A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 4 of 27

• This theorem is due to Gauss, General Theory of Terrestrial Magnetism, § 38. VOL. II. D

34 MAGNETIC SOLENOIDS AND SHELLS. [411-

have the same bounding- edge and do not include between them any centre of force, the action of the magnetic shell depends only on the form of its edge.

Now suppose the field of force to be that due to a magnetic pole of strength m. We have seen (Art. 76, Cor.) that the surface- integral over a surface bounded by a given edge is the product of the strength of the pole and the solid angle subtended by the edge at the pole. Hence the energy due to the mutual action of the pole and the shell is

and this (by Green's theorem. Art. 100) is equal to the product of the strength of the pole into the potential due to the shell at the pole. The potential due to the shell is therefore 4> co.

411.] If a magnetic pole m starts from a point on the negative surface of a magnetic shell, and travels along any path in space so as to come round the edge to a point close to where it started but on the positive side of the shell, the solid angle will vary continuously, and will increase by 4 TT during the process. The work done by the pole will be 4 TT 4> m, and the potential at any point on the positive side of the shell will exceed that at the neighbouring point on the negative side by 4 TT 4>.

If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4 TT 4>, being positive when the positive side of the shell is inward. Hence such a shell exerts no action on any magnet placed either outside or inside the shell.

412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, the distribution of magnetism is called Lamellar. If <£ is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point xy z by a line drawn within the magnet, then the conditions of lamellar magnetization are

,_<Z<I> d<}> d(f>

A = — =— , JD = -r— , L> = — T~ *

dx dy dz

The quantity, <J>, which thus completely determines the magnet ization at any point may be called the Potential of Magnetization. It must be carefully distinguished from the Magnetic Potential.

413.] A magnet which can be divided into complex magnetic shells is said to have a complex lamellar distribution of mag netism. The condition of such a distribution is that the lines of

415.] POTENTIAL DUE TO A LAMELLAE MAGNET. 35

magnetization must be such that a system of surfaces can be drawn cutting them at right angles. This condition is expressed by the well-known equation

Aff__<lB} ^A_<IC ^_<U ^dy dz> ^dz dx' ^dx dy '

Forms of the Potentials of Solenoidal and Lamellar Magnets. 414.] The general expression for the scalar potential of a magnet

where p denotes the potential at (#, y, z) due to a unit magnetic pole placed at f, TJ, £ or in other words, the reciprocal of the distance between (f, r;, Q, the point at which the potential is measured, and (#, y> z), the position of the element of the magnet to which it is due.

This quantity may be integrated by parts, as in Arts. 96, 386.

where I, m, n are the direction-cosines of the normal drawn out wards from dS, an element of the surface of the magnet.

When the magnet is solenoidal the expression under the integral sign in the second term is zero for every point within the magnet, so that the triple integral is zero, and the scalar potential at any point, whether outside or inside the magnet, is given by the surface- integral in the first term.

The scalar potential of a solenoidal magnet is therefore com pletely determined when the normal component of the magnet ization at every point of the surface is known, and it is independent of the form of the solenoids within the magnet.

415.] In the case of a lamellar magnet the magnetization is determined by c/>, the potential of magnetization, so that dcf) d<j> d$

•**• -— ~^ — j .£> = —7— , <-/ = — ; — •

ax ay dz

The expression for V may therefore be written

= fff, JJJ \

dp .

'

dx dx dy dy dz dz Integrating this expression by parts, we find

D 2

36 MAGNETIC SOLENOIDS AND SHELLS.

The second term is zero unless the point (f, r/, f) is included in the magnet, in which case it becomes 4 TT (<£) where (<£) is the value of <p at the point £, 77, f The surface-integral may be expressed in terms of rt the line drawn from (x, y, z] to (f, rj, f ), and 0 the angle which this line makes with the normal drawn outwards from dSt so that the potential may be written

where the second term is of course zero when the point (f, TJ, f) is not included in the substance of the magnet.

The potential, F, expressed by this equation, is continuous even at the surface of the magnet, where $ becomes suddenly zero, for if we write

fit =

and if £1L is the value of H at a point just within the surface, and 122 that at a point close to the first but outside the surface,

fla = ^ + 477^),

r2 = r,.

The quantity H is not continuous at the surface of the magnet.

The components of magnetic induction are related to 12 by the equations

d& da da

a= -- =— , 0= -- =-, c — -- -j- •

dx dy dz

416.] In the case of a lamellar distribution of magnetism we may also simplify the vector-potential of magnetic induction. Its ^-component may be written

By integration by parts we may put this in the form of the surface-integral

or F .

The other components of the vector-potential may be written down from these expressions by making the proper substitutions.

On Solid Angles. 417.] We have already proved that at any point P the potential

4 1 8.] SOLID ANGLES. 37

due to a magnetic shell is equal to the solid angle subtended by the edge of the shell multiplied by the strength of the shell. As we shall have occasion to refer to solid angles in the theory of electric currents, we shall now explain how they may be measured.

Definition. The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. This area is to be reckoned positive or negative according as it lies on the left or the right- hand of the path of the radius vector as seen from the given point.

Let (£, r], f) be the given point, and let (#, y, z) be a point on the closed curve. The coordinates- x, y, z are functions of s, the length of the curve reckoned from a given point. They are periodic functions of s, recurring whenever s is increased by the whole length of the closed curve.

We may calculate the solid angle o> directly from the definition thus. Using spherical coordinates with centre at (£, 77, Q, and putting

x — f = r sin0cos$, y— rj = r sin 0 sin^, z — C=rcos0, we find the area of any curve on the sphere by integrating

co = /(I— cos0) d$, or, using the rectangular coordinates,

the integration being extended round the curve s.

If the axis of z passes once through the closed curve the first term is 2 IT. If the axis of z does not pass through it this term is zero.

418.] This method of calculating a solid angle involves a choice of axes which is to some extent arbitrary, and it does not depend solely on the closed curve. Hence the following method, in which no surface is supposed to be constructed, may be stated for the sake of geometrical propriety.

As the radius vector from the given point traces out the closed curve, let a plane passing through the given point roll on the closed curve so as to be a tangent plane at each point of the curve in succession. Let a line of unit-length be drawn from the given point perpendicular to this plane. As the plane rolls round the

38 MAGNETIC SOLENOIDS AND SHELLS. [4 1 9.

closed curve the extremity of the perpendicular will trace a second closed curve. Let the length of the second closed curve be o-, then the solid angle subtended by the first closed curve is

00 = 27T — (7.

This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circum ference of the polar curve, is numerically equal to the circumference of a great circle of the sphere.

This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem.

419.] A closed curve s is given in space, and we have to find the solid angle subtended by s at a given point P.

If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if cr is the path of the pole as it approaches the point P, the potential must be the result of a line-integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and a.

When P is at an infinite distance, the solid angle is evidently zero. As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point ap proaches.

As the point P moves from P to P' over the element do-, the element QQ' of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ' will sweep over an area on the spherical surface, which we may write

da = Udsdcr. (I)

To find FT let us suppose P fixed while the closed curve is moved parallel to itself through a distance da- equal to PPf but in the opposite direction. The relative motion of the point P will be the same as in the real case.

420.]

GENERATION OF A SOLID ANGLE.

39

During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to Q Q' and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment d& which we are in search of.

To determine the value of this solid angle, let 6 and tf be the angles which ds and dcr make with PQ respect ively, and let <£ be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds .dcr on a. plane per pendicular to PQ or r will be

ds dcr sin Q sin 6' sin and since this is equal to r2 d<a, we find

Fig. 3.

Hence

du> = II ds dcr = -g sin Q sin 6' sin </> ds dcr. n = — - sin 6 sin 0' sin <>.

(2) (3)

420.] We may express the angles 6, 6', and $ in terms of and its differential coefficients with respect to s and o-, for

cos0= -=-,

/»/

cos<9'= •-=-, dcr

and sin 6 sin 6' cos cp = r

dsdcr

(4)

We thus find the following value for D2,

(5)

A third expression for II in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is d& and whose axis is r is J r* do) = J r* FT ds dcr.

But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and dcr on the axis of #, y and zt as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of n,

n = -^

-=— > -^— > -=—

c— *i

T\—y>

<* -.

l— *>

-7— > dcr

drj

-j— > dcr

T«'

dx Ts*

d_y_

7 ^

ds

dz ~ds"

(6)

40 MAGNETIC SOLENOIDS AND SHELLS. [421.

This expression gives the value of FT free from the ambiguity of sign introduced by equation (5).

421.] The value of o>, the solid angle subtended by the closed curve at the point P, may now be written

a) = ndsdv-i-WQ, (7)

where the integration with respect to s is to be extended completely round the closed curve, and that with respect to <r from A a fixed point on the curve to the point P. The constant <o0 is the value of the solid angle at the point A. It is zero if A is at an infinite distance from the closed curve.

The value of o> at any point P is independent of the form of the curve between A and P provided that it does not pass through the magnetic shell itself. If the shell be supposed infinitely thin, and if P and Pf are two points close together, but P on the positive and P' on the negative surface of the shell, then the curves AP and AP/ must lie on opposite sides of the edge of the shell, so that PAP' is a line which with the infinitely short line PP forms a closed circuit embracing the edge. The value of o> at P exceeds that at P' by 47T, that is, by the surface of a sphere of radius unity.

Hence, if a closed curve be drawn so as to pass once through the shell, or in other words, if it be linked once with the edge

of the shell, the value of the integral I lUdsdv extended round

both curves will be 47r.

This integral therefore, considered as depending only on the closed curve s and the arbitrary curve AP, is an instance of a _ function of multiple values, since, if we pass from A to P along different paths the integral will have different values according to the number of times which the curve AP is twined round the curve s.

If one form of the curve between A and P can be transformed into another by continuous motion without intersecting the curve s, the integral will have the same value for both curves, but if during the transformation it intersects the closed curve n times the values of the integral will differ by 47m.

If s and a- are any two closed curves in space, then, if they are not linked together, the integral extended once round both is zero.

If they are intertwined n times in the same direction, the value of the integral is 4iTn. It is possible, however, for two curves

422.] VECTOR- POTENTIAL OF A CLOSED CURVE. 41

to be intertwined alternately in opposite directions, so that they are inseparably linked together though the value of the integral is zero. See Fig. 4.

It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while de scribing a closed curve in presence of a closed electric current, and indicating the geometrical connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibnitz, Euler and Vandermonde. We have now, how- Flg> 4>

ever, some progress to report, chiefly due to Riemann, Helmholtz and Listing.

422.] Let us now investigate the result of integrating with respect to s round the closed curve.

One of the terms of FT in equation (7) is

f — x dri dz _ di) d A dz^ , .

r3 da- ds ~~ da d£ W ds' If we now write for brevity

^ f 1 dx 7 „ f 1 dy .. TT f 1 dz F — I - -r- ds, G = I - -f- ds, R—- ~ ds, (9)

J r ds J r ds J r ds

the integrals being taken once round the closed curve s, this term of FT may be written

da- d£ds and the corresponding term of / n ds will be

da- d£ Collecting all the terms of n, we may now write

This quantity is evidently the rate of decrement of co, the magnetic potential, in passing along the curve a-, or in other words, it is the magnetic force in the direction of da:

By assuming da- successively in the direction of the axes of x, y and z, we obtain for the values of the components of the magnetic force

42 MAGNETIC SOLENOIDS AND SHELLS. [423-

do> _ dH dG

Ot — ~~~ 7 f. — ~~j " T"T

dt, d-r] d£

d<* _ dF dH

dr] d£ d£

do> _ dG dF

y =~ JT> — ,7 /• ~j

(11)

The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential o>, functions having a series of values, but are perfectly determinate for every point in space.

The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction :

Let a point Q travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When Q has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P.

Potential Energy of a Magnetic Shell placed in a Magnetic Field.

423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength <£ placed in a magnetic field whose potential is T9 is

rffidV d7 dY\ 70

x-tJJ ('is +?+•)** ^

where I, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell.

Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we

-+cf+^,

where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell.

If we now suppose that the magnetic field is that due to a

423.] POTENTIAL OF TWO CLOSED CURVES. 43

second magnetic shell whose strength is <£', the values of F, G, H will be

where the integrations are extended once round the curve /, which forms the edge of this shell.

Substituting these values in the expression for M we find

, ff I fdx dx dy dy dz dz^ .

Jf = —$$'// - (-J- -j-' + ir j' + -j--,,)dsds', (15) ^ JJ r ^ds ds ds ds ds ds'

where the integration is extended once round s and once round /. This expression gives the potential energy due to the mutual action of the two shells, and is, as it ought to be, the same when s and / are interchanged. This expression with its sign reversed, when the strength of each shell is unity, is called the potential of the two closed curves s and /. It is a quantity of great importance in the theory of electric currents. If we write e for the angle between the directions of the elements ds and ds', the potential of s and / may be written

(16)

It is evidently a quantity of the dimension of a line.

CHAPTER IV.

INDUCED MAGNETIZATION.

424.] WE have hitherto considered the actual distribution of magnetization in a magnet as given explicitly among the data of the investigation. We have not made any assumption as to whether this magnetization is permanent or temporary, except in those parts of our reasoning in which we have supposed the magnet broken up into small portions, or small portions removed from the magnet in such a way as not to alter the magnetization of any part.

We have now to consider the magnetization of bodies with respect to the mode in which it may be produced and changed. A bar of iron held parallel to the direction of the earth's magnetic force is found to become magnetic, with its poles turned the op posite way from those of the earth, or the same way as those of a compass needle in stable equilibrium.

Any piece of soft iron placed in a magnetic field is found to exhibit magnetic properties. If it be placed in a part of the field where the magnetic force is great, as between the poles of a horse-shoe magnet, the magnetism of the iron becomes intense. If the iron is removed from the magnetic field, its magnetic properties are greatly weakened or disappear entirely. If the magnetic properties of the iron depend entirely on the magnetic force of the field in which it is placed, and vanish when it is removed from the field, it is called Soft iron. Iron which is soft in the magnetic sense is also soft in the literal sense. It is easy to bend it and give it a permanent set, and difficult to break it.

Iron which retains its magnetic properties when removed from the magnetic field is called Hard iron. Such iron does not take

425.] SOFT AND HARD STEEL. 45

up the magnetic state so readily as soft iron. The operation of hammering-, or any other kind of vibration, allows hard iron under the influence of magnetic force to assume the magnetic state more readily, and to part with it more readily when the magnetizing force is removed. Iron which is magnetically hard is also more stiff to bend and more apt to break.

The processes of hammering, rolling, wire-drawing, and sudden cooling tend to harden iron, and that of annealing tends to soften it.

The magnetic as well as the mechanical differences between steel of hard and soft temper are much greater than those between hard and soft iron. Soft steel is almost as easily magnetized and de magnetized as iron, while the hardest steel is the best material for magnets which we wish to be permanent.

Cast iron, though it contains more carbon than steel, is not so retentive of magnetization.

If a magnet could be constructed so that the distribution of its magnetization is not altered by any magnetic force brought to act upon it, it might be called a rigidly magnetized body. The only known body which fulfils this condition is a conducting circuit round which a constant electric current is made to flow.

Such a circuit exhibits magnetic properties, and may therefore be called an electromagnet, but these magnetic properties are not affected by the other magnetic forces in the field. We shall return to this subject in Part IV.

All actual magnets, whether made of hardened steel or of load stone, are found to be affected by any magnetic force which is brought to bear upon them.

It is convenient, for scientific purposes, to make a distinction between the permanent and the temporary magnetization, defining the permanent magnetization as that which exists independently of the magnetic force, and the temporary magnetization as that which depends on this force. We must observe, however, that this distinction is not founded on a knowledge of the intimate nature of magnetizable substances : it is only the expression of an hypothesis introduced for the sake of bringing calculation to bear on the phenomena. We shall return to the physical theory of magnetization in Chapter VI.

425.] At present we shall investigate the temporary magnet ization on the assumption that the magnetization of any particle of the substance depends solely on the magnetic force acting on

46 INDUCED MAGNETIZATION. [425.

that particle. This magnetic force may arise partly from external causes, and partly from the temporary magnetization of neigh bouring particles.

A body thus magnetized in virtue of the action of magnetic force, is said to be magnetized by induction, and the magnetization is said to be induced by the magnetizing force.

The magnetization induced by a given magnetizing force differs in different substances. It is greatest in the purest and softest iron, in which the ratio of the magnetization to the magnetic force may reach the value 32, or even 45 *.

Other substances, such as the metals nickel and cobalt, are capable of an inferior degree of magnetization, and all substances when subjected to a sufficiently strong magnetic force, are found to give indications of polarity.

When the magnetization is in the same direction as the magnetic force, as in iron, nickel, cobalt, &c., the substance is called Para magnetic, Ferromagnetic, or more simply Magnetic. When the induced magnetization is in the direction opposite to the magnetic force, as in bismuth, &c., the substance is said to be Diamagnetic.

In all these substances the ratio of the magnetization to the magnetic force which produces it is exceedingly small, being only about — 4 o (H) o Q m the case °f bismuth, which is the most highly diamagnetic substance known.

In crystallized, strained, and organized substances the direction of the magnetization does not always coincide with that of the magnetic force which produces it. The relation between the com ponents of magnetization, referred to axes fixed in the body, and those of the magnetic force, may be expressed by a system of three linear equations. Of the nine coefficients involved in these equa tions we shall shew that only six are independent. The phenomena of bodies of this kind are classed under the name of Magnecrystallic phenomena.

When placed in a field of magnetic force, crystals tend to set themselves so that the axis of greatest paramagnetic, or of least diamagnetic, induction is parallel to the lines of magnetic force. See Art. 435.

In soft iron, the direction of the magnetization coincides with that of the magnetic force at the point, and for small values of the magnetic force the magnetization is nearly proportional to it.

• Thaten, Nova Ada, Reg. Soc. Sc., Upsal., 1863.

427.] PROBLEM OF INDUCED MAGNETIZATION. 47

As the magnetic force increases, however, the magnetization in creases more slowly, and it would appear from experiments described in Chap. VI, that there is a limiting value of the magnetization, beyond which it cannot pass, whatever be the value of the magnetic force.

In the following outline of the theory of induced magnetism, we shall begin by supposing the magnetization proportional to the magnetic force, and in the same line with it.

Definition of the Coefficient of Induced Magnetization.

426.] Let $ be the magnetic force, defined as in Art. 398, at any point of the body, and let 3 be the magnetization at that point, then the ratio of 3 to § is called the Coefficient of Induced Magnetization.

Denoting this coefficient by K, the fundamental equation of induced magnetism is

The coefficient K is positive for iron and paramagnetic substances, and negative for bismuth and diamagnetic substances. It reaches the value 32 in iron, and it is said to be large in the case of nickel and cobalt, but in all other cases it is a very small quantity, not greater than 0.00001.

The force <£) arises partly from the action of magnets external to the body magnetized by induction, and partly from the induced magnetization of the body itself, Both parts satisfy the condition of having a potential.

427.] Let V be the potential due to magnetism external to the body, let X2 be that due to the induced magnetization, then if U is the actual potential due to both causes

u= r+a. (2)

Let the components of the magnetic force «£), resolved in the directions of x, y, z, be a, /3, y, and let those of the magnetization 3 be A, B, C, then by equation (1),

A = K a,

*=K/3, (3)

C = Ky.

Multiplying these equations by dx, dy, dz respectively, and adding, we find

Adx + Bdy+Cdz = K(

48 INDUCED MAGNETIZATION. [427.

But since a, (3 and y are derived from the potential U, we may write the second member —KdU.

Hence, if /c is constant throughout the substance, the first member must also be a complete differential of a function of #, y and z, which we shall call $, and the equation becomes

i A d(b „ d(b d(b

where A = -f- , B = ~- , C — — - . (5)

ax dy dz

The magnetization is therefore lamellar, as defined in Art. 412.

It was shewn in Art. 386 that if p is the volume-density of free magnetism,

(dA dB dC.

P- — (-J- +-J- + T-}' x## dy dz '

which becomes in virtue of equations (3),

/da d(3 dy\

\lx dy dz' But, by Art. 77,

da dj3 dy _

dx dy dz ~

Hence (l+47r*)p = 0,

whence p = 0 (6)

throughout the substance, and the magnetization is therefore sole- noidal as well as lamellar. See Art. 407.

There is therefore no free magnetism except on the bounding surface of the body. If v be the normal drawn inwards from the surface, the magnetic surface-density is

d^> (-^

a- = j-- (7)

dv

The potential II due to this magnetization at any point may therefore be found from the surface-integral

«-//=

dS. (8)

The value of £1 will be finite and continuous everywhere, and will satisfy Laplace's equation at every point both within and without the surface. If we distinguish by an accent the value of H outside the surface, and if v be the normal drawn outwards, we have at the surface

Of =0.1 (9)

428.] POISSON'S METHOD. 49

da da'

— + ^ = -4™, by Art. 78,

= 4*8.^). , ., -..: : .

dU = -47rKj;> bF(4)»

fdV d^ , = -47rK(^+^),by(2).

We may therefore write the surface-condition

Hence the determination of the magnetism induced in a homo geneous isotropic body, bounded by a surface S, and acted upon by external magnetic forces whose potential is V9 may be reduced to the following mathematical problem.

We must find two functions H and H' satisfying the following conditions :

Within the surface S9 XI must be finite and continuous, and must satisfy Laplace's equation.

Outside the surface S, Of must be finite and continuous, it must vanish at an infinite distance, and must satisfy Laplace's equation.

At every point of the surface itself, H = Of, and the derivatives of H, Of and V with respect to the normal must satisfy equation (10). _

This method of treating the problem of induced magnetism is due to Poisson. The quantity k which he uses in his memoirs is not the same as *, but is related to it as follows :

47TK(£-l)+3/&= 0. (11)

The coefficient K which we have here used was introduced by J. Neumann.

428.] The problem of induced magnetism may be treated in a different manner by introducing the quantity which we have called, with Faraday, the Magnetic Induction.

The relation between 23, the magnetic induction, «£j, the magnetic force, and 3> the magnetization, is expressed by the equation

53 = $ + 471 3. (12)

The equation which expresses the induced magnetization in terms of the magnetic force is

3 = K$. (13)

VOL. IT. E

50 INDUCED MAGNETIZATION. [428.

Hence, eliminating- 3, we find

$ = (1+47TK)£ (14)

as the relation between the magnetic induction and the magnetic force in substances whose magnetization is induced by magnetic force.

In the most general case K may be a function, not only of the position of the point in the substance, but of the direction of the vector «jp, but in the case which we are now considering K is a numerical quantity.

If we next write ^ = I + 4 -n K} (15)

we may define /x as the ratio of the magnetic induction to the magnetic force, and we may call this ratio the magnetic inductive capacity of the substance, thus distinguishing it from K, the co efficient of induced magnetization.

If we write U for the total magnetic potential compounded of T7, the potential due to external causes, and 12 for that due to the induced magnetization, we may express a, b, c, the components of magnetic induction, and a, (3, y, the components of magnetic force, as follows : dU

~}

a = "0 = -M'

dU e = ™=-*&'j

The components #, d, c satisfy the solenoidal condition

£+!+£=«• (17>

Hence, the potential U must satisfy Laplace's equation

at every point where /ot is constant, that is, at every point within the homogeneous substance, or in empty space.

At the surface itself, if v is a normal drawn towards the magnetic substance, and v one drawn outwards, and if the symbols of quan tities outside the substance are distinguished by accents, the con dition of continuity of the magnetic induction is

dv , dv dv , dv ,, dv , dv a-j- +6-j- +0-=- +a'-j- +V -r- +<f -j- = 0; (19) dx dy dz dx dy dz

429.] FARADAY'S THEORY OF MAGNETIC INDUCTION. 51 or, by equations (16),

fjf, the coefficient of induction outside the magnet, will be unity unless the surrounding medium be magnetic or diamagnetic.

If we substitute for U its value in terms of V and H, and for fj> its value in terms of K, we obtain the same equation (10) as we arrived at by Poisson's method.

The problem of induced magnetism, when considered with respect to the relation between magnetic induction and magnetic force, corresponds exactly with the problem of the conduction of electric currents through heterogeneous media, as given in Art. 309.

The magnetic force is derived from the magnetic potential, pre cisely as the electric force is derived from the electric potential.

The magnetic induction is a quantity of the nature of a flux, and satisfies the same conditions of continuity as the electric current does.

In isotropic media the magnetic induction depends on the mag netic force in a manner which exactly corresponds with that in which the electric current depends on the electromotive force.

The specific magnetic inductive capacity in the one problem corre sponds to the specific conductivity in the other. Hence Thomson, in his Theory of Induced Magnetism (Reprint, 1872, p. 484), has called this quantity the permeability of the medium.

We are now prepared to consider the theory of induced magnetism from what I conceive to be Faraday's point of view.

When magnetic force acts on any medium, whether magnetic or diamagnetic, or neutral, it produces within it a phenomenon called Magnetic Induction.

Magnetic induction is a directed quantity of the nature of a flux, and it satisfies the same conditions of continuity as electric currents and other fluxes do.

In isotropic media the magnetic force and the magnetic induction are in the same direction, and the magnetic induction is the product of the magnetic force into a quantity called the coefficient of induction, which we have expressed by p.