# A Treatise on Electricity and Magnetism/Part IV/Chapter XXII

​CHAPTER XXII.
FERROMAGNETISM AND DIAMAGNETISM EXPLAINED BY MOLECULAR CURRENTS.
On Electromagnetic Theories of Magnetism.
832.] WE have seen (Art. 380) that the action of magnets on one another can be accurately represented hy the attractions and repulsions of an imaginary substance called * magnetic matter. We have shewn the reasons why we must not suppose this magnetic matter to move from one part of a magnet to another through a sensible distance, as at first sight it appears to do when we magnetize a bar, and we were led to Poisson s hypothesis that the magnetic matter is strictly confined to single molecules of the mag netic substance, so that a magnetized molecule is one in which the opposite kinds of magnetic matter are more or less separated to wards opposite poles of the molecule, but so that no part of either can ever be actually separated from the molecule (Art. 430).
These arguments completely establish the fact, that magnetiza tion is a phenomenon, not of large masses of iron, but of molecules, that is to say, of portions of the substance so small that we cannot by any mechanical method cut one of them in two, so as to obtain a north pole separate from a south pole. But the nature of a mag netic molecule is by no means determined without further investi gation. We have seen (Art. 442) that there are strong reasons for believing that the act of magnetizing iron or steel does not consist in imparting magnetization to the molecules of which it is com posed, but that these molecules are already magnetic, even in un- magnetized iron, but with their axes placed indifferently in all directions, and that the act of magnetization consists in turning the molecules so that their axes are either rendered all parallel to one direction, or at least. are deflected towards that direction.
�� � ​834-] AMPERE S THEORY. 419
833.] Still, however, we have arrived at no explanation of the nature of a magnetic molecule, that is, we have not recognized its likeness to any other thing of which we know more. We have therefore to consider the hypothesis of Ampere, that the magnetism of the molecule is due to an electric current constantly circulating in some closed path within it.
It is possible to produce an exact imitation of the action of any magnet on points external to it, by means of a sheet of electric currents properly distributed on its outer surface. But the action of the magnet on points in the interior is quite different from the action of the electric currents on corresponding points. Hence Am pere concluded that if magnetism is to be explained by means of electric currents, these currents must circulate within the molecules of the magnet, and must not flow from one molecule to another. As we cannot experimentally measure the magnetic action at a point in the interior of a molecule, this hypothesis cannot be dis proved in the same way that we can disprove the hypothesis of currents of sensible extent within the magnet.
Besides this, we know that an electric current, in passing from one part of a conductor to another, meets with resistance and gene rates heat ; so that if there were currents of the ordinary kind round portions of the magnet of sensible size, there would be a constant expenditure of energy required to maintain them, and a magnet would be a perpetual source of heat. By confining the circuits to the molecules, within which nothing is known about resistance, we may assert, without fear of contradiction, that the current, in cir culating within the molecule, meets with no resistance.
According to Ampere s theory, therefore, all the phenomena of magnetism are due to electric currents, and if we could make ob servations of the magnetic force in the interior of a magnetic mole cule, we should find that it obeyed exactly the same laws as the force in a region surrounded by any other electric circuit.
834.] In treating of the force in the interior of magnets, we have supposed the measurements to be made in a small crevasse hollowed out of the substance of the magnet, Art. 395. We were thus led to consider two different quantities, the magnetic force and the magnetic induction, both of which are supposed to be observed in a space from which the magnetic matter is removed. We were not supposed to be able to penetrate into the interior of a mag netic molecule and to observe the force within it.
If we adopt Ampere s theory, we consider a magnet, not as a
E e 2
�� � ​420 ELECTE1C THEORY OF MAGNETISM. [8 35.
continuous substance, the magnetization of which varies from point to point according to some easily conceived law,, but as a multitude of molecules, within each of which circulates a system of electric currents, giving rise to a distribution of magnetic force of extreme complexity, the direction of the force in the interior *pf a molecule being generally the reverse of that of the average force in its neigh bourhood, and the magnetic potential, where it exists at all, being a function of as many degrees of multiplicity as there are molecules in the magnet.
835.J But we shall find, that, in spite of this apparent complexity, which, however, arises merely from the coexistence of a multitude of simpler parts, the mathematical theory of magnetism is greatly simplified by the adoption of Ampere s theory, and by extending our mathematical vision into the interior of the molecules.
In the first place, the two definitions of magnetic force are re duced to one, both becoming the same as that for the space outside the magnet. In the next place, the components of the magnetic force everywhere satisfy the condition to which those of induction are subject, namely, j a JQ j
��In other words, the distribution of magnetic force is of the same nature as that of the velocity of an incompressible fluid, or, as we have expressed it in Art. 25, the magnetic force has no convergence.
Finally, the three vector functions the electromagnetic momen tum, the magnetic force, and the electric current become more simply related to each other. They are all vector functions of no convergence, and they are derived one from the other in order, by the same process of taking the space-variation, which is denoted by Hamilton by the symbol V.
836.] But we are now considering magnetism from a physical point of view, and we must enquire into the physical properties of the molecular currents. We assume that a current is circulating in a molecule, and that it meets with no resistance. If L is the coefficient of self-induction of the molecular circuit, and M the co efficient of mutual induction between this circuit and some other circuit, then if y is the current in the molecule, and y that in the other circuit, the equation of the current y is
-Ry, (2)
�� � ​838.] CIRCUITS OF NO RESISTANCE. 421
and since by the hypothesis there is no resistance, R = 0, and we get by integration
Ly + My =. constant, = Zy , say. (3)
Let us suppose that the area of the projection of the molecular circuit on a plane perpendicular to the axis of the molecule is A, this axis being defined as the normal to the plane on which the projection is greatest. If the action of other currents produces a magnetic force, X, in a direction whose inclination to the axis of the molecule is 6, the quantity My becomes XA cos0, and we have as the equation of the current
Ly + XAco$0 Ly , (4)
where y is the value _of y when X = 0.
It appears, therefore, that the strength of the molecular current depends entirely on its primitive value y , and on the intensity of the magnetic force due to other currents.
837.] If we suppose that there is no primitive current, but that the current is entirely due to induction, then
y = -- -COS0. (o)
��The negative sign shews that the direction of the induced cur rent is opposite to that of the inducing current, and its magnetic action is such that in the interior of the circuit it acts in the op posite direction to the magnetic force. In other words, the mole cular current acts like a small magnet whose poles are turned towards the poles of the same name of the inducing magnet.
Now this is an action the reverse of that of the molecules of iron under magnetic action. The molecular currents in iron, therefore, are not excited by induction. But in diamagnetic substances an action of this kind is observed, and in fact this is the explanation of diamagnetic polarity which was first given by Weber.
Weber s Theory of Diamagnetism.
838.] According to Weber s theory, there exist in the molecules of diamagnetic substances certain channels round which an electric current can circulate without resistance. It is manifest that if we suppose these channels to traverse the molecule in every direction, this amounts to making the molecule a perfect conductor.
Beginning with the assumption of a linear circuit within the mo lecule, we have the strength of the current given by equation (5).
�� � ​422 ELECTRIC 'IIIEDHY OF mrow/1*1SlL [839. Thu moguulic moment af the eimvet in the product ol` its etmngdi lg lhs arm ot' the circuit, or yu, and the mwlvonl part ui this in the dlractiun of the iuagnutizing iowa ix yi} our 0, or, hy (S), ¤ - # sm a, (M) lf them are u such mnlseulns in nnitaf volume, and if their axes nin distritnxtnd inditforeiitly in all directions, than the avuznga vsluc of eo¤’0 will hs Q, and tha intensity of magnetization of tho sabslnnes will he 1# (7) Neumann; couliicieut r»l'mngar:ti.uhl¤n is therefore wil s - Mg T - wt Thu inngimtimtion ol' the sulxstanu is therefore iu the opposite tlireeticn to the magnetiziug A`or¤s_,or, in other wnnls, the sulmannn is rlinmzgnutie. It is also exactly proportional to the ningautizing fame, and llocs not will to it linits limit., as iu the casa uf ordinary magnetic indizctiun, Sse Arts. M2, Sac. E39.] It tho directions af the axes ul' the mcleeular channels ara nrinngerl, not inrliii`ei·eatly in all diieelions, but with n pi·epanrlt·i»- ating numbor in certain directions, than thu sum s we it extended to all the molnsules will have tliflhmni: values aceording tu the dirautiou of the line from which 0 is mcasumnl, and the Aim trihutiou of Lhnsc values in dillbrent ilirootians will hu similar to tho distrilzation of the values of moments el' inertia nhoub nm: in dif. fsmnf. directions through the same paint, Such in distribution will explain the magnetic plienomann related to axes in the body, described by Pltiokor, which Faraday has collorl Magracrystollis plisaoinaim. See Art. U5. 840,] Let us naw consider whnt would lm the effect, it inawnrl ol' the eluetiiu current being cantinell to it certain channel within thc molecule, the whale molecule wm snppnretl a parfeet eemluemr. Lot us Login with tho earn of n lmdy the Form of which is ncyclie, that is tn any, which is not in the ibrm of a ring; or peifuiatcll body, and lot us suppose that this livxly is cveryxvlierc surrenndul by u thin shall of pcri"¤¤tl_v ouxidaoting matter. l We have proved in Art, GM, that a closed sheet of perfectly l canrlueting matter of any {`urm, originally free hom enrrunts, he· ` ​842.] PERFECTLY CONDUCTING MOLECULES. 423
comes, when exposed to external magnetic force, a current-sheet, the action of which on every point of the interior is such as to make the magnetic force zero.
It may assist us in understanding this case if we observe that the distribution of magnetic force in the neighbourhood of such a body is similar to the distribution of velocity in an incompressible fluid in the neighbourhood of an impervious body of the same form.
It is obvious that if other conducting shells are placed within the first, since they are not exposed to magnetic force, no currents will be excited in them. Hence, in a solid of perfectly conducting material, the effect of magnetic force is to generate a system of currents which are entirely confined to the surface of the body.
841.] If the conducting body is in the form of a sphere of radius r, its magnetic moment is
-ir&gt;JT,
and if a number of such spheres are distributed in a medium, so that in unit of volume the volume of the conducting matter is , then, by putting ^=1, and ^ = in equation (1 7), Art. 314, we find the coefficient of magnetic permeability,
2 ~ U/ , (9)
��whence we obtain for Poisson s magnetic coefficient
and for Neumann s coefficient of magnetization by induction
��_
Since the mathematical conception of perfectly conducting bodies leads to results exceedingly different from any phenomena which we can observe in ordinary conductors, let us pursue the subject somewhat further.
842.] Returning to the case of the conducting channel in the form of a closed curve of area A, as in Art. 836, we have, for the moment of the electromagnetic force tending to increase the angle 0,
sme , (12)
smflcosfl. (13)
j
This force is positive or negative according as is less or greater than a right angle. Hence the effect of magnetic force on a per fectly conducting channel tends to turn it with its axis at right
�� � ​424 ELECTRIC THEORY OF MAGNETISM. [843.
angles to the line of magnetic force, that is, so that the plane of the channel becomes parallel to the lines of force.
An effect of a similar kind may be observed by placing a penny or a copper ring between the poles of an electromagnet. At the instant that the magnet is excited the ring turns its plane towards the axial direction, but this force vanishes as soon as the currents are deadened by the resistance of the copper *.
843.] We have hitherto considered only the case in which the molecular currents are entirely excited by the external magnetic force. Let us next examine the bearing of Weber s theory of the magneto-electric induction of molecular currents on Ampere s theory of ordinary magnetism. According to Ampere and Weber, the molecular currents in magnetic substances are not excited by the external magnetic force, but are already there, and the molecule itself is acted on and deflected by the electromagnetic action of the magnetic force on the conducting circuit in which the current flows. When Ampere devised this hypothesis, the induction of electric cur rents was not known, and he made no hypothesis to account for the existence, or to determine the strength, of the molecular currents.
We are now, how r ever, bound to apply to these currents the same laws that Weber applied to his currents in diamagnetic molecules. We have only to suppose that the primitive value of the current y, when no magnetic force acts, is not zero but y . The strength of the current when a magnetic force, X, acts on a molecular current of area A, whose axis is inclined Q to the line of magnetic force, is
.A. A. ( -I A\
��and the moment of the couple tending to turn the molecule so as
to increase is X 2 A 2
sin2&lt;9. (15)
��2L Hence, putting A
Ay Q = m, /- = *, (16)
L y
in the investigation in Art. 443, the equation of equilibrium becomes Xsin0-.X 2 sm0cos&lt;9 = J9sin(a-0). (17)
The resolved part of the magnetic moment of the current in the direction of X is
XA 2
yAcosO = y ^cos0 -- ^-cos 2 0, (18)
= mcos8(l-3Xco80). (19)
�� � ​845-] MODIFIED THEORY OF INDUCED MAGNETISM. 425
844.] These conditions differ from those in Weber s theory of magnetic induction by the terms involving the coefficient B. If BX is small compared with unity, the results will approximate to those of Weber s theory of magnetism. If BX is large compared with unity, the results will approximate to those of Weber s theory of diamagnetism.
Now the greater y , the primitive value of the molecular current,, the smaller will B become, and if L is also large, this will also diminish B. Now if the current flows in a ring channel, the value
T&gt;
of L depends on log , where R is the radius of the mean line of
the channel, and r that of its section. The smaller therefore the section of the channel compared with its area, the greater will be Z, the coefficient of self-induction, and the more nearly will the phe nomena agree with Weber s original theory. There will be this difference, however, that as X, the magnetizing force, increases, the temporary magnetic moment will not only reach a maximum, but will afterwards diminish as X increases.
If it should ever be experimentally proved that the temporary magnetization of any substance first increases, and then diminishes as the magnetizing force is continually increased, the evidence of the existence of these molecular currents would, I think, be raised almost to the rank of a demonstration.
845.] If the molecular currents in diamagnetic substances are confined to definite channels, and if the molecules are capable of being deflected like those of magnetic substances, then, as the mag netizing force increases, the diamagnetic polarity will always increase, but, when the force is great, not quite so fast as the magnetizing force. The small absolute value of the diamagnetic coefficient shews, however, that the deflecting force on each molecule must be small compared with that exerted on a magnetic molecule, so that any result due to this deflexion is not likely to be perceptible.
If, on the other hand, the molecular currents in diamagnetic bodies are free to flow through the whole substance of the molecules, the diamagnetic polarity will be strictly proportional to the mag netizing force, and its amount will lead to a determination of the whole space occupied by the perfectly conducting masses, and, if we know the number of the molecules, to the determination of the size of each,