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

by James Clerk Maxwell

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 opposite 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 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 demagnetized 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 magnetization on the assumption that the magnetization of any particle of the substance depends solely on the magnetic force acting on that particle. This magnetic force may arise partly from external causes, and partly from the temporary magnetization of neighbouring 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 ^[1].

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 Paramagnetic, 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 -1/400000 in the case of 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 components 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 equations 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. As the magnetic force increases, however, the magnetization in creases more slowly, and it would appear from experiments described in Chapter 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 ${\mathfrak {H}}$ be the magnetic force, denned as in Art. 398, at any point of the body, and let ${\mathfrak {J}}$ be the magnetization at that point, then the ratio of ${\mathfrak {J}}$ to ${\mathfrak {H}}$ is called the Coefficient of Induced Magnetization.

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

${\mathfrak {J}}=\kappa {\mathfrak {H}}.$

(1)

The coefficient κ 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 ${\mathfrak {H}}$ 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 Ω be that due to the induced magnetization, then if U is the actual potential due to both causes

$U=V+\Omega .\,$

(2)

Let the components of the magnetic force ${\mathfrak {H}}$, resolved in the directions of x, y, z, be α, β, γ, and let those of the magnetization ${\mathfrak {J}}$ be A, B, C, then by equation (1),

${\begin{aligned}A=\kappa \alpha ,\\B=\kappa \beta ,\\C=\kappa \gamma .\end{aligned}}$

(3)

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

$Adx+Bdy+Cdz=\kappa (\alpha dx+\beta dy+\gamma dz).\,$

But since α, β, γ, are derived from the potential U, we may write the second member –κ dU.

Hence, if κ is constant throughout the substance, the first member must also be a complete differential of a function of x, y and z, which we shall call φ, and the equation becomes

$d\phi =-\kappa dU,\,$

(4)

where

$A={\frac {d\phi }{dx}},\quad B={\frac {d\phi }{dy}},\quad C={\frac {d\phi }{dz}}.$

(5)

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

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

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

which becomes in virtue of equations (3),

$\rho =-\kappa \left({\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}\right).$

But, by Art. 77,

${\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}=4\pi \rho .\,$

Hence

$(1+4\pi \kappa )\rho =0,\,$

whence

$\rho =0\,$

(6)

throughout the substance, and the magnetization is therefore solenoidal as well as lamellar. See Art. 407.

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

$\sigma =-{\frac {d\phi }{d\nu }}.$

(7)

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

$\Omega =\iint {{\frac {d\sigma }{r}}dS}.$

(8)

The value of Ω 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 Ω outside the surface, and if ν' be the normal drawn outwards, we have at the surface

$\Omega '=\Omega ;\,$

(9)

${\begin{aligned}{\frac {d\Omega }{d\nu }}+{\frac {d\Omega '}{d\nu '}}&=-4\pi \sigma ,{\text{ by Art. 78}},\\&=4\pi {\frac {d\phi }{d\nu }},{\text{ by (7)}},\\&=-4\pi \kappa {\frac {dU}{d\nu }},{\text{ by (4)}},\\&=-4\pi \kappa \left({\frac {dV}{d\nu }}+{\frac {d\Omega }{d\nu }}\right),{\text{ by (2)}}.\end{aligned}}$

We may therefore write the surface-condition

$(1+4\pi \kappa ){\frac {d\Omega }{d\nu }}+{\frac {d\Omega '}{d\nu '}}+4\pi \kappa {\frac {dV}{d\nu }}=0.$

(10)

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

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

Within the surface S, Ω must be finite and continuous, and must satisfy Laplace's equation.

Outside the surface S, Ω' 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, Ω = Ω', and the derivatives of Ω, Ω' 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 κ, but is related to it as follows:

$4\pi \kappa (k-1)+3k=0.\,$

(11)

The coefficient κ 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 ${\mathfrak {B}}$, the magnetic induction, ${\mathfrak {H}}$, the magnetic force, and ${\mathfrak {J}}$, the magnetization, is expressed by the equation

${\mathfrak {B}}={\mathfrak {H}}+4\pi {\mathfrak {J}}.$

(12)

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

${\mathfrak {J}}=\kappa {\mathfrak {H}}.$

(13)

Hence, eliminating ${\mathfrak {J}}$, we find

${\mathfrak {B}}=(1+4\pi \kappa ){\mathfrak {J}}$

(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 κ may be a function, not only of the position of the point in the substance, but of the direction of the vector ${\mathfrak {H}}$, but in the case which we are now considering κ is a numerical quantity.

If we next write

$\mu =1+4\pi \kappa ,$

(15)

we may define μ 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 κ, the co efficient of induced magnetization.

If we write U for the total magnetic potential compounded of V, the potential due to external causes, and Ω for that due to the induced magnetization, we may express a, b, c, the components of magnetic induction, and α, β, γ, the components of magnetic force, as follows:

${\begin{aligned}&a=\mu \alpha =-\mu {\frac {dU}{dx}},\\&b=\mu \beta =-\mu {\frac {dU}{dy}},\\&c=\mu \gamma =-\mu {\frac {dU}{dz}}\end{aligned}}$

(16)

The components a, b, c satisfy the solenoidal condition

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}=0.$

(17)

Hence, the potential U must satisfy Laplace's equation

${\frac {d^{2}U}{dx^{2}}}+{\frac {d^{2}U}{dy^{2}}}+{\frac {d^{2}U}{dz^{2}}}=0$

(18)

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

At the surface itself, if ν is a normal drawn towards the magnetic substance, and ν' one drawn outwards, and if the symbols of quantities outside the substance are distinguished by accents, the condition of continuity of the magnetic induction is

$a{\frac {d\nu }{dx}}+b{\frac {d\nu }{dy}}+c{\frac {d\nu }{dz}}+a'{\frac {d\nu }{dx}}+b'{\frac {d\nu }{dy}}+c'{\frac {d\nu }{dz}}=0;$

(19)

or, by equations (16),

$\mu {\frac {dU}{d\nu }}+\mu '{\frac {dU'}{d\nu '}}=0.$

(20)

μ', 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 Ω, and for μ its value in terms of κ, 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, precisely 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 magnetic 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 corresponds 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 μ.

In empty space the coefficient of induction is unity. In bodies capable of induced magnetization the coefficient of induction is 1 + 4πκ = μ, where κ is the quantity already defined as the coefficient of induced magnetization.

429.] Let μ, μ' be the values of μ on opposite sides of a surface separating two media, then if V, V' are the potentials in the two media, the magnetic forces towards the surface in the two media are ${\frac {dV}{d\nu }}$ and ${\frac {dV'}{d\nu '}}$.

The quantities of magnetic induction through the element of surface dS are $\mu {\frac {dV}{d\nu }}dS$ and $\mu '{\frac {dV'}{d\nu '}}dS$ in the two media respectively reckoned towards dS.

Since the total flux towards dS is zero,

$\mu {\frac {dV}{d\nu }}+\mu '{\frac {dV'}{d\nu '}}=0.$

But by the theory of the potential near a surface of density σ,

${\frac {dV}{d\nu }}+{\frac {dV'}{d\nu '}}+4\pi \sigma =0.$

Hence

${\frac {dV}{d\nu }}\left(1-{\frac {\mu }{d\mu '}}\right)+4\pi \sigma =0.$

If κ₁ is the ratio of the superficial magnetization to the normal force in the first medium whose coefficient is μ, we have

$4\pi \kappa _{1}={\frac {\mu -\mu '}{\mu '}}.$

Hence κ₁ will be positive or negative according as μ is greater or less than μ'. If we put μ = 4πκ + 1 and μ' = 4πκ' + 1,

$\kappa _{1}={\frac {\kappa -\kappa _{1}}{4\pi \kappa +1}}.$

In this expression κ and κ' are the coefficients of induced magnetization of the first and second medium deduced from experiments made in air, and κ₁ is the coefficient of induced magnetization of the first medium when surrounded by the second medium.

If κ' is greater than κ, then κ₁ is negative, or the apparent magnetization of the first medium is in the opposite direction from the magnetizing force.

Thus, if a vessel containing a weak aqueous solution of a paramagnetic salt of iron is suspended in a stronger solution of the same salt, and acted on by a magnet, the vessel moves as if it were magnetized in the opposite direction from that in which a magnet would set itself if suspended in the same place.

This may be explained by the hypothesis that the solution in the vessel is really magnetized in the same direction as the magnetic force, but that the solution which surrounds the vessel is magnetized more strongly in the same direction. Hence the vessel is like a weak magnet placed between two strong ones all magnetized in the same direction, so that opposite poles are in contact. The north pole of the weak magnet points in the same direction as those of the strong ones, but since it is in contact with the south pole of a stronger magnet, there is an excess of south magnetism in the neighbourhood of its north pole, which causes the small magnet to appear oppositely magnetized.

In some substances, however, the apparent magnetization is negative even when they are suspended in what is called a vacuum.

If we assume κ = 0 for a vacuum, it will be negative for these substances. No substance, however, has been discovered for which κ has a negative value numerically greater than ${\frac {1}{4\pi }}$, and therefore for all known substances μ. is positive.

Substances for which κ is negative, and therefore μ less than unity, are called Diamagnetic substances. Those for which is κ positive, and μ greater than unity, are called Paramagnetic, Ferromagnetic, or simply magnetic, substances.

We shall consider the physical theory of the diamagnetic and paramagnetic properties when we come to electromagnetism, Arts. 831-845.

430.] The mathematical theory of magnetic induction was first given by Poisson^[2]. The physical hypothesis on which he founded his theory was that of two magnetic fluids, an hypothesis which has the same mathematical advantages and physical difficulties as the theory of two electric fluids. In order, however, to explain the fact that, though a piece of soft iron can be magnetized by induction, it cannot be charged with unequal quantities of the two kinds of magnetism, he supposes that the substance in general is a non-conductor of these fluids, and that only certain small portions of the substance contain the fluids under circumstances in which they are free to obey the forces which act on them. These small magnetic elements of the substance contain each precisely equal quantities of the two fluids, and within each element the fluids move with perfect freedom, but the fluids can never pass from one magnetic element to another.

The problem therefore is of the same kind as that relating to a number of small conductors of electricity disseminated through a dielectric insulating medium. The conductors may be of any form provided they are small and do not touch each other.

If they are elongated bodies all turned in the same general direction, or if they are crowded more in one direction than another, the medium, as Poisson himself shews, will not be isotropic. Poisson therefore, to avoid useless intricacy, examines the case in which each magnetic element is spherical, and the elements are disseminated without regard to axes. He supposes that the whole volume of all the magnetic elements in unit of volume of the substance is k.

We have already considered in Art. 314 the electric conductivity of a medium in which small spheres of another medium are distributed.

If the conductivity of the medium is μ₁, and that of the spheres μ₂, we have found that the conductivity of the composite system is

$\mu =\mu _{1}{\frac {2\mu _{1}+\mu _{2}+2k(\mu _{2}-\mu _{1})}{2\mu _{1}-\mu _{2}-k(\mu _{2}-\mu _{1})}}.$

Putting μ₁ = 1 and μ₂x = ∞, this becomes

$\mu ={\frac {1+2k}{1-k}}.$

This quantity μ is the electric conductivity of a medium consisting of perfectly conducting spheres disseminated through a medium of conductivity unity, the aggregate volume of the spheres in unit of volume being k.

The symbol μ also represents the coefficient of magnetic induction of a medium, consisting of spheres for which the permeability is infinite, disseminated through a medium for which it is unity.

The symbol k, which we shall call Poisson's Magnetic Coefficient, represents the ratio of the volume of the magnetic elements to the whole volume of the substance.

The symbol κ is known as Neumann's Coefficient of Magnetization by Induction. It is more convenient than Poisson's.

The symbol μ we shall call the Coefficient of Magnetic Induction. Its advantage is that it facilitates the transformation of magnetic problems into problems relating to electricity and heat.

The relations of these three symbols are as follows:

${\begin{aligned}&k={\frac {4\pi \kappa }{4\pi \kappa +3}},\quad k={\frac {\mu -1}{\mu +2}}\\&\kappa ={\frac {\mu -1}{4\pi }},\quad \kappa ={\frac {3k}{4\pi (1-k)}},\\&\mu ={\frac {1+2k}{1-k}},\quad \mu =4\pi \kappa +1.\end{aligned}}$

If we put κ = 32, the value given by Thalen's^[3] experiments on soft iron, we find $k={\frac {134}{135}}$. This, according to Poisson's theory, is the ratio of the volume of the magnetic molecules to the whole volume of the iron. It is impossible to pack a space with equal spheres so that the ratio of their volume to the whole space shall be so nearly unity, and it is exceedingly improbable that so large a proportion of the volume of iron is occupied by solid molecules whatever be their form. This is one reason why we must abandon Poisson's hypothesis. Others will be stated in Chapter VI. Of course the value of Poisson s mathematical investigations remains unimpaired, as they do not rest on his hypothesis, but on the experimental fact of induced magnetization.

1. * Thatén, Nova cta, Reg. Soc. Sc., Upsal., 1863.
2. Mémoires de l'Institut, 1824.
3. Recherches sur les Propriétes Magnétiques du fer, Nova Acta, Upsal, 1863.