DEPARTMENT OF PHYSICS

INDIAN INSTITUTE OF TECHNOLOGY, MADRAS

PH1020 Physics II Magnetostatics in matter 24.3.2014

Magnetism in material media

Magnetostatics in a material medium is somewhat more complicated than elec-trostatics in a material medium. Magnetism in matter is a phenomenon withmany subtleties, and the variety of magnetic behaviour that materials displayis remarkably diverse. Magnetism and magnetic materials have played a cru-cial role, and continue to do so, in the development of whole areas of physicsranging all the way from theoretical approaches in statistical physics to practicalapplications in device technology.

• As we have already mentioned in class, all magnetic fields ultimately arisefrom the motion of electric charges. At an atomic scale, the motion thatfirst comes to mind is that of electrons moving around atomic nuclei, thusforming tiny current loops. It appears as if this could conceivably be asource of bulk magnetism in matter. More will be said about this furtheron.

• The actual source of magnetism in matter is certainly the existence of el-ementary magnetic dipoles at the atomic and sub-atomic levels. Thesedipoles, however, are not associated exclusively with the orbital motion ofelectrons. Elementary particles such as electrons, protons, neutrons, etc.,have a certain intrinsic angular momentum, called the “spin” of theparticle concerned, that is quite independent of its motion (i.e., its linearmomentum). This spin is an angular momentum that each such particlepossesses even when its linear momentum is zero. It is an intrinsically

quantum mechanical property, and cannot be adequately describedwithin the purview of classical physics. What is of relevance to our im-mediate purpose is that the total angular momentum of objects such aselectrons, atoms, etc., is the resultant of the orbital and spin con-

tributions. Moreover, there is a magnetic dipole moment associated

with the angular momentum of atomic and sub-atomic particles.This is a basic fact that leads to an important contrast with the situationin electrostatics.

• In most dielectric media, the atomic or molecular dipole moments are in-

duced by an applied electric field. As we have mentioned earlier, there doexist materials in which the molecules have permanent electric dipole mo-ments (water is a common example), but this is not always the case. More-over, these permanent dipole moments are generally very weak in strength.

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In contrast, magnetic dipole moments at the atomic and sub-atomic levelare the rule rather than the exception, and their strength can be significant.

• The behaviour of any material under an applied magnetic field is dependenton these elementary magnetic dipoles and on their mutual interaction

with one another. This interaction can be quite complicated, and in-volves many subtle effects. Before that, we develop a basic formalism formagnetostatics in a material medium along the lines followed in the case ofdielectrics.

Analogy with electrostatics

The following useful analogues exist between electrostatics in matter (i.e., di-electrics) and magnetostatics in matter.

Electrostatics in matter Magnetostatics in matter

p – electric dipole moment m – magnetic dipole momentFor a loop of area A in the yz-planewith current I flowing anti-clockwise,

m = IAexP – Polarization M – Magnetization

≡ elec. dipole moment/unit volume ≡ magnetic dipole moment/unit volumeForce on a dipole in an electric field, Force on a dipole in a magnetic field,Felec = ∇(p · E) = (p · ∇)E Fmag = ∇(m ·B) = (m · ∇)B+m× (∇×B)

since ∇× E = 0 if the loop is infinitesimal.But unlike in electrostatics, ∇×B = µ0J

and thus ∇(m ·B) 6= (m · ∇)B unless Jhappens to vanish at the location of m

Torque on a dipole in an electric field, ‘ Torque on a dipole in a magnetic field,τ = p× E N = m×B

It tends to align p along E and It tends to align m along B.The scalar potential Φ has a The vector potential A has amultipole expansion. multipole expansion. The monopole term

The vector potential of asingle dipole isA = (µ0/4π)(m× er)/r

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as shown earlier. Therefore, each volumeelement dV carries a dipole moment MdVwith the total vector potential

A = (µ0/4π)∫

dV (M× er)/r2.

As in dielectrics, we can do an integrationby parts and getA = µ0

4π

∫

1r(∇×M)dV + µ0

4π

∮

1r(M× dS)

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Electrostatics Magnetostatics

ρp and σp are the bound volume Similarly, from the last equation aboveand surface charge densities one can define bound volume andgiven by surface currents given by

ρp = −∇ ·P Jb = ∇×M , Kb = M× n ,σp = P · n . where n is the unit outward normal to the

surface bounding the volume. We see that∇ · Jb = 0 since ‘div-curl’ is zero.

These are bound currents because eachcharge is bound to a single atom in whichit moves over a tiny loop, but in effect amacroscopic resultant current flows due toall such charges.

ρ = ρp + ρf J = Jb + Jf

where ρf is the free volume where Jf is the volume currentcharge density due to free charges. Thus,

1µ0

(∇×B) = J = Jf + (∇×M)

or ∇× [ 1µ0

B−M] = Jf

D = ǫ0E+P H ≡ 1µ0

B−M

H is called the auxiliary field.Gauss’ law for dielectrics, Ampere’s law in magnetostatics in matter,

∇ ·D = ρf ∇×H = Jf .Thus,

∮

D · da = (Qf )enclosed. Thus,∮

H · dl = (If )enclosed, where(If)enclosed is the total free current throughan Amperian loop.

For LIH dielectrics, For LIH magnetic materialsE = (1/ǫ)D = (1/ǫ0)(D−P) B = µH = µ0(H+M)dielectric permittivity: ǫ = ǫ0κ µ is the magnetic permeability

dielectric constant: κ = (1 + χ) We can write M = χmH, where χm is theχ = dielectric susceptibility. magnetic susceptibility. It is positive for

paramagnets and negative for diamagnets.

In vacuum In vacuumǫ = ǫ0, κ = 1 ⇒ χ = 0 µ = µ0 ⇒ χm = 0

For linear dielectrics, ρp is related to ρf In linear magnetic materials,If ρf = 0, then ρp = 0. Jb = ∇×M = χmJf

Jb = 0 if Jf = 0.

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Electrostatics Magnetostatics

Fundamental equations Fundamental equations∇×E = 0, ∇ ·B = 0,

since E is conservative. since there are no magnetic monopoles.Further, in a region without Further in a region withoutfree electric charges free currents,

∇ ·D = 0 . ∇×H = 0 .However, ∇×D 6= 0 in general – However, ∇ ·H 6= 0 in general –for instance at the interface separating for instance at interfaces separatingtwo dielectric media. two magnetic media.In the presence of free charges In the presence of free currents

∇ ·D = ρf . ∇×H = Jf .As a result, the normal component of As a result, the tangential component ofD may be discontinuous at an interface H may be discontinuous at an interfaceseparating two dielectrics separating two magnetic media(D1n −D2n) ∝ surface charge density (H1t −H2t) ∝ surface current density

E1t = E2t B1n = B2n

Note: The above equations are subject to the the continuity equation

∇ · Jf = 0 .

Note that∂ρf∂t

= 0 (because we are dealing with the static case at present).

Paramagnetism, diamagnetism, ferromagnetism

We have said that the magnetic behaviour of a material medium is dependent onthe mutual interaction between the elementary magnetic dipoles in themedium. We now describe in brief the most common situations in this regard, inthe most simplified version possible.

♠ The role of temperature Whatever be the nature of this interaction, thereis some characteristic energy scale associated with it, in general. Suppose thisenergy per atom or dipole has a magnitude ε. Then, in a medium at an abso-lute temperature T , the behaviour of the medium depends on whether kBT ≫ ε(“high temperatures”) or kBT ≪ ε (“low temperatures”). We may expect a“phase transition” from one kind of behaviour to another when kBT is of theorder of ε, under suitable circumstances.

♠ Paramagnetism At sufficiently high temperatures, the effect of the mutualinteraction between the elementary magnetic dipoles may be neglected. We thenhave the simplest case: a collection of independent magnetic dipoles that canbe influenced by an applied magnetic field. This situation, called paramag-

netism, is the immediate analogue of dielectric polarization: here the atomic

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magnetic dipole moments in the medium are aligned by an applied magneticfield to produce a net dipole moment per unit volume, i.e., the magnetizationM, just as a dielectric polarization P is produced by an applied electric field E;and M vanishes when B is absent, just as P vanishes when E is absent. In thedemonstrations we saw the paramagnetic behavior of liquid oxygen.

We reiterate that the linear relationship between M and H is only true forsufficiently small values of the magnitudes of these quantities, in a paramagnet.At higher values of the magnitude of H, for a fixed temperature, the magneti-zation of the medium tends to its saturation value: namely, the value it wouldhave if all the elementary dipoles in it were free to align themselves along thedirection of the applied field. (As in the case of dielectric polarization, at anynon-zero temperature the thermal vibrations of the atoms of the medium preventcomplete alignment of all the dipoles under the influence of any finite appliedmagnetic field.) Figure 1 shows schematically how Mi , the component of themagnetization in a given direction (labelled by the Cartesian index i), varieswith the corresponding component Hi of the auxiliary field. The two plots corre-spond to two different temperatures T1 and T2 , where T1 > T2 . It is obvious onphysical grounds that it takes a larger field to produce the same magnetization(or degree of alignment of the elementary dipoles) at a higher temperature, be-cause the field has to compete against the disorder caused by thermal agitation.

T2

T1

T2

T1

O Hi

Mi

Figure 1: Schematic plot of Mi vs Hi at two different temperatures T1 > T2.

¶ Correct definition of susceptibility The definition of magnetic suscepti-bility in LIH materials is

M = χmH . (1)

Since Eq. (1) is based on the assumption that M is directly proportional to H . Itis obvious from Fig. 1 that this linearity is only valid for sufficiently small values ofthe field strength, i.e., near the origin. Therefore the slope of the magnetizationat the origin would be a more correct definition of the susceptibility. As this

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slope depends on the temperature, we must call it the isothermal magneticsusceptibility and define it as

χm =

[(

∂Mi

∂Hi

)

T

]

Hi=0

(2)

It can be shown on very general grounds that, at very high temperatures (suchthat kBT ≫ ε), the paramagnetic susceptibility χm decreases like 1/T with in-creasing T . This is called Curie’s Law.

♠ Diamagnetism In most materials, the application of an external magneticfield sets up currents at the atomic or molecular level, by affecting the motionof charges. Recall that a magnetic field leads to a force on a charge q movingwith a velocity v that is given by q (v×B). This causes a looping motion of thecharges, and the resultant of all these atomic level motions is a net current. Thiscurrent, in turn, sets up a magnetic field that counters the effect of the originalapplied field, in accordance with Le Chatelier’s Principle. In other words, themagnetization induced in the medium as a result if this effect is in a directionopposite to that of the applied magnetic field.

In general, this effect is a weak one, and is masked by stronger paramagneticmagnetization. In certain materials, however, it is the dominant effect. In thedemonstations we saw the diamagnetic behavior of liquid nitrogen. These mate-rials therefore have a negative magnetic susceptitbility. Such materials are saidto be diamagnetic.

♠ Ferromagnetism At sufficiently low temperatures (such that kBT ≪ ε), thebehaviour of the elementary dipoles in a medium is controlled by the interactionbetween them. In many substances, this leads to an ordered arrangement ofthe dipoles. If this ordered arrangement is such that neighbouring dipoles arealigned parallel to each other, on the average, the result is a net magnetizationof the medium. This magnetization continues to exist even if the applied fieldis switched off. This is called “permanent magnetization” or “spontaneousmagnetization”, and the medium is said to be ferromagnetic. Figure 2 showsthe behaviour of M as a function of H for a typical ferromagnetic material, inthe ideal case. Figure 3 shows the behaviour of M when the phenomenon ofmagnetic hysteresis is taken into account. Both curves are highly schematic. Realmagnets display more complicated features which we do not go into here.

♠ Phase transition We have described in general terms what happens in amagnetic medium at both high and low temperatures. This suggests that atsome intermediate temperature, such that kBT is of the order of ε, the materialundergoes a change or phase transition from a paramagnet to a ferromagnet.The temperature at which the transition occurs is called theCurie temperature

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O H

M

Figure 2: Schematic plot of M vs H for an ideal ferromagnet

OC C

R

R

H

M

Figure 3: Schematic plot of M vs H for an ideal ferromagnet with hysteresis.The magnitude of M (points labelled R) at zero H is called Remanence whilethe magnitude of H that is needed to reverse the sign of M (points labelled C)is called Coercivity.

or critical temperature, and is denoted by TC . At temperatures above TC ,the effects of thermal agitation dominate, and the material is paramagnetic. Attemperatures below TC , the ordering effect of the interaction between elemen-tary dipoles dominates, and the material is ferromagnetic. The basic property ofa ferromagnetic material is that it can have a permanent magnetization. Iron,cobalt and nickel are materials that exhibit ferromagnetism. The Curie tempera-ture of Nickel is around 630K. In the demonstrations, we saw a Nickel permanentmagnet lose its ferromagnetism on being heated (using a blow torch) past itsCurie temperature.

¶ Real ferromagnets are also subject to the formation of so-called domains. Ingeneral, the permanent magnetization differs from domain to domain, and thenet permanent magnetization may cancel out in a physical specimen, unless othereffects such as hysteresis prevent it from vanishing.

¶ The interaction between elementary dipoles in material media can also lead

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to several forms of ordered arrangements of these dipoles other than the ferro-magnetic one. These correspond to phenomena such as anti-ferromagnetism,ferrimagnetism, etc.

¶ Finally, it is necessary to add that, almost invariably, the interaction betweenelementary dipoles that leads to magnetic ordering has a purely quantum

mechanical origin. This interaction is not a direct dipole-dipole interactionbetween pairs of elementary dipoles at the classical level. The phenomenon ofmagnetism may be describable under suitable circumstances in terms of classicalphysics, to a good approximation. But we must bear in mind that its origin isquantum mechanical. No bulk magnetism is possible strictly within the

laws of classical physics! This is called the Bohr-van Leeuwen Theorem.

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Summary of magnetostatics in a medium

♠ Magnetization The magnetic dipole moment per unit volume, or the mag-netization M , is related to the volume and surface bound current densitiesaccording to

∇×M = Jb and M× n = Kb , (3)

where the second equation is valid at points on any surface separating the mediumand free space, and n is the unit outward normal from the medium to free space.Further, ∇ · Jb ≡ 0 since Jb is the curl of a vector field.

♠ The auxiliary field H The auxiliary field H is defined in terms of themagnetic induction B and the magnetization M by

H =B

µ0

−M . (4)

♠ Maxwell’s Equations The two basic equations that determine magnetostat-ics in a medium are

∇ ·B(r) = 0 and ∇×H(r) = Jf , (5)

where Jf is the free volume current density. Both B and H are pseudovectorsrather than vectors: they do not change sign under a parity transformation ofthe coordinates.

♠ Boundary conditions The boundary conditions at an interface with a sur-face density of free current Kf are

(B2 −B1) · n = 0 and n× (H2 −H1) = Kf , (6)

where n is the unit normal to the interface directed from interface 1 to 2.

♠ Energy density The magnetostatic energy density is given by

w =1

2B ·H . (7)

♠ Constitutive relation For a linear, isotropic, homogeneous (LIH) magneticmedium, H is directly proportional to B :

B = µH . (8)

µ is the permeability of the medium..

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