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

by James Clerk Maxwell

CHAPTER VI.

WEBER'S THEORY OF INDUCED MAGNETISM.

442.] We have seen that Poisson supposes the magnetization of iron to consist in a separation of the magnetic fluids within each magnetic molecule. If we wish to avoid the assumption of the existence of magnetic fluids, we may state the same theory in another form, by saying that each molecule of the iron, when the magnetizing force acts on it, becomes a magnet.

Weber's theory differs from this in assuming that the molecules of the iron are always magnets, even before the application of the magnetizing force, but that in ordinary iron the magnetic axes of the molecules are turned indifferently in every direction, so that the iron as a whole exhibits no magnetic properties.

When a magnetic force acts on the iron it tends to turn the axes of the molecules all in one direction, and so to cause the iron, as a whole, to become a magnet.

If the axes of all the molecules were set parallel to each other, the iron would exhibit the greatest intensity of magnetization of which it is capable. Hence Weber's theory implies the existence of a limiting intensity of magnetization, and the experimental evidence that such a limit exists is therefore necessary to the theory. Experiments shewing an approach to a limiting value of magnetization have been made by Joule^[1] and by J. Müller^[2].

The experiments of Beetz^[3] on electrotype iron deposited under the action of magnetic force furnish the most complete evidence of this limit,—

A silver wire was varnished, and a very narrow line on the metal was laid bare by making a fine longitudinal scratch on the varnish. The wire was then immersed in a solution of a salt of iron, and placed in a magnetic field with the scratch in the direction of a line of magnetic force. By making the wire the cathode of an electric current through the solution, iron was deposited on the narrow exposed surface of the wire, molecule by molecule. The filament of iron thus formed was then examined magnetically. Its magnetic moment was found to be very great for so small a mass of iron, and when a powerful magnetizing force was made to act in the same direction the increase of temporary magnetization was found to be very small, and the permanent magnetization was not altered. A magnetizing force in the reverse direction at once reduced the filament to the condition of iron magnetized in the ordinary way.

Weber's theory, which supposes that in this case the magnetizing force placed the axis of each molecule in the same direction during the instant of its deposition, agrees very well with what is observed.

Beetz found that when the electrolysis is continued under the action of the magnetizing force the intensity of magnetization of the subsequently deposited iron diminishes. The axes of the molecules are probably deflected from the line of magnetizing force when they are being laid down side by side with the molecules already deposited, so that an approximation to parallelism can be obtained only in the case of a very thin filament of iron.

If, as Weber supposes, the molecules of iron are already magnets, any magnetic force sufficient to render their axes parallel as they are electrolytically deposited will be sufficient to produce the highest intensity of magnetization in the deposited filament.

If, on the other hand, the molecules of iron are not magnets, but are only capable of magnetization, the magnetization of the deposited filament will depend on the magnetizing force in the same way in which that of soft iron in general depends on it. The experiments of Beetz leave no room for the latter hypothesis.

443.] We shall now assume, with Weber, that in every unit of volume of the iron there are n magnetic molecules, and that the magnetic moment of each is m. If the axes of all the molecules were placed parallel to one another, the magnetic moment of the unit of volume would be

$M=nm,\,$

and this would be the greatest intensity of magnetization of which the iron is capable.

In the unmagnetized state of ordinary iron Weber supposes the axes of its molecules to be placed indifferently in all directions.

To express this, we may suppose a sphere to be described, and a radius drawn from the centre parallel to the direction of the axis of each of the n molecules. The distribution of the extremities of these radii will express that of the axes of the molecules. In the case of ordinary iron these n points are equally distributed over every part of the surface of the sphere, so that the number of molecules whose axes make an angle less than a with the axis of x is

${\frac {n}{2}}(1-\cos \alpha ),$

and the number of molecules whose axes make angles with that of x, between α and α + dα is therefore

${\frac {n}{2}}\sin \alpha \,d\alpha .$

This is the arrangement of the molecules in a piece of iron which has never been magnetized.

Let us now suppose that a magnetic force X is made to act on the iron in the direction of the axis of x, and let us consider a molecule whose axis was originally inclined a to the axis of x.

If this molecule is perfectly free to turn, it will place itself with its axis parallel to the axis of x, and if all the molecules did so, the very slightest magnetizing force would be found sufficient to develope the very highest degree of magnetization. This, however, is not the case.

The molecules do not turn with their axes parallel to x, and this is either because each molecule is acted on by a force tending to preserve it in its original direction, or because an equivalent effect is produced by the mutual action of the entire system of molecules.

Weber adopts the former of these suppositions as the simplest, and supposes that each molecule, when deflected, tends to return to its original position with a force which is the same as that which a magnetic force D, acting in the original direction of its axis, would produce.

The position which the axis actually assumes is therefore in the direction of the resultant of X and D.

Let APB represent a section of a sphere whose radius represents, on a certain scale, the force D. Let the radius OP be parallel to the axis of a particular molecule in its original position.

Let SO represent on the same scale the magnetizing force X which is supposed to act from S towards 0. Then, if the molecule is acted on by the force X in the direction SO, and by a force D in a direction parallel to OP, the original direction of its axis, its axis will set itself in the direction SP, that of the resultant of X and D.

Since the axes of the molecules are originally in all directions, P may be at any point of the sphere indifferently. In Fig. 5, in which X is less than D, SP, the final position of the axis, may be in any direction whatever, but not indifferently, for more of the molecules will have their axes turned towards A than towards B. In Fig. 6, in which X is greater than D, the axes of the molecules will be all confined within the cone STT' touching the sphere.

Hence there are two different cases according as X is less or greater than D.

Let

α = AOP, the original inclination of the axis of a molecule to the axis of x.

θ = ASP, the inclination of the axis when deflected by the force X.

β = 'SPO', the angle of deflexion.

SO = X, the magnetizing force.

OP = D, the force tending towards the original position.

SP = R, the resultant of X and D.

m = magnetic moment of the molecule.

Then the moment of the statical couple due to X, tending to diminish the angle θ, is

$mL=mX\sin \theta ,\,$

and the moment of the couple due to D, tending to increase θ, is

$mL=mD\sin \beta .\,$

Equating these values, and remembering that β = α - θ, we find

$\tan \theta ={\frac {D\sin \alpha }{X+D\cos \alpha }}$

(1)

to determine the direction of the axis after deflexion.

We have next to find the intensity of magnetization produced in the mass by the force X, and for this purpose we must resolve the magnetic moment of every molecule in the direction of x, and add all these resolved parts.

The resolved part of the moment of a molecule in the direction of x is

$m\cos \theta .\,$

The number of molecules whose original inclinations lay between α and α + dα is

${\frac {n}{2}}\sin \alpha \,d\alpha .$

We have therefore to integrate

$I=\int _{0}^{\pi }{{\frac {mn}{2}}\cos \theta \sin \alpha \,d\alpha ,}$

(2)

remembering that θ is a function of α.

We may express both θ and α in terms of R, and the expression to be integrated becomes

${\frac {mn}{4X^{2}D}}(R^{2}+X^{2}-D^{2})dR,$

(3)

the general integral of which is

${\frac {mnR}{12X^{2}D}}(R^{2}+3X^{2}-D^{2})+C.$

(4)

In the first case, that in which X is less than D, the limits of integration are R = D + X and R = D – X. In the second case, in which X is greater than D, the limits are R = X + D and R = X – D.

When X is less than D,

$I={\frac {2}{3}}{\frac {mn}{D}}X.$

(5)

When X is equal to D,

$I={\frac {2}{3}}mn.$

(6)

When X is greater than D,

$I=mn\left(1-{\frac {1}{3}}{\frac {D^{2}}{X^{2}}}\right);$

(7)

and when X becomes infinite

$I=mn.\,$

(7)

According to this form of the theory, which is that adopted by Weber^[4], as the magnetizing force increases from 0 to D, the magnetization increases in the same proportion. When the magnetizing force attains the value D, the magnetization is two-thirds of its limiting value. When the magnetizing force is further increased, the magnetization, instead of increasing indefinitely, tends towards a finite limit.

The law of magnetization is expressed in Fig. 7, where the magnetizing force is reckoned from towards the right and the magnetization is expressed by the vertical ordinates. Weber's own experiments give results in satisfactory accordance with this law. It is probable, however, that the value of D is not the same for all the molecules of the same piece of iron, so that the transition from the straight line from O to E to the curve beyond E may not be so abrupt as is here represented.

444.] The theory in this form gives no account of the residual magnetization which is found to exist after the magnetizing force is removed. I have therefore thought it desirable to examine the results of making a further assumption relating to the conditions under which the position of equilibrium of a molecule may be permanently altered.

Let us suppose that the axis of a magnetic molecule, if deflected through any angle β less than β₀, will return to its original position when the deflecting force is removed, but that if the deflexion β exceeds β₀, then, when the deflecting force is removed, the axis will not return to its original position, but will be permanently deflected through an angle β - β₀, which may be called the permanent set of the molecule.

This assumption with respect to the law of molecular deflexion is not to be regarded as founded on any exact knowledge of the intimate structure of bodies, but is adopted, in our ignorance of the true state of the case, as an assistance to the imagination in following out the speculation suggested by Weber.

Let

$L=D\sin \beta _{0},\,$

(9)

then, if the moment of the couple acting on a molecule is less than mL, there will be no permanent deflexion, but if it exceeds mL there will be a permanent change of the position of equilibrium.

To trace the results of this supposition, describe a sphere whose centre is O and radius OL = L.

As long as X is less than L everything will be the same as in the case already considered, but as soon as X exceeds L it will begin to produce a permanent deflexion of some of the molecules.

Let us take the case of Fig. 8, in which X is greater than L but less than D. Through S as vertex draw a double cone touching the sphere L. Let this cone meet the sphere D in P and Q. Then if the axis of a molecule in its original position lies between OA and OP, or between OS and OQ, it will be deflected through an angle less than β₀, and will not be permanently deflected. But if the axis of the molecule lies originally between OP and OQ, then a couple whose moment is greater than L will act upon it and will deflect it into the position SP, and when the force X ceases to act it will not resume its original direction, but will be permanently set in the direction OP.

⁠
Fig. 8.	Fig. 9.

Let us put

$L=X\sin \theta _{0}\quad {\text{when}}\quad \theta =PSA{\text{ or }}QSB,$

then all those molecules whose axes, on the former hypotheses, would have values of θ between θ₀ and π - θ₀ will be made to have the value θ₀ during the action of the force X.

During the action of the force X, therefore, those molecules whose axes when deflected lie within either sheet of the double cone whose semivertical angle θ₀ is will be arranged as in the former case, but all those whose axes on the former theory would lie outside of these sheets will be permanently deflected, so that their axes will form a dense fringe round that sheet of the cone which lies towards A.

As X increases, the number of molecules belonging to the cone about B continually diminishes, and when X becomes equal to D all the molecules have been wrenched out of their former positions of equilibrium, and have been forced into the fringe of the cone round A, so that when X becomes greater than D all the molecules form part of the cone round A or of its fringe.

When the force X is removed, then in the case in which X is less than L everything returns to its primitive state. When X is between L and D then there is a cone round A whose angle

$AOP=\theta _{0}+\beta _{0},\,$

and another cone round B whose angle

$BOQ=\theta _{0}-\beta _{0}.\,$

Within these cones the axes of the molecules are distributed uniformly. But all the molecules, the original direction of whose axes lay outside of both these cones, have been wrenched from their primitive positions and form a fringe round the cone about A.

If X is greater than D, then the cone round B is completely dispersed, and all the molecules which formed it are converted into the fringe round A, and are inclined at the angle θ₀ + β₀.

445.] Treating this case in the same way as before, we find for the intensity of the temporary magnetization during the action of the force X, which is supposed to act on iron which has never before been magnetized,

When X is less than L,

$I={\frac {2}{3}}M{\frac {X}{D}}.$

When X is equal to L,

$I={\frac {2}{3}}M{\frac {L}{D}}.$

When X is between L and D

$I=M\left\{{\frac {2}{3}}{\frac {X}{D}}+\left(1-{\frac {L^{2}}{X^{2}}}\right)\left[{\sqrt {1-{\frac {L^{2}}{D^{2}}}}}-{\frac {2}{3}}{\sqrt {{\frac {X^{2}}{D^{2}}}-{\frac {L^{2}}{D^{2}}}}}\right]\right\}.$

When X is equal to D,

$I=M\left\{{\frac {2}{3}}+{\frac {1}{3}}\left(1-{\frac {L^{2}}{X^{2}}}\right)^{\frac {3}{2}}\right\}.$

When X is greater than D,

$I=M\left\{{\frac {1}{3}}{\frac {X}{D}}+{\frac {1}{2}}-{\frac {1}{6}}{\frac {D}{X}}+{\frac {(D^{2}-L^{2})^{\frac {3}{2}}}{6X^{2}D}}-{\frac {\sqrt {X^{2}-L^{2}}}{6X^{2}D}}(2X^{2}-3XD+L^{2})\right\}.$

When X is infinite, I = M.

When X is less than L the magnetization follows the former law, and is proportional to the magnetizing force. As soon as X exceeds L the magnetization assumes a more rapid rate of increase on account of the molecules beginning to be transferred from the one cone to the other. This rapid increase, however, soon comes to an end as the number of molecules forming the negative cone diminishes, and at last the magnetization reaches the limiting value M.

If we were to assume that the values of L and of D are different for different molecules, we should obtain a result in which the different stages of magnetization are not so distinctly marked.

The residual magnetization, I' , produced by the magnetizing force X, and observed after the force has been removed, is as follows:

When X is less than L,

No residual magnetization.

When X is between L and D,

$I'=M\left(1-{\frac {L^{2}}{D^{2}}}\right)\left(1-{\frac {L^{2}}{X^{2}}}\right).$

When X is equal to D,

$I'=M\left(1-{\frac {L^{2}}{D^{2}}}\right)^{2}.$

When X is greater than D,

$I'={\frac {1}{4}}\left\{1-{\frac {L^{2}}{XD}}+{\sqrt {1-{\frac {L^{2}}{D^{2}}}}}{\sqrt {1-{\frac {L^{2}}{X^{2}}}}}\right\}^{2}.$

When X is infinite,

$I'={\frac {1}{4}}\left\{1-{\frac {L^{2}}{XD}}+{\sqrt {1-{\frac {L^{2}}{D^{2}}}}}\right\}^{2}.$

If we make

$M=1000,\quad L=3,\quad D=5,\,$

we find the following values of the temporary and the residual magnetization:—

Magnetizing Force.	Temporary Magnetization.	Residual Magnetization.
X	I	I'
0	0	0
1	133	0
2	267	0
3	400	0
4	729	280
5	837	410
6	864	485
7	882	537
8	897	574
≈	1000	810

These results are laid down in Fig. 10.

Fig. 10.

The curve of temporary magnetization is at first a straight line from X = 0 to X = L. It then rises more rapidly till X = 2), and as X increases it approaches its horizontal asymptote.

The curve of residual magnetization begins when X = L, and approaches an asymptote at a distance = .81 M.

It must be remembered that the residual magnetism thus found corresponds to the case in which, when the external force is removed, there is no demagnetizing force arising from the distribution of magnetism in the body itself. The calculations are therefore applicable only to very elongated bodies magnetized longitudinally. In the case of short, thick bodies the residual magnetism will be diminished by the reaction of the free magnetism in the same way as if an external reversed magnetizing force were made to act upon it.

446.] The scientific value of a theory of this kind, in which we make so many assumptions, and introduce so many adjustable constants, cannot be estimated merely by its numerical agreement with certain sets of experiments. If it has any value it is because it enables us to form a mental image of what takes place in a piece of iron during magnetization. To test the theory, we shall apply it to the case in which a piece of iron, after being subjected to a magnetizing force X, is again subjected to a magnetizing force X₁.

If the new force X_l acts in the same direction in which X acted, which we shall call the positive direction, then, if X₁ is less than X₀, it will produce no permanent set of the molecules, and when X_l is removed the residual magnetization will be the same as that produced by X₀. If X₁ is greater than X₀, then it will produce exactly the same effect as if X₀ had not acted.

But let us suppose X₁ to act in the negative direction, and let us suppose X₀ = Z cosec θ₀, and X₁ = –L coesc θ₀.

As X₁ increases numerically, θ₁ diminishes. The first molecules on which X₁ will produce a permanent deflexion are those which form the fringe of the cone round A, and these have an inclination when undeflected of θ₀ + β₀.

As soon as θ₁ – β₀ becomes less than θ₀ + β₀ the process of demagnetization will commence. Since, at this instant, θ₁ = θ₀ + 2β₀, X₁, the force required to begin the demagnetization, is less than XQ, the force which produced the magnetization.

If the value of D and of L were the same for all the molecules, the slightest increase of X₁ would wrench the whole of the fringe of molecules whose axes have the inclination θ₀ + β₀ into a position in which their axes are inclined θ₁ + β₀ to the negative axis OB.

Though the demagnetization does not take place in a manner so sudden as this, it takes place so rapidly as to afford some confirmation of this mode of explaining the process.

Let us now suppose that by giving a proper value to the reverse force X_l we have exactly demagnetized the piece of iron.

The axes of the molecules will not now be arranged indiffer ently in all directions, as in a piece of iron which has never been magnetized, but will form three groups.

(1) Within a cone of semiangle θ₁ – β₀ surrounding the positive pole, the axes of the molecules remain in their primitive positions.

(2) The same is the case within a cone of semiangle θ₀ – β₀ surrounding the negative pole.

(3) The directions of the axes of all the other molecules form a conical sheet surrounding the negative pole, and are at an inclination θ₁ + β₀.

When X₀ is greater than D the second group is absent. When X₁ is greater than D the first group is also absent.

The state of the iron, therefore, though apparently demagnetized, is in a different state from that of a piece of iron which has never been magnetized.

To shew this, let us consider the effect of a magnetizing force X₂ acting in either the positive or the negative direction. The first permanent effect of such a force will be on the third group of molecules, whose axes make angles = θ₁ + β₀ with the negative axis.

If the force X₂ acts in the negative direction it will begin to produce a permanent effect as soon as θ₂ + β₂ becomes less than θ₁ + β₀, that is, as soon as X₂ becomes greater than X₁. But if X₂ acts in the positive direction it will begin to remagnetize the iron as soon as θ₂ + β becomes less than θ₁ + β₀, that is, when θ₂ = θ₁ + 2 β₀, or while X₂ is still much less than X₁.

It appears therefore from our hypothesis that—

When a piece of iron is magnetized by means of a force X₀, its magnetism cannot be increased without the application of a force greater than X₀. A reverse force, less than X₀, is sufficient to diminish its magnetization.

If the iron is exactly demagnetized by a reversed force X₁, then it cannot be magnetized in the reversed direction without the application of a force greater than X₁, but a positive force less than X₁ is sufficient to begin to remagnetize the iron in its original direction.

These results are consistent with what has been actually observed by Ritchie^[5], Jacobi^[6], Marianini^[7], and Joule^[8].

A very complete account of the relations of the magnetization of iron and steel to magnetic forces and to mechanical strains is given by Wiedemann in his Galvanismus. By a detailed comparison of the effects of magnetization with those of torsion, he shews that the ideas of elasticity and plasticity which we derive from experiments on the temporary and permanent torsion of wires can be applied with equal propriety to the temporary and permanent magnetization of iron and steel.

447.] Matteucci^[9] found that the extension of a hard iron bar during the action of the magnetizing force increases its temporary magnetism. This has been confirmed by Wertheim. In the case of soft bars the magnetism is diminished by extension.

The permanent magnetism of a bar increases when it is extended, and diminishes when it is compressed.

Hence, if a piece of iron is first magnetized in one direction, and then extended in another direction, the direction of magnetization will tend to approach the direction of extension. If it be compressed, the direction of magnetization will tend to become normal to the direction of compression.

This explains the result of an experiment of Wiedemann's. A current was passed downward through a vertical wire. If, either during the passage of the current or after it has ceased, the wire be twisted in the direction of a right-handed screw, the lower end becomes a north pole.

Here the downward current magnetizes every part of the wire in a tangential direction, as indicated by the letters NS.

The twisting of the wire in the direction of a right-handed screw causes the portion ABCD to be extended along the diagonal AC and compressed along the diagonal BD. The direction of magnetization therefore tends to approach AC and to recede from BD, and thus the lower end becomes a north pole and the upper end a south pole.

Effect of Magnetization on the Dimensions of the Magnet.

448.] Joule^[10], in 1842, found that an iron bar becomes lengthened when it is rendered magnetic by an electric current in a coil which surrounds it. He afterwards^[11] shewed, by placing the bar in water within a glass tube, that the volume of the iron is not augmented by this magnetization, and concluded that its transverse dimensions were contracted.

Finally, he passed an electric current through the axis of an iron tube, and back outside the tube, so as to make the tube into a closed magnetic solenoid, the magnetization being at right angles to the axis of the tube. The length of the axis of the tube was found in this case to be shortened.

He found that an iron rod under longitudinal pressure is also elongated when it is magnetized. When, however, the rod is under considerable longitudinal tension, the effect of magnetization is to shorten it.

This was the case with a wire of a quarter of an inch diameter when the tension exceeded 600 pounds weight.

In the case of a hard steel wire the effect of the magnetizing force was in every case to shorten the wire, whether the wire was under tension or pressure. The change of length lasted only as long as the magnetizing force was in action, no alteration of length was observed due to the permanent magnetization of the steel.

Joule found the elongation of iron wires to be nearly proportional to the square of the actual magnetization, so that the first effect of a demagnetizing current was to shorten the wire.

On the other hand, he found that the shortening effect on wires under tension, and on steel, varied as the product of the magnetization and the magnetizing current.

Wiedemann found that if a vertical wire is magnetized with its north end uppermost, and if a current is then passed downwards through the wire, the lower end of the wire, if free, twists in the direction of the hands of a watch as seen from above, or, in other words, the wire becomes twisted like a right-handed screw.

In this case the magnetization due to the action of the current on the previously existing magnetization is in the direction of a left-handed screw round the wire. Hence the twisting would indicate that when the iron is magnetized it contracts in the direction of magnetization and expands in directions at right angles to the magnetization. This, however, seems not to agree with Joule's results.

For further developments of the theory of magnetization, see Arts. 832-845.

1. Annals of Electricity, iv. p. 131, 1839; Phil. Mag. [4] ii. p. 316.
2. Pogg., Ann. Lxxix. p. 337, 1850
3. Pogg. cxi. 1860.
4. There is some mistake in the formula given by Weber (Trans. Acad. Sax. i. p. 572 (1852), or Pogg., Ann. Lxxxvii. p. 167 (1852)) as the result of this integration, the steps of which are not given by him. His formula is

   $I=mn{\frac {X}{\sqrt {x^{2}+D^{2}}}}{\frac {x^{4}+{\frac {7}{6}}X^{2}D^{2}+{\frac {2}{3}}D^{4}}{X^{4}+X^{2}D^{2}+D^{4}}}.$

5. Phil. Mag., 1833.
6. Pog., Ann., 1834.
7. Ann. de Chimie et de Physique, 1846.
8. Phil. Trans., 1855, p. 287
9. Ann. de Chimie et de Physique, 1858.
10. Sturgeon's Annals of Electricity, vol. viii. p. 219.
11. Phil. Mag., 1847.