A Treatise on Electricity and Magnetism/Part III/Chapter I

by James Clerk Maxwell

PART III.

MAGNETISM.

CHAPTER I.

ELEMENTARY THEORY OF MAGNETISM.

371.] Certain bodies, as, for instance, the iron ore called load stone, the earth itself, and pieces of steel which have been subjected to certain treatment, are found to possess the following properties, and are called Magnets.

If, near any part of the earth's surface except the Magnetic Poles, a magnet be suspended so as to turn freely about a vertical axis, it will in general tend to set itself in a certain azimuth, and if disturbed from this position it will oscillate about it. An unmagnetized body has no such tendency, but is in equilibrium in all azimuths alike.

372.] It is found that the force which acts on the body tends to cause a certain line in the body, called the Axis of the Magnet, to become parallel to a certain line in space, called the Direction of the Magnetic Force.

Let us suppose the magnet suspended so as to be free to turn in all directions about a fixed point. To eliminate the action of its weight we may suppose this point to be its centre of gravity. Let it come to a position of equilibrium. Mark two points on the magnet, and note their positions in space. Then let the magnet be placed in a new position of equilibrium, and note the positions in space of the two marked points on the magnet.

Since the axis of the magnet coincides with the direction of magnetic force in both positions, we have to find that line in the magnet which occupies the same position in space before and after the motion. It appears, from the theory of the motion of bodies of invariable form, that such a line always exists, and that a motion equivalent to the actual motion might have taken place by simple rotation round this line.

To find the line, join the first and last positions of each of the marked points, and draw planes bisecting these lines at right angles. The intersection of these planes will be the line required, which indicates the direction of the axis of the magnet and the direction of the magnetic force in space.

The method just described is not convenient for the practical determination of these directions. We shall return to this subject when we treat of Magnetic Measurements.

The direction of the magnetic force is found to be different at different parts of the earth's surface. If the end of the axis of the magnet which points in a northerly direction be marked, it has been found that the direction in which it sets itself in general deviates from the true meridian to a considerable extent, and that the marked end points on the whole downwards in the northern hemisphere and upwards in the southern.

The azimuth of the direction of the magnetic force, measured from the true north in a westerly direction, is called the Variation, or the Magnetic Declination. The angle between the direction of the magnetic force and the horizontal plane is called the Magnetic Dip. These two angles determine the direction of the magnetic force, and, when the magnetic intensity is also known, the magnetic force is completely determined. The determination of the values of these three elements at different parts of the earth's surface, the discussion of the manner in which they vary according to the place and time of observation, and the investigation of the causes of the magnetic force and its variations, constitute the science of Terrestrial Magnetism.

373.] Let us now suppose that the axes of several magnets have been determined, and the end of each which points north marked. Then, if one of these be freely suspended and another brought near it, it is found that two marked ends repel each other, that a marked and an unmarked end attract each other, and that two unmarked ends repel each other.

If the magnets are in the form of long rods or wires, uniformly and longitudinally magnetized, see below, Art. 384, it is found that the greatest manifestation of force occurs when the end of one magnet is held near the end of the other, and that the phenomena can be accounted for by supposing that like ends of the magnets repel each other, that unlike ends attract each other, and that the intermediate parts of the magnets have no sensible mutual action.

The ends of a long thin magnet are commonly called its Poles. In the case of an indefinitely thin magnet, uniformly magnetized throughout its length, the extremities act as centres of force, and the rest of the magnet appears devoid of magnetic action. In all actual magnets the magnetization deviates from uniformity, so that no single points can be taken as the poles. Coulomb, however, by using long thin rods magnetized with care, succeeded in establishing the law of force between two magnetic poles^[1].

The repulsion between two magnetic poles is in the straight line joining them, and is numerically equal to the product of the strengths of the poles divided by the square of the distance between them.

374.] This law, of course, assumes that the strength of each pole is measured in terms of a certain unit, the magnitude of which may be deduced from the terms of the law.

The unit-pole is a pole which points north, and is such that, when placed at unit distance from another unit-pole, it repels it with unit of force, the unit of force being defined as in Art. 6. A pole which points south is reckoned negative.

If $m_{1}$ and $m_{2}$ are the strengths of two magnetic poles, $l$ the distance between them, and $f$ the force of repulsion, all expressed numerically, then

$f={\frac {m_{1}m_{2}}{l^{2}}}$.

But if $[m]$, $[L]$ and $[F]$ be the concrete units of magnetic pole, length and force, then

$f[F]=\left[{\frac {m}{L}}\right]^{2}{\frac {m_{1}m_{2}}{l^{2}}}$

whence it follows that

$[m^{2}]=[L^{2}F]=\left[L^{2}{\frac {ML}{T^{2}}}\right]$,

or $[m]=[L^{\frac {3}{2}}T^{-1}M^{\frac {1}{2}}]$.

The dimensions of the unit pole are therefore 3/2 as regards length, (-1) as regards time, and 1/2 as regards mass. These dimensions are the same as those of the electrostatic unit of electricity, which is specified in exactly the same way in Arts. 41, 42.

375.] The accuracy of this law may be considered to have been established by the experiments of Coulomb with the Torsion Balance, and confirmed by the experiments of Gauss and Weber, and of all observers in magnetic observatories, who are every day making measurements of magnetic quantities, and who obtain results which would be inconsistent with each other if the law of force had been erroneously assumed. It derives additional support from its consistency with the laws of electromagnetic phenomena.

376.] The quantity which we have hitherto called the strength of a pole may also be called a quantity of Magnetism, provided we attribute no properties to Magnetism except those observed in the poles of magnets.

Since the expression of the law of force between given quantities of 'Magnetism' has exactly the same mathematical form as the law of force between quantities of 'Electricity' of equal numerical value, much of the mathematical treatment of magnetism must be similar to that of electricity. There are, however, other properties of magnets which must be borne in mind, and which may throw some light on the electrical properties of bodies.

Relation between the Poles of a Magnet.

377.] The quantity of magnetism at one pole of a magnet is always equal and opposite to that at the other, or more generally thus:—

In every Magnet the total quantity of Magnetism (reckoned algebraically) is zero.

Hence in a field of force which is uniform and parallel throughout the space occupied by the magnet, the force acting on the marked end of the magnet is exactly equal, opposite and parallel to that on the unmarked end, so that the resultant of the forces is a statical couple, tending to place the axis of the magnet in a determinate direction, but not to move the magnet as a whole in any direction.

This may be easily proved by putting the magnet into a small vessel and floating it in water. The vessel will turn in a certain direction, so as to bring the axis of the magnet as near as possible to the direction of the earth's magnetic force, but there will be no motion of the vessel as a whole in any direction; so that there can be no excess of the force towards the north over that towards the south, or the reverse. It may also be shewn from the fact that magnetizing a piece of steel does not alter its weight. It does alter the apparent position of its centre of gravity, causing it in these latitudes to shift along the axis towards the north. The centre of inertia, as determined by the phenomena of rotation, remains unaltered.

378.] If the middle of a long thin magnet be examined, it is found to possess no magnetic properties, but if the magnet be broken at that point, each of the pieces is found to have a magnetic pole at the place of fracture, and this new pole is exactly equal and opposite to the other pole belonging to that piece. It is impossible, either by magnetization, or by breaking magnets, or by any other means, to procure a magnet whose poles are unequal.

If we break the long thin magnet into a number of short pieces we shall obtain a series of short magnets, each of which has poles of nearly the same strength as those of the original long magnet. This multiplication of poles is not necessarily a creation of energy, for we must remember that after breaking the magnet we have to do work to separate the parts, in consequence of their attraction for one another.

379.] Let us now put all the pieces of the magnet together as at first. At each point of junction there will be two poles exactly equal and of opposite kinds, placed in contact, so that their united action on any other pole will be null. The magnet, thus rebuilt, has therefore the same properties as at first, namely two poles, one at each end, equal and opposite to each other, and the part between these poles exhibits no magnetic action.

Since, in this case, we know the long magnet to be made up of little short magnets, and since the phenomena are the same as in the case of the unbroken magnet, we may regard the magnet, even before being broken, as made up of small particles, each of which has two equal and opposite poles. If we suppose all magnets to be made up of such particles, it is evident that since the algebraical quantity of magnetism in each particle is zero, the quantity in the whole magnet will also be zero, or in other words, its poles will be of equal strength but of opposite kind.

Theory of Magnetic 'Matter'

380.] Since the form of the law of magnetic action is identical with that of electric action, the same reasons which can be given for attributing electric phenomena to the action of one 'fluid' or two 'fluids' can also be used in favour of the existence of a magnetic matter, or of two kinds of magnetic matter, fluid or otherwise. In fact, a theory of magnetic matter, if used in a purely mathematical sense, cannot fail to explain the phenomena, provided new laws are freely introduced to account for the actual facts.

One of these new laws must be that the magnetic fluids cannot pass from one molecule or particle of the magnet to another, but that the process of magnetization consists in separating to a certain extent the two fluids within each particle, and causing the one fluid to be more concentrated at one end, and the other fluid to be more concentrated at the other end of the particle. This is the theory of Poisson.

A particle of a magnetizable body is, on this theory, analogous to a small insulated conductor without charge, which on the two- fluid theory contains indefinitely large but exactly equal quantities of the two electricities. When an electromotive force acts on the conductor, it separates the electricities, causing them to become manifest at opposite sides of the conductor. In a similar manner, according to this theory, the magnetizing force causes the two kinds of magnetism, which were originally in a neutralized state, to be separated, and to appear at opposite sides of the magnetized particle.

In certain substances, such as soft iron and those magnetic substances which cannot be permanently magnetized, this magnetic condition, like the electrification of the conductor, disappears when the inducing force is removed. In other substances, such as hard steel, the magnetic condition is produced with difficulty, and, when produced, remains after the removal of the inducing force.

This is expressed by saying that in the latter case there is a Coercive Force, tending to prevent alteration in the magnetization, which must be overcome before the power of a magnet can be either increased or diminished. In the case of the electrified body this would correspond to a kind of electric resistance, which, unlike the resistance observed in metals, would be equivalent to complete insulation for electromotive forces below a certain value.

This theory of magnetism, like the corresponding theory of electricity, is evidently too large for the facts, and requires to be restricted by artificial conditions. For it not only gives no reason why one body may not differ from another on account of having more of both fluids, but it enables us to say what would be the properties of a body containing an excess of one magnetic fluid. It is true that a reason is given why such a body cannot exist, but this reason is only introduced as an after-thought to explain this particular fact. It does not grow out of the theory.

331.] We must therefore seek for a mode of expression which shall not be capable of expressing too much, and which shall leave room for the introduction of new ideas as these are developed from new facts. This, I think, we shall obtain if we begin by saying that the particles of a magnet are Polarized.

Meaning of the term 'Polarization'.

When a particle of a body possesses properties related to a certain line or direction in the body, and when the body, retaining these properties, is turned so that this direction is reversed, then if as regards other bodies these properties of the particle are reversed, the particle, in reference to these properties, is said to be polarized, and the properties are said to constitute a particular kind of polarization.

Thus we may say that the rotation of a body about an axis constitutes a kind of polarization, because if, while the rotation continues, the direction of the axis is turned end for end, the body will be rotating in the opposite direction as regards space.

A conducting particle through which there is a current of electricity may be said to be polarized, because if it were turned round, and if the current continued to flow in the same direction as regards the particle, its direction in space would be reversed.

In short, if any mathematical or physical quantity is of the nature of a vector, as defined in Art. 11, then any body or particle to which this directed quantity or vector belongs may be said to be Polarized^[2], because it has opposite properties in the two opposite directions or poles of the directed quantity.

The poles of the earth, for example, have reference to its rotation, and have accordingly different names.

Meaning of the term 'Magnetic Polarization.'

382.] In speaking of the state of the particles of a magnet as magnetic polarization, we imply that each of the smallest parts into which a magnet may be divided has certain properties related to a definite direction through the particle, called its Axis of Magnetization, and that the properties related to one end of this axis are opposite to the properties related to the other end.

The properties which we attribute to the particle are of the same kind as those which we observe in the complete magnet, and in assuming that the particles possess these properties, we only assert what we can prove by breaking the magnet up into small pieces, for each of these is found to be a magnet.

Properties of a Magnetized Particle.

383.] Let the element dxdydz be a particle of a magnet, and let us assume that its magnetic properties are those of a magnet the strength of whose positive pole is m, and whose length is ds. Then if P is any point in space distant r from the positive pole and r' from the negative pole, the magnetic potential at P will be ${\frac {m}{r}}$ due to the positive pole, and $-{\frac {m}{r'}}$ due to the negative pole, or

$V={\frac {m}{rr'}}(r-r').$

(1)

If dS, the distance between the poles, is very small, we may put

$(r-r')ds\cos \epsilon \,.$

(2)

where ε is the angle between the vector drawn from the magnet to P and the axis of the magnet, or

$V={\frac {m}{rr'}}\cos \epsilon .$

(3)

Magnetized Moment.

384.] The product of the length of a uniformly and longitudinally magnetized bar magnet into the strength of its positive pole is called its Magnetic Moment.

Intensity of Magnetization.

The intensity of magnetization of a magnetic particle is the ratio of its magnetic moment to its volume. We shall denote it by $I$.

The magnetization at any point of a magnet may be defined by its intensity and its direction. Its direction may be defined by

its direction-cosines λ, μ, ν.

Components of Magnetization.

The magnetization at a point of a magnet (being a vector or directed quantity) may be expressed in terms of its three components referred to the axes of coordinates. Calling these A, B, C

$A=I\lambda ,B=I\mu ,C=I\nu ,\,$

and the numerical value of $I$ is given by the equation (4)

$I^{2}=A2+B^{2}+C^{2}.\,$

(5)

385.] If the portion of the magnet which we consider is the differential element of volume dxdydz, and if $I$ denotes the intensity of magnetization of this element, its magnetic moment is $Idxdydz$. Substituting this for m in equation (3), and remembering that

$\cos \epsilon =\lambda (\xi -x)+\mu (\eta -y)+\nu (\zeta -z),\,$

(6)

where $\xi$, $\eta$, $\zeta$ are the coordinates of the extremity of the vector r drawn from the point (x, y, z), we find for the potential at the point ($\xi$, $\eta$, $\zeta$) due to the magnetized element at (x, y, z),

$\delta V=\left(A(\xi -x)+B(\eta -y)+C(\zeta -z)\right){\frac {1}{r^{3}}}dxdydz.$

(7)

To obtain the potential at the point ($\xi$, $\eta$, $\zeta$) due to a magnet of finite dimensions, we must find the integral of this expression for every element of volume included within the space occupied by the magnet, or

$V=\iiint \left(A(\xi -x)+B(\eta -y)+C(\zeta -z)\right){\frac {1}{r^{3}}}dxdydz.$

(8)

Integrating by parts, this becomes

$V=\iint {A{\frac {1}{r}}dydz}+\iint {B{\frac {1}{r}}dxdz}+\iint {C{\frac {1}{r}}dxdy}$

$-\iiint {{\frac {1}{r}}\left({\frac {dA}{dx}}+{\frac {dB}{dy}}+{\frac {dC}{dz}}\right)dxdydz},$

where the double integration in the first three terms refers to the surface of the magnet, and the triple integration in the fourth to the space within it.

If $l$, m, n denote the direction-cosines of the normal drawn outwards from the element of surface dS, we may write, as in Art. 21, the sum of the first three terms,

$V=\iint {\left(lA+mB+MnC\right){\frac {1}{r}}dS}$,

where the integration is to be extended over the whole surface of the magnet.

If we now introduce two new symbols σ and ρ, denned by the equations

$\sigma =lA+mB+nC\,$,

$\rho =-\left({\frac {dA}{dx}}+{\frac {dB}{dy}}+{\frac {dC}{dz}}\right)$,

the expression for the potential may be written

$V=\iint {{\frac {\sigma }{r}}dS}+\iiint {{\frac {\rho }{r}}dxdydz}.$

386.] This expression is identical with that for the electric potential due to a body on the surface of which there is an electrification whose surface-density is σ, while throughout its substance there is a bodily electrification whose volume-density is ρ. Hence, if we assume σ and ρ to be the surface- and volume-densities of the distribution of an imaginary substance, which we have called 'magnetic matter', the potential due to this imaginary distribution will be identical with that due to the actual magnetization of every element of the magnet.

The surface-density σ is the resolved part of the intensity of magnetization I in the direction of the normal to the surface drawn outwards, and the volume-density ρ is the 'convergence' (see Art. 25) of the magnetization at a given point in the magnet.

This method of representing the action of a magnet as due to a distribution of 'magnetic matter' is very convenient, but we must always remember that it is only an artificial method of representing the action of a system of polarized particles.

On the Action of one Magnetic Molecule on another.

387.] If, as in the chapter on Spherical Harmonics, Art. 129, we make

${\frac {d}{dh}}=l{\frac {d}{dx}}+m{\frac {d}{dy}}+n{\frac {d}{dz}}$,

(1)

where l, m, n are the direction-cosines of the axis h, then the potential due to a magnetic molecule at the origin, whose axis is parallel to h₁, and whose magnetic moment is m₁, is

$V_{1}=-{\frac {d}{dh_{1}}}={\frac {m1}{r^{2}}}\lambda _{1}$,

(2)

where λ₁ is the cosine of the angle between h₁ and r.

Again, if a second magnetic molecule whose moment is m₂ and whose axis is parallel to h₂, is placed at the extremity of the radius vector r, the potential energy due to the action of the one magnet on the other is

$W=-m_{2}{\frac {dV_{1}}{dh_{2}}}=m_{1}m_{2}{\frac {d^{2}}{dh_{1}dh_{2}}}\left({\frac {1}{r}}\right),$

(3)

$\quad ={\frac {m_{1}m_{2}}{r^{3}}}(\mu _{12}-3\lambda _{1}\lambda _{2}),$

(4)

where μ₁₂ is the cosine of the angle which the axes make with each other, and λ₁, λ₂ are the cosines of the angles which they make with r.

Let us next determine the moment of the couple with which the first magnet tends to turn the second round its centre.

Let us suppose the second magnet turned through an angle dφ in a plane perpendicular to a third axis h₃, then the work done against the magnetic forces will be ${\frac {dW}{d\varphi }}d\varphi$ and the moment of the magnet in this plane will be

$-{\frac {dW}{d\varphi }}=-{\frac {m_{1}m_{2}}{r^{3}}}\left({\frac {d\mu _{12}}{d\varphi }}-3\lambda _{1}{\frac {d\lambda _{2}}{d\varphi }}\right).$

(5)

The actual moment acting on the second magnet may therefore be considered as the resultant of two couples, of which the first acts in a plane parallel to the axes of both magnets, and tends to increase the angle between them with a force whose moment is

${\frac {m_{1}m_{2}}{r^{3}}}\sin(h_{1}h_{2}),$

(6)

while the second couple acts in the plane passing through r and the axis of the second magnet, and tends to diminish the angle between these directions with a force

${\frac {3m_{1}m_{2}}{r^{3}}}\cos(rh_{1})\sin(rh_{2}),$

(7)

where (rh₁), (rh₂), (h₁h₂) denote the angles between the lines r, h₁, h₂.

To determine the force acting on the second magnet in a direction parallel to a line h₃, we have to calculate

${\frac {dW}{dh_{3}}}=m_{1}m_{2}{\frac {d^{3}}{dh_{1}\,dh_{2}\,dh_{3}}}\left({\frac {1}{r}}\right),$

(8)

$=3{\frac {m_{1}m_{2}}{r^{4}}}\left(\lambda _{1}\mu _{23}+\lambda _{2}\mu _{31}+\lambda _{3}\mu _{12}-5\lambda _{1}\lambda _{2}\lambda _{3}\right),$

(9)

$=3\lambda _{3}{\frac {m_{1}m_{2}}{r^{4}}}\left(\mu _{12}-5\lambda _{1}\lambda _{2}\right)+3\mu _{13}{\frac {m_{1}m_{2}}{r^{4}}}\lambda _{2}+3\mu _{23}{\frac {m_{1}m_{2}}{r^{4}}}\lambda _{1}.$

(10)

If we suppose the actual force compounded of three forces, R, H₁ and H₂, in the directions of r, h₁ and h₂ respectively, then the force in the direction of h₃ is

$\lambda _{3}R+\mu _{13}H_{1}+\mu _{23}H_{2}.\,$

(11)

Since the direction of h₃ is arbitrary, we must have

${\begin{matrix}R={\frac {3m_{1}m_{2}}{r^{4}}}(m_{12}-5\lambda _{1}\lambda _{2}),\\H_{1}={\frac {3m_{1}m_{2}}{r^{4}}}\lambda _{2},\quad H_{2}={\frac {3m_{1}m_{2}}{r^{4}}}\lambda _{1}.\end{matrix}}$

(12)

The force R is a repulsion, tending to increase r; H_l and H₂ act on the second magnet in the directions of the axes of the first and second magnet respectively.

This analysis of the forces acting between two small magnets was first given in terms of the Quaternion Analysis by Professor Tait in the Quarterly Math. Journ. for Jan. 1860. See also his work on Quaternions, Art. 414.

Particular Positions.

388.] (1) If λ₁ and λ₂ are each equal to 1, that is, if the axes of the magnets are in one straight line and in the same direction, μ₁₂ = 1, and the force between the magnets is a repulsion

$R+H_{1}+H_{2}=-{\frac {6m_{1}m_{2}}{r^{4}}}.$

(13)

The negative sign indicates that the force is an attraction.

(2) If λ₁ and λ₂ are zero, and μ₁₂ unity, the axes of the magnets are parallel to each other and perpendicular to r, and the force is a repulsion

${\frac {3m_{1}m_{2}}{r^{4}}}.$

(14)

In neither of these cases is there any couple.

(3) If

$\lambda _{1}=1{\text{ and }}\lambda _{2}=0,{\text{ then }}\mu _{12}=0.\,$

(15)

The force on the second magnet will be ${\frac {3m_{1}m_{3}}{r^{4}}}$ in the direction of its axis, and the couple will be ${\frac {2m_{1}m_{2}}{r^{4}}}$, tending to turn it parallel to the first magnet. This is equivalent to a single force ${\frac {3m_{1}m_{2}}{r^{4}}}$ acting parallel to the direction of the axis of the second magnet, and cutting r at a point two-thirds of its length from m₂.

Fig. 1.

Thus in the figure (1) two magnets are made to float on water, m₂ being in the direction of the axis of m₁, but having its own axis at right angles to that of m₁. If two points, A, B, rigidly connected with m₁ and m₂ respectively, are connected by means of a string T, the system will be in equilibrium, provided T cuts the line m₁m₂ at right angles at a point one-third of the distance from m₁ to m₂.

(4) If we allow the second magnet to turn freely about its centre till it comes to a position of stable equilibrium, W will then be a minimum as regards h₂, and therefore the resolved part of the force due to m₂, taken in the direction of h₁, will be a maximum. Hence, if we wish to produce the greatest possible magnetic force at a given point in a given direction by means of magnets, the positions of whose centres are given, then, in order to determine the proper directions of the axes of these magnets to produce this effect, we have only to place a magnet in the given direction at the given point, and to observe the direction of stable equilibrium of the axis of a second magnet when its centre is placed at each of the other given points. The magnets must then be placed with their axes in the directions indicated by that of the second magnet.

Of course, in performing this experiment we must take account of terrestrial magnetism, if it exists.

Let the second magnet be in a position of stable equilibrium as regards its direction, then since the couple acting on it vanishes, the axis of the second magnet must be in the same plane with that of the first. Hence

$(h_{1}h_{2})=(h_{1}r)+(rh_{2})\,$

(16)

and the couple being

${\frac {m_{1}m_{2}}{r^{3}}}\left(\sin(h_{1}h_{2})-3\cos(h_{1}r)\sin(rh_{2})\right),$

(17)

we find when this is zero

$\tan(h_{1}r)=2\tan(rh_{2}),\,$

(18)

or

$\tan \,H_{1}\,m_{2}\,R=2\tan \,R\,m_{2}\,H_{2}.$

(19)

When this position has been taken up by the second magnet the value of W becomes

$-m_{2}{\frac {dV}{dh_{2}}},$

where h₂ is in the direction of the line of force due to m₁ at m₂. Hence

$W=-m_{2}{\sqrt {{\frac {\overline {dV}}{dx}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dy}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dz}}{\Bigg |}^{2}}},$

(20)

Hence the second magnet will tend to move towards places of greater resultant force.

The force on the second magnet may be decomposed into a force R, which in this case is always attractive towards the first magnet, and a force H₁ parallel to the axis of the first magnet, where

$R=-3{\frac {m_{1}m_{3}}{r^{4}}}{\frac {4\lambda _{1}^{2}+1}{\sqrt {3\lambda _{1}^{2}+1}}},\quad H_{1}=3{\frac {m_{1}m_{3}}{r^{4}}}{\frac {\lambda _{1}}{\sqrt {3\lambda _{1}^{2}+1}}}.$

(21)

In Fig. XVII, at the end of this volume, the lines of force and equipotential surfaces in two dimensions are drawn. The magnets which produce them are supposed to be two long cylindrical rods the sections of which are represented by the circular blank spaces, and these rods are magnetized transversely in the direction of the arrows.

If we remember that there is a tension along the lines of force, it is easy to see that each magnet will tend to turn in the direction of the motion of the hands of a watch.

That on the right hand will also, as a whole, tend to move towards the top, and that on the left hand towards the bottom of the page.

On the Potential Energy of a Magnet placed in a Magnetic Field.

389.] Let V be the magnetic potential due to any system of magnets acting on the magnet under consideration. We shall call V the potential of the external magnetic force.

If a small magnet whose strength is m, and whose length is ds, be placed so that its positive pole is at a point where the potential is V, and its negative pole at a point where the potential is V' the potential energy of this magnet will be m(V – V'), or, if ds is measured from the negative pole to the positive,

$m{\frac {dV}{ds}}ds.$

(1)

If I is the intensity of the magnetization, and λ, μ, ν its direction-cosines, we may write,

$m\,ds=I\,dxdydz,$

${\text{and }}{\frac {dV}{ds}}=\lambda {\frac {dV}{dx}}+\mu {\frac {dV}{dy}}+\nu {\frac {dV}{dz}},$

and, finally, if A, B, C are the components of magnetization,

$A=\lambda I,\quad B=\mu I,\quad C=\nu I,$

so that the expression (1) for the potential energy of the element

of the magnet becomes

$\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz.$

(2)

To obtain the potential energy of a magnet of finite size, we must integrate this expression for every element of the magnet. We thus obtain

$W=\iiint {\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz}$

(3)

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

$W=\iint {(Al+Bm+Cn)V\,dS}-\iiint V{\left(A{\frac {dV}{dx}}+B{\frac {dV}{dy}}+C{\frac {dV}{dz}}\right)dxdydz},$

(4)

where l, 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

$W=\iint {V\sigma \,dS}+\iiint {V\rho dxdydz}.$[3]

(5)

We may write equation (3) in the form

$W=-\iiint {A\alpha +B\beta +C\gamma )dxdydz}.$

(6)

where α, β, γ 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 α, β, γ will be constant quantities, and if we write

$\iiint {Adxdydz}=lK,\iiint {Bdxdydz}=mK,\iiint {Cdxdydz}=nK$

(7)

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

$W=-K(l\alpha +m\beta +n\gamma ).\,$

(8)

In this expression l, m, n are the direction-cosines of the axis of the magnet, and K is the magnetic moment of the magnet. If ε is the angle which the axis of the magnet makes with the direction of the magnetic force ${\mathfrak {H}}$, the value of W may be written

$W=-K{\mathfrak {H}}\cos \epsilon .\,$

(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 $\phi$, and let it be inclined $\theta$ to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is $\delta$ and dip $\zeta$, then

$\alpha ={\mathfrak {H}}\cos \zeta \cos \delta ,\quad \beta ={\mathfrak {H}}\cos \zeta \sin \delta ,\quad \gamma ={\mathfrak {H}}\sin \zeta ;$

(10)

$l=\cos \theta ,\quad m=\cos \theta \sin \phi ,\quad n=\sin \theta ;$

(11)

whence W K$ (cos cos0 cos(< 8) + sin <>in 0). (12)

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

$-{\frac {dW}{d\phi }}=-K{\mathfrak {H}}\cos \zeta \sin(\phi -\delta ).$

(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 (ξ, η, ζ). The value of V at the point x, y, z is

$V=\left[(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}\right]^{-1/2}.$

(1)

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

$V=V_{0}+V_{1}+\,$&c.,

(2)

when

$V_{0}={\frac {1}{r}},$ r being the distance of (ξ, η, ζ) from the origin,

(3)

$V_{1}={\frac {\xi x+\eta y+\zeta z}{r^{3}}},$

(4)

$V_{2}={\frac {3(\xi x+\eta y+\zeta z)^{2}-(x^{2}+y^{2}+z^{2})(\xi ^{2}+\eta ^{2}+\zeta ^{2})}{2r^{5}}},$

(5)

&c.

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 ξ, η, ζ as constants.

If we consider only the terms introduced by V₀, V₁ and V₂ the result will depend on the following volume-integrals,

$lK=\iiint {Adxdydz},\quad mK=\iiint {Bdxdydz},\quad nK=\iiint {Cdxdydz};$

(6)

$L=\iiint {Axdxdydz},\quad M=\iiint {Bydxdydz},\quad N=\iiint {Cxdxdydz};$

(7)

$P=\iiint {(Bz+Cy)dxdydz},\quad Q=\iiint {(Cx+Az)dxdydz},\quad R=\iiint {(Ay+Bz)dxdydz}.$

(8)

We thus find for the value of the potential energy of the magnet placed in presence of the unit pole at the point (ξ, η, ζ),

$W=K{\frac {l\xi +m\eta +n\zeta }{r^{3}}}-{\frac {\xi ^{2}(2L-M-N)+\eta ^{2}(2M-N-L)+\zeta ^{2}(2N-L-M)+3(P\eta \zeta +Q\zeta \xi +R\zeta \eta )}{r^{5}}}.$

(9)

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 ξ, η, ζ due to the magnet.

On the 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

$l=1,\quad m=0,\quad n=0.$

(10)

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

$L'=L-lKx',\quad M'=M-mKy',\quad N'=N-nKz';$

(11)

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

(12)

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

$x'={\frac {2L-M-N}{2K}},\quad y'={\frac {R}{K}},\quad z'={\frac {Q}{K}}$

(13)

then for the new axes M and N have their values unchanged, and the value of L' becomes ${\frac {1}{2}}(M+N)$. P remains unchanged, and Q and R vanish. We may therefore write the potential thus,

$K{\frac {\xi }{r^{3}}}+{\frac {{\frac {3}{2}}(\eta ^{2}-\zeta ^{2})(M-N)+3P\eta \zeta }{r^{5}}}.$[4]

(14)

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 ${\frac {P}{M-N}}$. This will cause P to become zero, and the final form of the potential may be written

$K{\frac {\xi }{r^{3}}}+{\frac {3}{2}}{\frac {(\eta ^{2}-\zeta ^{2})(M-N)}{r^{5}}}.$

(15)

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,

$4(l^{2}+M^{2}+N^{2}-MN-NL-LM)+3(P^{2}+Q^{2}+R^{2}).\,$

(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

${\begin{matrix}2l(2L-M-N)+3nQ+3mR=0,\\2m(2M-N-L)+3lR+3nP=0,\\2n(2N-L-M)+3mP+3lQ=0.\end{matrix}}$

(17)

If we assume l = 1, m = 0, n = 0, these conditions become

$2L-M-N=0,\quad Q=0,\quad R=0,$

(18)

which are the conditions made use of in the previous investigation.

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, 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 L, M, N correspond to the moments of inertia, and P, Q, R to the products of inertia of a material body, except that L, 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 imaginary 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.

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

$V=\sum {\left({\frac {m}{r}}\right)},$

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

1. His experiments on magnetism with the Torsion Balance are contained in the Memoirs of the Academy of Paris, 1780-9, and in Biot's Traité de Physique, tom. iii.
2. The word Polarization has been used in a sense not consistent with this in Optics, where a ray of light is said to be polarized when it has properties relating to its sides, which are identical on opposite sides of the ray. This kind of polarization refers to another kind of Directed Quantity, which may be called a Dipolar Quantity, in opposition to the former kind, which may be called Unipolar.
   When a dipolar quantity is turned end for end it remains the same as before. Tensions and Pressures in solid bodies, Extensions, Compressions and Distortions and most of the optical, electrical, and magnetic properties of crystallized bodies are dipolar quantities.
   The property produced by magnetism in transparent bodies of twisting the plane of polarization of the incident light, is, like magnetism itself, a unipolar property. The rotatory property referred to in Art. 303 is also unipolar.
3. There is a typo in the second integral in the original: the integration there is over dS
4. The denominator in the second term in the original is r, obviously an error.