes

a quantity which vanishes when the ratio of b to a is made infinite. Hence, when the cavity is a very narrow cylinder with its axis parallel to the direction of magnetization, the magnetic force within the cavity is not affected by the surface distribution on the ends of the cylinder, and the components of this force are simply a, /3, y, where

dV dV dV ,,.

a = -- 7-, 0 = — -=-, y= — -—. (4)

dx dy dz

We shall define the force within a cavity of this form as the magnetic force within the magnet. Sir William Thomson has called this the Polar definition of magnetic force. When we have occasion to consider this force as a vector we shall denote it

*>7$.

399.] Case II. Let the length of the cylinder be very small

compared with its diameter, so that the cylinder becomes a thin disk. Expanding the expression for R in terms of - , it becomes

_ £+££-*..}, (5)

a 2 #3 3

the ultimate value of which, when the ratio of a to b is made infinite, is 4 TT J.

Hence, when the cavity is in the form of a thin disk, whose plane is normal to the direction of magnetization, a unit magnetic pole

24 MAGNETIC FORCE AND MAGNETIC INDUCTION. [400.

placed at the middle of the axis experiences a force 4 IT I in the direction of magnetization arising from the superficial magnetism on the circular surfaces of the disk *.

Since the components of J are A, B and (7, the components of this force are 4 -n A, 4 TT B and 4 TT C. This must be compounded with the force whose components are a, {3, y.

400.] Let the actual force on the unit pole be denoted by the vector 35, and its components by a, b and c, then a = a + 4 TT A,

0=/3 + 47T.£, (6)

C = y -f 4 TT C.

We shall define the force within a hollow disk, whose plane sides are normal to the direction of magnetization, as the Magnetic Induction within the magnet. Sir William Thomson has called this the Electromagnetic definition of magnetic force.

The three vectors, the magnetization 3, the magnetic force <!fj, and the magnetic induction S3 are connected by the vector equation

47:3. (7)

Line-Integral of Magnetic Force.

401.] Since the magnetic force, as denned in Art. 398, is that due to the distribution of free magnetism on the surface and through the interior of the magnet, and is not affected by the surface- magnetism of the cavity, it may be derived directly from the general expression for the potential of the magnet, and the line- integral of the magnetic force taken along any curve from the point A to the point B is

where VA and V^ denote the potentials at A and B respectively.

• On the force within cavities of other forms.

1. Any narrow crevasse. The force arising from the surface-magnetism is 47r/cos€ in the direction of the normal to the plane of the crevasse, where 6 is the angle between this normal and the direction of magnetization. When the crevasse is parallel to the direction of magnetization the force is the magnetic force £ ; when the crevasse is perpendicular to the direction of magnetization the force is the magnetic induction 93.

2. In an elongated cylinder, the axis of which makes an angle « with the direction of magnetization, the force arising from the surface-magnetism is 27r/sin e, perpendicular to the axis in the plane containing the axis and the direction of magnetization.

3. In a sphere the force arising from surface-magnetism is f IT I in the direction of magnetization.

402.] SURF ACE -INTEGRAL. 25

Surface-Integral of Magnetic Induction.

402.] The magnetic induction through the surface 8 is defined as the value of the integral

Q = ff%cos€dS, (9)

where 23 denotes the magnitude of the magnetic induction at the element of surface clS, and e the angle between the direction of the induction and the normal to the element of surface, and the integration is to be extended over the whole surface, which may be either closed or bounded by a closed curve.

If a, b, c denote the components of the magnetic induction, and /, m, n the direction-cosines of the normal, the surface-integral may be written

q = jj(la+mb+nG)d8. (10)

If we substitute for the components of the magnetic induction their values in terms of those of the magnetic force, and the magnetization as given in Art. 400, we find

Q = n(la + mp + ny)dS + 4 TT (lA + m£ + nC)dS. (11)

We shall now suppose that the surface over which the integration extends is a closed one, and we shall investigate the value of the two terms on the right-hand side of this equation.

Since the mathematical form of the relation between magnetic force and free magnetism is the same as that between electric force and free electricity, we may apply the result given in Art. 77 to the first term in the value of Q by substituting a, ft, y, the components of magnetic force, for X, Y, Z, the components of electric force in Art. 77, and M, the algebraic sum of the free magnetism within the closed surface, for e, the algebraic sum of the free electricity.

We thus obtain the equation

ny)48*x 4irM. (12)

Since every magnetic particle has two poles, which are equal in numerical magnitude but of opposite signs, the algebraic sum of the magnetism of the particle is zero. Hence, those particles which are entirely within the closed surface S can contribute nothing to the algebraic sum of the magnetism within S. The

26 MAGNETIC FORCE AND MAGNETIC INDUCTION. [403.

value of M must therefore depend only on those magnetic particles which are cut by the surface S.

Consider a small element of the magnet of length s and trans verse section kz, magnetized in the direction of its length, so that the strength of its poles is m. The moment of this small magnet will be ms, and the intensity of its magnetization, being the ratio of the magnetic moment to the volume, will be

/=£• (13)

Let this small magnet be cut by the surface S, so that the direction of magnetization makes an angle e' with the normal drawn outwards from the surface, then if dS denotes the area of the section, p = ds cos e/t ( 1 4)

The negative pole — m of this magnet lies within the surface S.

Hence, if we denote by dM the part of the free magnetism within S whic*h is contributed by this little magnet,

IS. (15)

To find M, the algebraic sum of the free magnetism within the closed surface S, we must integrate this expression over the closed

surface, so that

M=-

or writing A, .Z?, C for the components of magnetization, and I, m, n for the direction-cosines of the normal drawn outwards,

(16)

This gives us the value of the integral in the second term of equation (11). The value of Q in that equation may therefore be found in terms of equations (12) and (16),

Q = 47r3/-47rl/= 0, (17)

or, the surface-integral of the magnetic induction through any closed surface is zero.

403.] If we assume as the closed surface that of the differential element of volume dx dy dz, we obtain the equation

*! + + = 0. (18)

dx dy dz

This is the solenoidal condition which is always satisfied by the components of the magnetic induction.

405.] LINES OF MAGNETIC INDUCTION. 27

Since the distribution of magnetic induction is solenoidal, the induction through any surface bounded by a closed curve depends only on the form and position of the closed curve, and not on that of the surface itself.

404.] Surfaces at every point of which

la + mb + nc = 0 (19)

are called Surfaces of no induction, and the intersection of two such surfaces is called a Line of induction. The conditions that a curve, Sj may be a line of induction are

1 dx 1 dy \ dz , .

= 'L = . (20)

a ds I ds c ds

A system of lines of induction drawn through every point of a closed curve forms a tubular surface called a Tube of induction.

The induction across any section of such a tube is the same. If the induction is unity the tube is called a Unit tube of in duction.

All that Faraday * says about lines of magnetic force and mag netic sphondyloids is mathematically true, if understood of the lines and tubes of magnetic induction.

The magnetic force and the magnetic induction are identical outside the magnet, but within the substance of the magnet they must be carefully distinguished. In a straight uniformly mag netized bar the magnetic force due to the magnet itself is from the end which points north, which we call the positive pole, towards the south end or negative pole, both within the magnet and in the space without.

The magnetic induction, on the other hand, is from the positive pole to the negative outside the magnet, and from the negative pole to the positive within the magnet, so that the lines and tubes of induction are re-entering or cyclic figures.

The importance of the magnetic induction as a physical quantity will be more clearly seen when we study electromagnetic phe nomena. When the magnetic field is explored by a moving wire, as in Faraday's Exp. Res. 3076, it is the magnetic induction and not the magnetic force which is directly measured.

The Vector-Potential of Magnetic Induction.

405.] Since, as we have shewn in Art. 403, the magnetic in duction through a surface bounded by a closed curve depends on

• Exp. Res., series xxviii.

28 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.

the closed curve, and not on the form of the surface which is bounded by it, it must be possible to determine the induction through a closed curve by a process depending only on the nature of that curve, and not involving the construction of a surface forming a diaphragm of the curve.

This may be done by finding a vector 21 related to 33, the magnetic induction, in such a way that the line-integral of SI, extended round the closed curve, is equal to the surface-integral of 33, extended over a surface bounded by the closed curve.

If, in Art. 24, we write F9 G, H for the components of SI, and a, b, c for the components of 33, we find for the relation between these components

dH dG dF dH dG dF

a= —

.j 7

dz dz ax ax ay

The vector SI, whose components are F, G, //, is called the vector- potential of magnetic induction. The vector-potential at a given point, due to a magnetized particle placed at the origin, is nume rically equal to the magnetic moment of the particle divided by the square of the radius vector and multiplied by the sine of the angle between the axis of magnetization and the radius vector, and the direction of the vector-potential is perpendicular to the plane of the axis of magnetization and the radius vector, and is such that to an eye looking in the positive direction along the axis of magnetization the vector-potential is drawn in the direction of rotation of the hands of a watch.

Hence, for a magnet of any form in which A^ B, C are the components of magnetization at the point xyz, the components of the vector-potential at the point f 77 £ are

(22)

where p is put, for conciseness, for the reciprocal of the distance between the points (f, 77, Q and (#, y, z), and the integrations are extended over the space occupied by the magnet.

406.] The scalar, or ordinary, potential of magnetic force, Art. 386, becomes when expressed in the same notation,

406.] VECTOR- POTENTIAL. 29

/v /y\ t-j /v\

Kemembering that ~ = — -~, and that the integral dx u/ £

has the value — 4 TT ( A) when the point (£, 77, f) is included within the limits of integration, and is zero when it is not so included, (A) being the value of A at the point (f, 77, (*), we find for the value of the ^-component of the magnetic induction,

dH _ dG_ dr] d£

f d^p dzp \ d'*p d2j) }

\dydr) dzdC' dx dr] dxd^S

7> r, ^ 7 7

-ri - ~ + B -/- -f- (7 7 \dxdydz d£jJJ ( dx dy d

The first term of this expression is evidently -- ^ , or a, the component of the magnetic force.

The quantity under the integral sign in the second term is zero for every element of volume except that in which the point (f, ry, £) is included. If the value of A at the point (f, r/, f) is (A), the value of the second term is 4 TT (A)9 where (A) is evidently zero at all points outside the magnet.

We may now write the value of the ^-component of the magnetic induction « = o+4w(^), (25)

an equation which is identical with the first of those given in Art. 400. The equations for b and c will also agree with those of Art. 400.

We have already seen that the magnetic force § is derived from the scalar magnetic potential V by the application of Hamilton's operator y , so that we may write, as in Art. 1 7,

£=-vF, (26)

and that this equation is true both without and within the magnet.

It appears from the present investigation that the magnetic induction S3 is derived from the vector-potential SI by the appli cation of the same operator, and that the result is true within the magnet as well as without it.

The application of this operator to a vector-function produces,

30 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.

in general, a scalar quantity as well as a vector. The scalar part, however, which we have called the convergence of the vector- function, vanishes when the vector-function satisfies the solenoidal

condition

dF dG dH

•Jl + -J~ + -7TF = °* df; dr] d£

By differentiating the expressions for F, G, If in equations (22), we find that this equation is satisfied by these quantities.

We may therefore write the relation between the magnetic induction and its vector-potential

23 = V %

which may be expressed in words by saying that the magnetic induction is the curl of its vector-potential. See Art. 25.

CHAPTER III

MAGNETIC SOLENOIDS AND SHELLS*.

On Particular Forms of Magnets.

407.] IF a long narrow filament of magnetic matter like a wire is magnetized everywhere in a longitudinal direction, then the product of any transverse section of the filament into the mean intensity of the magnetization across it is called the strength of the magnet at that section. If the filament were cut in two at the section without altering the magnetization, the two surfaces, when separated, would be found to have equal and opposite quan tities of superficial magnetization, each of which is numerically equal to the strength of the magnet at the section.

A filament of magnetic matter, so magnetized that its strength is the same at every section, at whatever part of its length the section be made, is called a Magnetic Solenoid.

If m is the strength of the solenoid, ds an element of its length, r the distance of that element from a given point, and e the angle which r makes with the axis of magnetization of the element, the potential at the given point due to the element is

m ds cos € m dr ..

— o = —s- r ds.

r2 r* ds

Integrating this expression with respect to s} so as to take into account all the elements of the solenoid, the potential is found

to be ,11^ V = m ( ) >

rl r2

T! being the distance of the positive end of the solenoid, and r^ that of the negative end from the point where V exists.

Hence the potential due to a solenoid, and consequently all its magnetic effects, depend only on its strength and the position of

• See Sir W. Thomson's 'Mathematical Theory of Magnetism,' Phil. Trans., 1850, or Reprint.

32 MAGNETIC SOLENOIDS AND SHELLS. [408.

its ends, and not at all on its form, whether straight or curved, between these points.

Hence the ends of a solenoid may be called in a strict sense its poles.

If a solenoid forms a closed curve the potential due to it is zero at every point, so that such a solenoid can exert no magnetic action, nor can its magnetization be discovered without breaking it at some point and separating the ends.

If a magnet can be divided into solenoids, all of which either form closed curves or have their extremities in the outer surface of the magnet, the magnetization is said to be solenoidal, and, since the action of the magnet depends entirely upon that of the ends of the solenoids, the distribution of imaginary magnetic matter will be entirely superficial.

Hence the condition of the magnetization being solenoidal is dA dB dC _ dx dy dz

where A, B, C are the components of the magnetization at any point of the magnet.

408.] A longitudinally magnetized filament, of which the strength varies at different parts of its length, may be conceived to be made up of a bundle of solenoids of different lengths, the sum of the strengths of all the solenoids which pass through a given section being the magnetic strength of the filament at that section. Hence any longitudinally magnetized filament may be called a Complex Solenoid.

If the strength of a complex solenoid at any section is m, then the potential due to its action is

ds where m is variable,

Cm dr

f% -

m\ mi /I

fll* 4* i 4*

/I /*> J I

l dm 7

ds

This shews that besides the action of the two ends, which may in this case be of different strengths, there is an action due to the distribution of imaginary magnetic matter along the filament with a linear density dm

/V. — - •"— j *

ds

Magnetic Shells. 409.] If a thin shell of magnetic matter is magnetized in a

SHELLS. 33

direction everywhere normal to its surface, the intensity of the magnetization at any place multiplied by the thickness of the sheet at that place is called the Strength of the magnetic shell at that place.

If the strength of a shell is everywhere equal, it is called a Simple magnetic shell; if it varies from point to point it may be conceived to be made up of a number of simple shells superposed and overlapping each other. It is therefore called a Complex magnetic shell.

Let dS be an element of the surface of the shell at Q, and 4> the strength of the shell, then the potential at any point, P, due to the element of the shell, is

d V = <J> — - dS cos €* r2

where e is the angle between the vector QP, or r and the normal drawn from the positive side of the shell.

But if du> is the solid angle subtended by dS at the point P

r2 da — dS cos e,

whence dF = <&da>,

and therefore in the case of a simple magnetic shell

or, the potential due to a magnetic shell at any point is the product of its strength into the solid angle subtended by its edge at the given point*.

410.] The same result may be obtained in a different way by supposing the magnetic shell placed in any field of magnetic force, and determining the potential energy due to the position of the shell.

If V is the potential at the element dS, then the energy due to this element is dy dy dy

• (^ -r- +m~j- + n ~r) <*** \ da dy dz'

or, the product of the strength of the shell into the part of the surface-integral of V due to the element dS of the shell.

Hence, integrating with respect to all such elements, the energy due to the position of the shell in the field is equal to the product of the strength of the shell and the surf ace -integral of the magnetic induction taken over the surface of the shell.

Since this surface-integral is the same for any two surfaces which