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Chapter 5

Magnetic Materials

In the previous chapter, we have considered currents that were able to move on a mesoscopicscale (i.e. on the order of µm). Such currents are called free currents. These are due to themovement of free charges.

However, in some media found in nature, one cannot neglect the influence of bound currents,that is to say currents which displacements occur on a maximum scale of the order of the nm.Indeed, although bound, these currents can significantly contribute to the mesoscopically-averagedcurrent densities.

Once again, Maxwell’s equations stated in chapter 1 are still valid if one considers both freeand bound charges and currents. Like in chapter 3, however, we will here see that it is often morepractical to express Maxwell’s equations under the form of what are called “Maxwell’s equationsin material media” where the electric and magnetic field are replaced by the electric displacement

field−→D and by the magnetizing field

−→H .

In this chapter, we will first examine the general expression and consequences of Maxwell’sequations. We will then look more specifically into non dielectric materials (εr = 1). Some media,however, are both dielectrics and magnetic materials, in which case the results obtained in chapter3 will have to be combined with those given in the present chapter in order to achieve a completedescription of the material behavior in electromagnetic field. We will end by giving an overview ofthe microscopic phenomena which can engender a magnetic behavior in materials.

5.1 Maxwell’s equations in magnetic materials

5.1.1 Magnetization vector

5.1.1.a Definition

Under the influence of an external magnetic field, bound charges can be set into motion ona nanometer scale. Thus appear nanoscopic current loops, i.e. nanoscopic magnetic dipoles.Materials capable of magnetizing themselves in such a way are called magnetic materials.

Definition : In a magnetic material, the magnetization vector−→M is the mesoscopically-

averaged 1 magnetic dipole moment per unit volume :

−→M =

d−→M

dV(5.1)

It is expressed in A.m−1.

1. See section 1.7.

108

5.1. MAXWELL’S EQUATIONS IN MAGNETIC MATERIALS 109

This quantity is directly related to the magnetic dipoles studied in section 4.3.

Indeed, let us consider an elemental volume dV containing a large number of magnetic dipoles

{−→Mk}k, then :

−→M =

1

dV

∑

k

−→Mk

=1

dVdN〈

−→Mk〉

=d−→M

dV

where dN is the number of dipoles contained in dV and 〈−→Mk〉 the average magnetic dipole moment

inside the volume dV .

♣ Example : Lattice of current loops

Let us illustrate the mechanism at play in mag-netic materials on a simple - and simplistic - exam-ple. We will consider the opposite 3D array of cur-rent loops, placed at abscissae xn = na, and throughwhich flows a current In = nI0.

Let us calculate the magnetization vector. Inthis case :

−→M =

d−→M

dV=

nI0a2

a3−→uz

=I0xn

a2−→uz

=I0x

a2−→uz

Note that the averaging for the calculation of−→M is done on a scale ≫ a.

The effective current - the magnetization current - can be calculated at point xn by summingthe branches of adjacent loops :

−→J M =

dI

dS−→uy = −

I0 × number of wires in dS

dS−→uy

= −I0

dSa2

dS−→uy = −

I0

a2−→uy

On this simple example, one can verify the relation−→J M =

−→rot

−→M which we will now prove in

the general case.

5.1.1.b Magnetization current density

In this paragraph, we would like to derive general formulae for the magnetization current den-sity. We will therefore consider a magnetic material of volume (V ) and surface (Σ), as schematizedfigure 5.1.

110 CHAPTER 5. Magnetic Materials

Figure 5.1: Two equivalent ways of dealing with a magnetic material : either consider the magne-tization or think in terms of magnetization current densities (in volume and on the surface).

At point M , the vector potential created by a single magnetic dipole−→M placed at point Q is

(equation 4.19):

−→A (M) =

µ0

4π

−→M∧

−−→QM

QM3

The vector potential created by the entire magnetic material can then be written as :

−→A (M) =

µ0

4π

˚

(V )

−→M∧

−−→QM

QM3

dN

dVdV

with dN the average number of magnetic dipoles per unit volume. Hence :

−→A (M) =

µ0

4π

˚

(V )

−→M ∧

−−→QM

QM3dV (5.2)

We have seen in section 3.1.1.b that :−−→QM

QM3= −

−−→gradM

(

1

QM

)

=−−→gradQ

(

1

QM

)

And we know from vector analysis that−−→curl (α

−→A ) = α

−−→curl

−→A +

−−→grad α ∧

−→A

which gives :−→M ∧

−−→QM

QM3=−→M ∧

−−→gradQ

(

1

QM

)

=

−−→curl

−→M

QM−−−→curl

( −→M

QM

)

So that :

−→A (M) =

µ0

4π

˚

(V )

−−→curl

−→M

QMdV −

µ0

4π

˚

(V )

−→rot

( −→M

QM

)

dV

=µ0

4π

˚

(V )

−−→curl

−→M

QMdV −

µ0

4π

‹

(Σ)

−→n ∧−→M

QMdS

−→A (M) =

µ0

4π

˚

(V )

−−→curl

−→M

QMdV +

µ0

4π

‹

(Σ)

−→M ∧ −→n

QMdS


As can be seen from the last expression, the vector potential created by the magnetized material

is equivalent to the one created by the combination of a volume current density−→J M and a surface

current density−→J s

M :

−→J M =

−−→curl

−→M (5.3)

−→J s

M =−→M ∧ −→n (5.4)

Figure 5.2: Schematization of surface currents in a magnet. The bulk current density is notrepresented.

⋆ A few comments on these expressions:• If the magnetization is non uniform, its curl gives the net current density appearing in thematerial.

• These currents are real currents. They are called “magnetization currents” to distinguishthem from “free” currents, but both represent a physical reality. In a magnet, for instance,there can be no free currents circulating through the sample, but the magnetization induces

surface currents (see figure 5.2) : if the magnetization−→M is constant,

−→J M =

−→0 , whereas

−→J s

M =−→M ∧ −→n =M−→uz ∧

−→ur = −M−→uθ.

• As stated on numerous occasions, Maxwell’s equations hold for magnetic materials. However,the volume current density which has to be considered is the total volume current density

which also includes the magnetization current density :−→J tot =

−→J free +

−→J M .

5.1.2 Maxwell’s equations in material media

5.1.2.a Charges and Current distributions

Taking into account the contributions of both free and bound charges, one has the total chargedensity :

ρ = ρfree + ρP

ρ = ρfree − div−→P (5.5)

and the total current density :

−→J =

−→J free +

−→J P +

−→J M

−→J =

−→J free +

∂−→P

∂t+−−→curl

−→M (5.6)

By substituting these total charge and current densities into Maxwell-Gauss and Maxwell-Ampere equations, one obtains :

div−→E =

ρ

ε0−div−→P

ε0

div−→D = ρfree


and :

−−→curl

−→B = µ0

−→J + ε0µ0

∂−→E

∂t

= µ0−→J free + µ0

−−→curl

−→M + µ0

∂−→P

∂t+ ε0µ0

∂−→E

∂t

−−→curl

(−→B

µ0−−→M

)

=−→J free +

∂−→D

∂t

5.1.2.b H-vector

This leads us to define the H-vector.

Definition : The magnetic excitation vector−→H is defined by :

−→H =

−→B

µ0−−→M (5.7)

It is expressed in A.m−1.

The H-vector represents the properties of bound charges appearing when a magnetic materialis magnetized.

The terminology for the magnetic-field related vectors is sometimes confused. The importantthing to remember are the equations and their meaning. The actual name given to the differentvectors does not really matter as long as you do not get confused. Here are however the namesthat can be found in the literature. In bold are the names we will adopt from now on:

Symbol Unit Possible name

M A.m−1 Magnetization

Magnetic polarization (Weber)B T Magnetic field (Feynman)

Magnetic induction (Slater, Pauli, Feynman)Magnetic flux intensity (Jackson, Weber)

H A.m−1 H-vector

Magnetic excitationMagnetic field (Slater, Jackson, Feynman)Magnetic field intensity (Pauli)Magnetic intensity (Slater, Weber)Magnetizing force (Weber)

5.1.2.c Maxwell’s equations

Maxwell’s equations in material media can therefore be written :

Maxwell-Gauss : div−→D =

−→∇ .−→D = ρfree (5.8)

Conservation of Magnetic Flux : div−→B =

−→∇ .−→B = 0 (5.9)

Maxwell-Faraday :−−→curl

−→E =

−→∇ ∧

−→E = −

∂−→B

∂t(5.10)

Maxwell-Ampere :−−→curl

−→H =

−→∇ ∧

−→H =

−→J free +

∂−→D

∂t(5.11)


⋆ A few comments on these equations :• Note that in these expressions, the charge and current densities are those corresponding to

the free charges. The bound charges and current densities are included in−→D and

−→H .

• One can either choose to write Maxwell’s equations in the general case (equations 1.4 to 1.7)

with ρ and−→J being the total charge and current densities OR use the above expressions

where only the free charge and current densities appear. Please do not mix both expressions...

5.1.3 Consequences of Maxwell’s equations

The conservation of magnetic flux and Maxwell-Faraday equations are unchanged so that :

‹

(S)

−→B.−→dS = 0 (equation 1.44)

and Faraday’s law still holds :

˛

(C)

−→E .−→dl = −

∂φ

∂t= e (equation 1.46)

Maxwell-Gauss equation is the same than what we saw in chapter 3, so that Gauss theorem forthe electric displacement field is still valid :

‹

(S)

−→D.−→dS = Qfree,int (equation 3.12)

5.1.3.a Generalized Ampere’s theorem

However Maxwell-Ampere’s law has been modified and Ampere’s theorem can therefore berephrased 2.

Let us consider a surface (S) delimited by a closed countour (C) which contains a free current

volume density−→J free, and a total free current Ifree. The generalized Ampere’s law can be

expressed :˛

(C)

−→H.−→dl = Ifree,(S) +

¨

(S)

∂−→D

∂t.−→dS (5.12)

♦ Proof :

˛

(C)

−→H.−→dl =

¨

(S)

−−→curl

−→H.−→dS

=

¨

(S)

(

−→J free +

∂−→D

∂t

)

.−→dS = Ifree,(S) +

¨

(S)

∂−→D

∂t.−→dS

In the quasi-steady state regime :

˛

(C)

−→H.−→dl = Ifree,(S) (5.13)

2. Once again, the original formulation seen in section 1.4 holds, provided the charge and current distributionsused are the total ones.


5.1.3.b Discontinuity equations

The discontinuity equations are obtained in a similar manner as in sections 1.5.2 and 1.5.3. Fortwo media (1) and (2) separated by an interface characterized by a unit vector −→n 1→2, with a free

surface charge density σfree and a free surface current density−→J s,free :

div−→D = ρfree →

−−→D2,n −

−−→D1,n = σfree

−→n 1→2 (5.14)

−→rot

−→E = −

∂−→B

∂t→

−−→E2,t −

−−→E1,t =

−→0 (5.15)

div−→B = 0 →

−−→B2,n −

−−→B1,n =

−→0 (5.16)

−→rot

−→H = µ0

−→J free +

∂−→D

∂t→

−−→H2,t −

−−→H1,t =

−→J s,free ∧

−→n 1→2 (5.17)

⋆ A few comments on these equations :• The bound surface charge density can still be expressed (equation 3.19) :

−−→P2,n −

−−→P1,n = −σP

−→n 1→2

• The bound surface current density can be retrieved :

−→J s

M =−→M ∧ −→n 1→2

♦ Proof :

−−→H2,t −

−−→H1,t =

−→J s,free ∧

−→n 1→2

−−→B2,t

µ0−

−−→B1,t

µ0−−−→M2,t +

−−→M1,t =

(−→J s,tot −

−→J s

M

)

∧ −→n 1→2

−−−→M2,t +

−−→M1,t = −

−→J s

M ∧ −→n 1→2

5.2 Linear magnetic materials

From this point on, we will concentrate on non-dielectric magnetic materials (εr = 1 ; µr 6= 1).

5.2.1 Constitutive relations

The above equations describe the relations between−→E ,−→H and

−→B . Unless

−→H can be expressed

as a function of−→E and

−→B , Maxwell’s equations cannot be solved. The connection between these

quantities are called the constitutive relations :

−→H =

−→H[−→E ,−→B]

We will now examine particular cases for these constitutive relations.

5.2. LINEAR MAGNETIC MATERIALS 115

5.2.2 Categorizing the magnetic media

5.2.2.a Linear magnetic materials

Definition : A magnetic medium is called linear if there is a magnetic tensor 3 such that:

−→B = [µ(ω)]

−→H (5.18)

⋆ A few comments :

– This property means that the different components of−→H ,

−→E and

−→B are related by linear

partial differential equations with constant coefficients.– The coefficient can however be a function of the frequency ω. In that case, it is useful tointroduce the complex notation.

– In linear magnetic materials, one assumes that the response of the material to an appliedfield is linear 4. This excludes the case of materials presenting a spontaneous magnetization.Outside of these materials, the linear response is reasonable as long as the fields are not toolarge.

5.2.2.b Linear Isotropic magnetic materials

Definition : A magnetic medium is called linear isotropic if there is a scalar magnetic

permeability µ(ω) such that :−→B = µ(ω)

−→B (5.19)

µr =µµ0

is then called the relative permeability. One also defines themagnetic susceptibility

χm such that:µ = µ0 (1 + χm) (5.20)

CAUTION : Electric and magnetic susceptibilities are often both noted χ. It is therefore impor-tant to know which one you are dealing with...

⋆ A few comments :– The relative permeability is a priori a complex number. We will see the physical meaning ofthe real and imaginary parts in a later chapter.

– In isotropic materials, the relative permeability does not depend on the directions.– In linear isotropic materials, the magnetic field and the magnetization are colinear.

5.2.2.c Linear Homogeneous Isotropic magnetic materials

Definition : A magnetic medium is called linear homogeneous isotropic (lhi) if it islinear, isotropic and if the magnetic permeability µ does not depend on the pointM of the material.The magnetic permeability is then characteristic of the material.

3. i.e. a matrix.4. This is more generally called the linear response theory.


Mu-metal (used for magnetic field shielding) 50’000Iron 5’000Ferrite 500Platinum 1.000265Teflon 1Air 1Bismuth 0.999834Superconductors 0

Table 5.1: Indicative relative permeability for a few materials.

5.2.3 Useful relations in lhi magnetic materials

Combining the previously seen relations for lhi magnetic materials, one obtains :

−→B = µ0

(−→H +

−→M)

= µ−→H = µ0 (1 + χm)

−→H = µ0µr

−→H (5.21)

−→B =

µ

χm

−→M =

µ0µr

µr − 1

−→M =

µ0 (1 + χm)

χm

−→M (5.22)

−→M =

χm

µ0 (1 + χm)

−→B =

µr − 1

µ0µr

−→B (5.23)

−→M = χm

−→H = (µr − 1)

−→H (5.24)

χm = µr − 1 (5.25)

Using:

−→J total =

−→J free +

−→J P +

−→J M

−−→curl

−→B = µ0

−→J total

−−→curl

−→H =

−→J free

and−−→curl

−→M =

−→J M

one could find relations between−→J total,

−→J free and

−→J M like what we have done in section 3.2.3 for

the charge densities. However, these would be true ONLY if−→J P =

−→0 , that is to say if

−→E =

−→0

or in the steady state regime... Otherwise, things are more complicated, due to the polarizationcurrent density.

⋆ Comment on the sign of χm:• If χm = 0, the material is non-magnetic.• If χm > 0, the material is either paramagnetic, ferromagnetic, ferrimagnetic or anti-

ferromagnetic. In those systems, the magnetic field within the material is increased due toa magnetization parallel to the applied field.

• If χm < 0, the material is diamagnetic. In those systems, the magnetic field within thematerial is weakened due to a magnetization anti-parallel to the applied field.

• Superconductors are perfectly diamagnetic: the magnetization completely compensatesthe applied field, so that the effective magnetic field within the material is exactly zero.

We will detail the different kind of magnetic behaviors in section 5.6.

5.2. LINEAR MAGNETIC MATERIALS 117

5.2.4 Snell-Descartes relations for a dielectric material

Figure 5.3: Evolution of the magnetic field at a boundary between two magnetic materials, in theabsence of free currents.

The discontinuity equations applied at the boundary between to lhi magnetic materials, in theabsence of free currents, gives, with the notations of figure 5.3 :

−−→B2,n −

−−→B1,n =

−→0 → B2 cosα2 = B1 cosα1

−−→H2,t −

−−→H1,t =

−→0 →

B2

µ2sinα2 =

B1

µ1sinα1

so that :tanα1

µ1=tanα2

µ2

When a magnetic field crosses from a region of low µ to a region of high µ, the field lines tend tomove apart, very much like what happens in optics when light crosses to a material with a higherrefractive index.

Hence, as seen figure 5.4, magnetic materials tend to bend the magnetic field lines. For high µr

materials, the field lines can be considered to be parallel to the incoming surfaces of the magnet.

Figure 5.4: Evolution of the magnetic field in magnetic rods, depending on their permeability µr.


♣ Application: Choke coil inductor - Choke coil inductors (as represented figure 5.2.4 are widelyused in electronic devices. They consist in a wire through which flows a current I, winded arounda high magnetic permeability material (iron, or ferrite for example). They enable to achieve highinductances while limiting magnetic field leakage 5.

Figure 5.5: Choke coil inductor. Picture taken from http://www.indiamart.com.

♦ Proof : Let us suppose that the magnetic material has a sufficiently high µr to confine themagnetic field in its core. The field lines are therefore circular and the ferrite is a flux tube. Theconservation of magnetic flux gives that the magnetic field is constant in all the material.

˛

−→H.−→dl = 2πRH = NI

B = µrH =µrNI

2πR

φB = N

¨

−→B.−→dS =

µrN2IS

2πR= LI

L =µrN

2S

2πR

Moreover, in series with resistors and capacitors, this inductance gives rise to a resonant circuit

which quality factor is given by Q = 1R

√

LC, meaning that a higher inductance renders the circuit

more selective and hence less prone to electronic noise.

5.3 Forces and momentum

We have seen in section 4.3.3.a that the force exerted by an external magnetic field−→B 0 on a

rigid dipole−→M can be expressed as :

−→F =

−→∇(−→M.−→B 0

)

A magnetic material submitted to an external magnetic field−→B 0 is therefore submitted to a volume

density of external force 6:

d−→F

dV=−→∇(−→M.−→B)

(5.26)

5. These device therefore limit the electromagnetic interferences which are an important issue in a society whereelectronic devices are numerous.

6. This expression does not take into account the internal forces within the magnetic material.

5.4. ENERGY IN MATERIAL MEDIA 119

where−→B is the magnetic field as seen by each one of the dipoles.

The corresponding angular momentum can be shown to be :

−→ΓO =

−→M ∧

−→B (5.27)

5.4 Energy in material media

We had seen in section 1.2.2.b, that :

∂Uem

∂t= −

˚

(V )

−→J .−→E dV −

‹

(S)

−→R.−→dS

with−→J the total current density, Uem =

˝

(V )

(

ε0E2

2 + B2

2µ0

)

dV the total electromagnetic energy

contained in a volume V and−→R =

−→E∧−→B

µ0

the Poynting vector.However, this expression - valid in all cases, including material media - involves the total current

density. In some case it is useful to derive a similar expression containing only the free currentdensity.

−→J free.

−→E =

−−→curl

−→H.−→E −

∂−→D

∂t.−→E

=(−−→curl

−→H.−→E −

−→H.−−→curl

−→E)

+−→H.−−→curl

−→E −

∂−→D

∂t.−→E

= −div(−→E ∧

−→H)

+−→H.

(

−∂−→B

∂t

)

−∂−→D

∂t.−→E

Hence the local energy conservation equation :

div(−→E ∧

−→H)

+−→H.

(

∂−→B

∂t

)

+∂−→D

∂t.−→E = −

−→J free.

−→E (5.28)

In the case of a lhi material :

−−→J free.

−→E = div

(−→E ∧

−→H)

+

−→B

µ.

(

∂−→B

∂t

)

+∂ε−→E

∂t.−→E

= div(−→E ∧

−→H)

+∂

∂t

(

B2

2µ+

εE2

2

)

One can then write :∂U ′em∂t

= −

˚

(V )

−→J free.

−→E dV −

‹

(S)

−→R′.−→dS (5.29)

with

U ′em =

˚

(V )

(

εE2

2+

B2

2µ

)

dV (5.30)

the total electromagnetic energy contained in a volume V,−→J free the free current density, and

−→R′ =

−→E ∧

−→H a version of the Poynting vector for material media. Note that the material properties

are taken into account in ε and µ.

⋆ A few comments :


– Energetics in dielectric or magnetic materials is a difficult and tricky problem. The discussionon this matter has here been kept to a minimum. For a more complete discussion on thesubtleties of this issue, see, for example, J.D. Jackson, Classical Electrodynamics, chapters 4,5 and 6.

– Let us note however that U ′em corresponds to the total electromagnetic energy density inthe medium. It corresponds to the total energy required to produce the magnetization inthe medium. It includes the energy for establishing the magnetization in a given field andthe energy for the creation of the magnetic moment and the energy for maintaining themagnetization.

– It can be compared to the magnetic energy −−→M.−→B seen in equation 5.26, which corresponds

only to the energy for establishing the magnetization in a given field.

5.5 Examples

5.5.1 Example #1 : Uniformly magnetized sphere

Let us consider a uniformly magnetized sphere

of magnetization−→M =M−→uz. Through equation 5.2,

we can express the vector potential and use an anal-ogy with a uniformly charged sphere and the calcu-lation of the corresponding electric field.

−→A (M) =

µ0

4π

˚

(V )

−→M ∧

−−→QM

QM3dV

=µ0

4π

−→M ∧

˚

(V )

−−→QM

QM3dV

Let us now make the analogy with the electric cre-ated by a uniformly charged sphere (of charge den-

sity ρ). In that case :

−→E =

¨

ρ−−→QM

4πε0QM3dV

which was solved via Gauss theorem :

−→E int =

rρ

3ε0

−→ur if r < R

−→E ext =

R3ρ

3ε0r2−→ur if r > R

Reintroducing these expressions in the initial problem yields :

−→A int =

µ0

4π

−→M ∧

4πε0ρ

rρ

3ε0

−→ur if r < R

−→A int =

µ0r

3

−→M ∧ −→ur

−→A ext =

µ0

4π

−→M ∧

4πε0ρ

R3ρ

3ε0r2−→ur if r > R

−→A ext =

µ0R3

3r2−→M ∧ −→ur

5.5. EXAMPLES 121

Since−→M = cos θ−→ur + sin θ−→uθ :

−→A int =

µ0rM sin θ

3−→uφ if r < R

−→A ext =

µ0R3M sin θ

3r2−→uφ if r > R

and using the expression of the curl in spherical coordinates :

−→B int =

1

r sin2 θ

∂

∂θ

(

µ0rM sin θ

3

)

−→ur −1

r

∂

∂r

(

µ0r2M sin θ

3

)

−→uθ if r < R

=2µ0M cos θ

3−→ur −

2µ0M sin θ

3−→uθ

=2µ0

3(M cos θ−→ur −M sin θ−→uθ)

=2µ0

3

−→M

−→H int =

−→B int

µ0−−→M = −

−→M

3if r < R

−→B ext =

1

r sin θ

∂

∂θ

(

µ0R3M sin2 θ

3r2

)

−→ur −1

r

∂

∂r

(

µ0R3M sin θ

3r

)

−→uθ if r > R

=2µ0R

3M cos θ

3r3−→ur +

µ0R3M sin θ

3r3−→uθ

=µ0R

3

3r3(2M cos θ−→ur +M sin θ−→uθ)

=µ0

4π

(

3(−→m.−→ur)−→ur −

−→m

r3

)

with −→m =4

3πR3−→M

−→H ext =

Bext

µ0if r > R

Inside the material−→H ‖ −

−→M . This can be generalized : The H-vector always points in the

direction opposite to that of the magnetization.

Outside the material, the sphere behaves like a magnetic moment −→m = 43πR

3−→M .Figure 5.6 gives the corresponding field lines.

Figure 5.6: Field lines for a uniformly polarized sphere. Taken from http://web.mit.edu/6.013_

book/www/chapter8/8.5.html.


5.5.2 Example #2: Quincke Tube

Figure 5.7: Experimental setup for Quincke’s Tube experiment.

Let us consider a magnetic liquid material of magnetization−→M , of mass per unit volume ρ, and

of magnetic susceptibility χ. The liquid surface is placed in the air gap of an electromagnet which

creates a magnetic field−→B = B0(z)

−→ux.−→B can be taken to be uniform in the air gap but decreases

rapidly outside the gap (see figure 5.7). When an uniform field is created in the electromagnet airgap, one observes that the liquid rises in the smaller tube (of radius r) from point A to point A′,while the liquid falls in the larger tube (of radius R) from point B to point B′. Explain.

First, let us examine the value of the magnetic field within the liquid. At the interface betweenair and the magnetic liquid, the tangential component of the H-vector is conserved :

−→H =

−→B0

µ0=

−→B liq

µ0µr

where−→B liq is the magnetic field within the magnetic liquid.

Moreover, due to volume conservation :

h′ = hr2

R2

When the magnetic liquid rises, the magnetic force exerted by the electromagnet on the liquidmust exactly compensate the weight of the liquid that is lifted up the smaller tube. The energyrequired for the creation of a magnetization within the liquid in the air gap can be understood asthe replacement of air by the liquid in the air gap :

Uem =1

2BliqHliqπr

2h−1

2BairHairπr

2h

=1

2πr2h

(

µH2 − µ0H2)

=1

2πr2hµ0H

2χm

Hence :

−→F magn = −

−−→grad (−Uem) =

1

2πr2µ0H

2χm−→uz

−→F grav = −mg−→uz = −πr

2(h+ h′)ρg−→uz

= −πr2h

(

1 +r2

R2

)

ρg−→uz ≃ −πr2hρg−→uz

5.6. CATEGORIZING MAGNETIC MATERIALS 123

Here, we have moreover neglected the weight of the displaced air and we have used the fact thatr ≪ R. Then, the condition for equilibrium gives :

πr2hρg =1

2πr2µ0H

2χm

h =µ0H

2χm

2ρg

h =B2

0χm

2µ0ρg

For a paramagnetic liquid with ρ = 103 kg.m−3 and χm = 10−4, one can compute h = 4 mm.For a diamagnetic liquid, since, as we will see χ < 0, the liquid lowers in the smaller tube.

5.6 Categorizing magnetic materials

In this section, we will discuss the origins and properties of some magnetic materials 7. Thesecan be evidenced by the experiment schematized figure 5.8: if a piece of material is suspendedbetween the poles of an electromagnet, different behaviors can be observed. Let us first note thatthe magnetic field is here strongest near the South pole, due to the convergence of field lines.Ferromagnets (section 5.6.4) will be strongly attracted towards the high-field region, whereas allother magnetic materials will experience a force of much smaller intensity. Paramagnets (section5.6.3) and anti-ferromagnets (section 5.6.5) will be weakly attracted towards the high-field region,whereas diamagnetic materials (section 5.6.2) will be repelled towards the low-field region.

Figure 5.8: Schematic representation of an experiment designed to evidence the magnetic propertiesof materials. Taken from R.P. Feynman, The Feynman Lectures.

5.6.1 Microscopic origin of magnetism

Magnetic effects are either due to the permanent magnetic moments carried by individualatoms or to the movement of electrons or nuclei. They can therefore only be rigorously explainedby quantum mechanics. However, semiclassical models can give us a flavor of what is going on inthese materials. We will here limit ourselves to this level of explanation 8.

7. All magnetic materials will not be treated. For instance, we will not talk about ferrimagnetic materials,helimagnets or spinels.

8. For a quantum mechanical treatment of magnetism - of which you have had an introduction in your QuantumMechanics course - see, for example, N.W. Ashcroft and N.D. Mermin, Solid State Physics, chapter 31.


Magnetic

moment

alig

nment

Typical

valueofχm

Magnetiz

atio

nEvolutio

nof

χm

with

temperatu

re

Comments

Examples

Diamagnetis

mAnti-p

arallelto

anextern

almagn

eticfield

χm

<0,

|χm|≃10−9-

10−5,

forsu-

percon

ductors

χm=1

χm=−

µ0nZe2

6m

〈r2〉

Bism

uth,

Mercu

ry,Silver,

Argo

n,Heliu

m,

Water,

N2 ,most

organic

compounds

Paramagnetis

mParallel

toan

exter-

nalmagn

eticfield

χm

>0,

χm

≃10−5-

10−3

Atlow

temperatu

re,onehas

Curie’s

law:

χm=

CT

Tungsten

,Sodium,

Magnesiu

m,Aluminum

Ferromagnetis

mSpontan

eousmagn

e-tization

;mom

ents

alignparallel

toone

anoth

er

χm

>0

For

T>

TCuriepara-

magnetic

behavior

;For

T<

TCuriefer-

romagnetic

behavior

Gadolinium

(TCurie

=292

K),

Cobalt

(TCurie

=1388

K),Iron

(TCurie=1043

K),

NdFeB

(TCurie=583

K)

Antife

rromagnetis

mMom

ents

alignanti-

parallel

toonean-

other

χm

>0

For

T>

θparam

ag-netic

behavior

;For

T<

θantiferrom

ag-netic

behavior

Chrom

ium,NiO

Table5.2:

Princip

alcharacteristics

ofsom

emagn

eticmaterials.


We have seen in section 4.3 how a semi-classical model could account for the relation betweenthe electronic orbital moment

−→L and the corresponding magnetic moment

−−→ML:

−−→ML = −

e

2m

−→L = γ

−→L (5.31)

with γ the gyromagnetic ratio.In the same way, one can derive from quantum mechanics the relation between the electronic

spin−→S and the corresponding magnetic moment

−−→MS :

−−→MS = −

e

m

−→S (5.32)

Note that the contribution of the spin moment is twice as important as the contribution of theorbital moment. Hence, depending on the authorized combinations of orbital and spin moments 9,one obtains the electronic magnetic moment:

−→M = −g

e

2m

−→J total (5.33)

where g is a dimensionless factor, of the order of 1, called the Lande factor. It is characteristicof the considered wavefunction. The direction of the magnetic moment is opposite to the directionof the angular momentum.

The nucleus can also contribute to the global magnetic moment in a similar manner :

−→Mnucl = +g

e

2mp

−→J total (5.34)

where mp is the proton mass and g the nuclear Lande factor. Since the magnetic moment isinversely proportional to the mass, a proton contributes 1000 times less than an electron to theaverage moment, all other things being equal.

The principal categories of magnetic materials are summarized table 5.2.

5.6.2 Diamagnetism

In numerous compounds, the electron spins motion are exactly counter-balanced by their orbital

motion, so that there is no average magnetic moment. In that case, an external magnetic field−→B

induces some small extra currents, which tend to oppose the applied field. Diamagnetic compoundsusually are made out of atoms where all electrons are paired, so that the net spin of each molecule

is zero. The resulting magnetic moment is hence of opposite direction than that of−→B 10. Hence

the diamagnetism.Let us consider once again the semi-classical model of electrons orbiting around the nucleus.

Since the atom bears no average magnetic moment in the absence of magnetic field, the only non-zero contribution to M is due to the Lorentz force acting on the electrons. The angular velocityof such motion is ω = γB = eB

2m . Then the current associated to the motion of Z electrons can bewritten :

I = (−Ze)ω

2π

9. That is one application of the moments composition seen in Quantum Mechanics.10. Diamagnetic material therefore tend to expel the magnetic field. This property can be used to make objects

levitate. This has been done by Andrey Geim - more famous for the discovery of Graphene, Nobel Prize 2010 -.He has explained it in numerous papers, among which “Everyone’s magnetism”, Physics Today, p. 36, sept 1998.The videos of the levitating frog/cricket/strawberry/tomato/... can be found at http://www.ru.nl/hfml/research/levitation/diamagnetic/. He, and M.V. Berry, have won the IgNobel for this work...


Hence:

M = IS−→n = −Ze2

4πmBS−→n

Since S is the surface of the orbit in the (x, y) plane :

S = π(

〈x2〉+ 〈y2〉)

S = π

(

2

3〈r2〉

)

with 〈r2〉 = 〈x2〉 + 〈y2〉 + 〈z2〉 the mean distance of the electrons to the nucleus. Hence, themagnetic susceptibility for one atom is :

χm =M

H=

µ0M

H

= −µ0Ze2

6m〈r2〉

and for n atoms per unit volume:

χm = −µ0nZe2

6m〈r2〉 (5.35)

⋆ Comment on the diamagnetism of superconductors : Superconductors are considered to be perfect

diamagnetic materials (χm = −1). This actually means that−→B = µ0 (1 + χm)

−→H =

−→0 . In other

words, a superconductor expels the magnetic field from its volume, thanks to non-dissipative so-called supercurrents that are the consequence of the pairing of electrons into Cooper pairs 11. Thisdiamagnetism explains the Meissner effect and originates from purely quantum mechanical effectsthat have nothing to do with the simplistic semi-classical model developed above.

5.6.3 Paramagnetism

In paramagnetic compounds, the atoms usually have at least one unpaired electron, i.e. a netspin larger than zero. In all cases, the molecules possess a non-zero magnetic moment. Whenan external magnetic field is applied, this moment tends to align with the field, with a statisticdistribution similar to what we have studied for the orientation polarization section 3.5.2. In thesematerial, the diamagnetic effect is always present but is negligible compared to paramagnetic effect.Let us examine a semi-classical model for paramagnetism due to Leon Brillouin.

5.6.3.a Case of a single spin

We will examine the case of a single spin and extend the obtained result in the general case.The hypotheses are twofold:

1. Each electron has a magnetic moment associated with its spin. The projection of the spin

along the direction of the applied magnetic field−→B can only take the values

−→M or −

−→M. The

corresponding potential energies are Um = −−→M.−→B .

2. Interactions between magnetic moments are neglected. Only thermal agitation and the cor-responding chocs between molecules can change the directions of the magnetic moments.

If the magnetic field is along the Oz axis (−→B = B−→uz), the projection Mz of the magnetic

moment of a molecule can only take the valuesMz =M orMz = −M. Each value corresponds

11. See section 5.7.


to a number of molecules : N+ = A exp−−MBkBT

and N− = A exp−MBkBT

. We moreover have the

total number of molecules N = N+ +N− = 2Acosh(

MBkBT

)

.

The mesoscopically averaged magnetic moment is :

〈M〉 =N+M−N−M

N

= M2Asinh

(

MBkBT

)

2Acosh(

MBkBT

)

〈M〉 = Mtanh

(

MB

kBT

)

And the magnetization can then be expressed as :

−→M = n〈M〉−→uz

−→M = nMtanh

(

MB

kBT

)

−→uz

where n is the molecular density. Let us now examine the order of magnitude of the involvedenergies :

MB

kBT=

e~B

2mkBT

≃10−19 × 10−34 × 1

10−31 × 10−23 × 300MB

kBT≃ 0.03≪ 1

We can therefore rightfully linearize the tanh is the previous expression :

−→M ≃

nM2B

kBT−→uz

=χm

µ0 (1 + χm)

−→B ≃

χm

µ0for χm ≪ 1

Hence :

χm =nM2µ0

kBT

5.6.3.b Extension to the case of a general magnetic moment

These results can then be extended to general case of a magnetic moment−→J . Then the possible

projections on the Oz axis are given by Quantum Mechanics: mJgµB where µB = ~e2m is the Bohr

magneton, g the Lande factor and mJ the eigenvalue of the projection of−→J on the Oz axis

(−J ≤ mJ ≤ J). The calculation then gives the magnetization:

−→M = ngµBJBJ(x) (5.36)

where BJ(x) is the Brillouin function (see figure 5.9) defined by:

BJ(x) =2J + 1

2Jcotanh

(

2J + 1

2Jx

)

−1

2Jcotanh

(

1

2Jx

)

(5.37)


and x is given by:

x =JgµBB

kBT

In the case where µBB ≪ kBT , we can linearize the previous expression to obtain Curie’s law

which states that the magnetic permeability is inversely proportional to the temperature :

χm =µ0nµ

2eff

3kBT=

C

T(5.38)

with C the so-called Curie’s constant:

µeff = gµB

√

J(J + 1) (5.39)

Note that in the high-field limit, the magnetization reaches a plateau of value:

M =Ms = ngµB (5.40)

which corresponds to the maximum magnetization reached when all magnetic moments are alignedwith the field.

Figure 5.9: The Brillouin function for for J = 12 to J = 5

2 . Taken from http://moxbee.blogspot.

fr/.

5.6.4 Ferromagnetism

Some materials, like iron, nickel or cobalt, display, below a characteristic temperature calledthe Curie temperature TCurie a spontaneous magnetization, even in the absence of externalmagnetic field. Although we will not review the detailed microscopic mechanisms that give rise tothis phenomenon 12, let us review its main features.

5.6.4.a Phenomenological description

• For temperatures larger than TCurie, ferromagnets behave like paramagnetic materials :

χm =C

T − TCurie

for T > TCurie (5.41)

12. For more details, see, for example, R.P. Feynman, The Feynman Lectures, chapters 36 and 37 and referencetherein.


• For temperatures smaller than TCurie, ferromagnets have a non-defined χm - since the magne-tization is spontaneous and does not depend on the applied magnetic field -. The idea is thatinteractions between magnetic moments are sufficiently strong - unlike in the case of param-agnets - to induce a magnetization. Once one magnetic moment has a certain directions, allneighboring moments tend to align with it.

5.6.4.b Calculation of the magnetization of a ferromagnet

In order to compute the magnetization of a ferromagnet, we will extend the semi-classical modelstudied in the case of paramagnets (section 5.6.3) in order to take into account interactions betweenmagnetic moments.

We will suppose that a moment−→Mi has an interaction energy with other moments:

Ep = −J∑

k

−→Mi.

−−→Mk = −

−→Mi.

(

J∑

k

−−→Mk

)

(5.42)

where J is a constant which positive sign models the fact that the magnetic moments tend to alignwith one another to minimize their energy.

All is as if an effective, virtual, magnetic field−→B eq = J

∑

k

−−→Mk acts on the magnetic moment

−→Mi. In the framework of the mean field theory developed by Pierre Weiss 13, we will suppose that:

−→B eq = λ

−→M (5.43)

with λ > 0.By applying the result found in section 5.6.3 in the case of the virtual field

−→B eq, one finds :

−→M = nMtanh

(

MBeq

kBT

)

−→uz (5.44)

−→M = nMtanh

(

MλM

kBT

)

−→uz (5.45)

This equation can be solved numerically or graphically (see figure 5.10) by plotting the inter-section of tanhx (with x = λMM

kBT) and M

nM= kBT

nM2λx = px. Then :

• For T > TCurie =λnM2

kB

, p > 1 and the only solution is M = 0.• For T < TCurie, p < 1 and there are two solutions : M = 0 which is an unstable solution ;andM 6= 0 which is the more stable solution, thermodynamically speaking. This correspondsto a spontaneous magnetization and hence to the ferromagnetic state.

• The transition between those two cases is achieved when T = TCurie. This critical tempera-ture defines a phase transition between a paramagnetic phase and a ferromagnetic phase.

13. The mean field theory was actually developed by Pierre Weiss and Pierre Curie to explain phase transitions.See Pierre Weiss, “L’hypothese du champ moleculaire et la propriete ferromagnetique”, J. Phys.Theor. Appl., 6,pp.661-690, 1907.


Figure 5.10: Graphic solution of the self-consistent equation for magnetization in ferromagnets.

5.6.4.c The magnetization curve

The magnetization curve of a ferromagnet is illustrated figure 5.11 in the case of soft iron. Thedifferent steps can be understood as follows :

– Let us start by a non-magnetized ferromagnet. As soon as the applied H-field is applied(curve a), the magnetization - given by B - increases.

– The magnetization eventually saturates at a value Msat = nM representing the maximumsaturation achieved when all magnetic moments are aligned.

– When the H-field is decreased (curve b), there is a competition between the applied H-fieldand the interactions between the magnetic moments inside the material. The magnetizationslowly decreases. At zero H-field, the magnetization is non-zero.

– When the polarity of the H-field is reversed, the magnetization eventually reverses until itreaches −Msat.

– Then, again, if the applied H-field is increased (curve c), the magnetization will progressivelyreverse.

Figure 5.11: Magnetization curve for soft iron. Taken from R.P. Feynman, The Feynman Lectures.

5.7. ADDITIONAL READING 131

5.6.5 Anti-ferromagnetism

• For temperatures larger than θ, the Neel temperature, antiferromagnets behave like para-magnetic materials :

χm =C

T − θfor T > θ (5.46)

• For temperatures smaller than θ, antiferromagnets tend to have their magnetic momentsorganize into a periodic array of parallel/antiparallel moments. This gives rise to a zeromagnetization at low field. More generally χm = M

Hdecreases with the temperature : at low

temperature, the thermal agitation is not strong enough to act against the anti-alignment ofmagnetic moments.

5.7 Additional reading

The following article 14, reviews Meissner effect in superconductors (the expulsion of the mag-netic field within the bulk of these materials) in link with classical electromagnetism.

14. Essen and Fiolhais, Am. J. Phys., 80, 164 2012.

Meissnereffect,diamagnetism,andclassicalphysics—areview

HannoEssen

DepartmentofMechanics,RoyalInstituteofTechnology(KTH),Stockholm

SE-10044,Sweden

MiguelC

.N.Fiolhais

LIP-Coimbra,DepartmentofPhysics,UniversityofCoimbra,Coimbra

3004-516,Portugal

(Received

8September

2011;accepted28October

2011)

Wereview

theliterature

onwhat

classicalphysics

saysabouttheMeissner

effect

andtheLondon

equations.Wediscuss

therelevance

oftheBohr-van

Leeuwen

theorem

fortheperfectdiamagnetism

ofsuperconductors

andconcludethat

thetheorem

isbased

oninvalid

assumptions.Wealso

point

outresultsin

theliterature

that

show

how

magnetic

fluxexpulsionfrom

asample

cooledto

superconductivitycanbeunderstoodas

anapproachto

themagnetostatic

energyminim

um.These

resultshavebeenpublished

severaltimes

butmanytextbooksonmagnetism

stillclaim

thatthereis

noclassicaldiamagnetism,andvirtually

allbooksonsuperconductivityrepeatMeissner’s

1933

statem

entthatfluxexpulsionhas

noclassicalexplanation.V

C2012AmericanAssociationofPhysicsTeachers.

[DOI:10.1119/1.3662027]

I.IN

TRODUCTIO

N

Itisnowacentury

since

superconductivitywas

discovered

byKam

merlingh-O

nnes

inLeiden

in1911.From

thebegin-

ning,therewas

considerable

interestfrom

theoreticalphysi-

cists.

Unfortunately

progress

has

been

slow

and

onecan

safely

saythatthephenomenonisstillnotcompletely

under-

stood,at

leastnotin

afundam

entalreductionistsense.Itis

therefore

importantthatthethingsthatcanbeunderstoodare

correctlypresentedin

textbooks.Tothecontrary,textbooks

often

repeattwomythswhichhavebecomeso

ingrained

inthemindsofphysiciststhatthey

ham

per

progress.Theseare:

•Thereisnoclassicaldiamagnetism

(Bohr,1van

Leeuwen

2).

•Thereis

noclassicalexplanationoffluxexpulsionfrom

asuperconductor(M

eissner

3).

Althoughboth

statem

ents

havebeendisproved

multiple

times

inthescientificliterature,thesemythscontinueto

be

spread.Wehopethisreviewwillim

provethesituation.

Meissner

founditnaturalthataweakmagnetic

fieldcould

notpenetrate

aType-Isuperconductor.Thefact

that

this

isalreadyin

conflictwithclassicalphysics,accordingto

the

Bohr-van

Leeuwen

theorem

israrely

mentioned.On

the

other

hand,Meissner

founditremarkablethatanorm

almetal

withamagnetic

fieldinsidewillexpelthisfieldwhen

cooled

tosuperconductivity.This

was

claimed

tohavenoclassical

explanation,and

consequently

superconductors

were

not

considered

perfectconductors.

Wewillnotpresentanynew

resultsin

thispaper.Instead,

wewillreview

someofthestrongevidence

inthearchival

literature

dem

onstratingthat

theabovetwostatem

ents

are

false.

Webegin

withtheBohr-van

Leeuwen

theorem

and

then

turn

totheclassicalexplanationoffluxexpulsion.After

discussingthearguments

forthetraditionalpointofview,

wewilldem

onstratewhythey

arewrong.

II.THEBOHR-V

ANLEEUW

ENTHEOREM

AND

CLASSIC

ALDIA

MAGNETISM

TheHam

iltonian

H,forasystem

ofcharged

particles

interactingviaa(scalar)potentialenergyUis

Hðr

k;p

kÞ¼

X

N j¼1

p2 j

2m

j

þUðr

kÞ:

(1)

The

particles

have

masses

mj,

position

vectors

r j,and

momenta

pj.Thepresence

ofan

external

magnetic

fieldwith

vectorpotentialA(r)alterstheHam

iltonianto

Hðr

k;p

kÞ¼

X

N j¼1

pjÿ

e j cAðr

jÞ�

�

2

2m

j

þUðr

kÞ;

(2)

wheree j

arethecharges

ofmassesmj.Usingthis

equation

onecan

show

that

statisticalmechanicspredicts

that

the

energy—thethermal

averageoftheHam

iltonian—does

not

dependontheexternal

field.Hence,thesystem

exhibitsnei-

ther

aparam

agneticnoradiamagneticresponse.

Inhis

1911doctoraldissertation,Niels

Bohr1

usedthe

aboveequationandconcluded

thereisnomagnetic

response

ofametal

accordingto

classicalphysics.In

1919Hendrika

Johannavan

Leeuwen

independentlycameto

thesamecon-

clusionin

her

Leiden

thesis.When

her

work

was

published

in1921,Missvan

Leeuwen

2notedthat

similar

conclusions

had

beenreached

byBohr.

Manybooksonmagnetism

referto

theaboveresultsas

theBohr-van

Leeuwen

theorem,orsimply

thevan

Leeuwen

theorem.Thetheorem

isoften

summarized

asstatingthereis

noclassicalmagnetism.Since

this

isobviousin

thecase

of

spin

oratomic

angularmomenta—thequantum

phenomena

responsible

for

param

agnetism—the

more

interesting

conclusionisthat

classicalstatisticalmechanicsandelectro-

magnetism

cannotexplain

diamagnetism.A

verythorough

treatm

entoftheBohr-van

Leeuwen

theorem

canbefound

inVan

Vleck.4

Other

books,

such

asMohn,5

Getzlaff,6

Aharoni,7and

Levy,8

mention

thetheorem

more

orless

briefly.Weaknessesin

classicalderivationsofdiamagnetism

inmoderntextbookshavebeenpointedoutbyO’D

elland

Zia.9A

discussionofthetheorem

canalso

befoundin

the

Feynman

lectures1

0(V

ol.2,Sec.34-6),whichpointsoutthat

aconstantexternal

fieldwillnotdoanywork

onasystem

of

charges.Therefore,theenergyofthissystem

cannotdepend

ontheexternalfield.In

addition,thefact

that

thediamag-

netic

response

ofmostmaterialsisverysm

alllendsem

piri-

calsupportto

thetheorem.

A.Theinadequacy

ofthebasicassumptions

IfweacceptthepopularversionoftheBohr-van

Leeuwen

theorem

astrue,

then

onemust

concludethat

theperfect

164

Am.J.Phys.80(2),February2012

http://aapt.org/ajp

VC2012American

AssociationofPhysics

Teachers

164

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art

icle

is c

opyrighte

d a

s indic

ate

d in t

he a

rtic

le.

Reuse

of A

AP

T c

onte

nt

is s

ubje

ct

to t

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erm

s a

t: h

ttp:/

/scitation.a

ip.o

rg/t

erm

sconditio

ns.

Dow

nlo

aded t

o I

P:

129.1

75.9

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Fri,

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2015 1

8:0

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5

diamagnetism

ofsuperconductors

cannothaveaclassicalex-

planation.A

studyoftheproofofthetheorem,however,

revealsthatitisonly

valid

under

assumptionsthatdonotnec-

essarily

hold.The

assumed

Ham

iltonian

ofEq.(2)only

includes

thevectorpotential

oftheexternal

magnetic

field.

Butithas

beenknownsince

1920that

thebestHam

iltonian

forasystem

ofclassicalcharged

particles

istheDarwin

Ham

iltonian.11In

thisHam

iltonianonetakes

into

accountthe

internalmagneticfieldsgenerated

bythemovingcharged

par-

ticles

ofthesystem

itself,plusanycorrespondinginteractions.

Thesimplestway

toseethat

thetotalenergyofasystem

ofcharged

particles

dependsontheexternal

fieldis

tonote

thatthetotalenergyincludes

amagneticenergy12

EB¼

1 8p

ð

B2ðrÞdV¼

1 8p

ð

ðBeþBiÞ2dV;

(3)

whereBeistheexternal

magnetic

fieldandBiistheinternal

field.UsingtheBiot-Savartlaw

this

fieldis

given,to

first

order

inv/c,by

BiðrÞ

¼X

N j¼1

BjðrÞ

¼X

N j¼1

e j c

vj�ðr

ÿr jÞ

jrÿr jj3

:(4)

Equation(3)makes

itobviousthat

tominim

izetheenergy

theinternal

fieldwillbe,as

much

aspossible,equal

inmag-

nitudeandopposite

indirectionto

theexternal

field.This

gives

diamagnetism.

CharlesGaltonDarwin

was

firstto

derivean

approxim

ate

Lagrangian

forasystem

ofcharged

particles(neglecting

radiation)thatiscorrectto

order

(v/c)2.11,13–16In

atimeinde-

pendentexternal

magnetic

field,thereis

then

aconserved

Darwin

energygiven

by

ED¼

X

N j¼1

mj

2v2 jþe j cvj�

1 2Aiðr j;r k;v

kÞþAeðr

jÞ

��

��

þUðr

kÞþEe:

(5)

Inthis

equation,vjarevelocity

vectors,Aeis

theexternal

vectorpotential,Eeis

the(constant)

energyoftheexternal

magnetic

field,andAiis

theinternal

vectorpotential

given

by

Aiðr j;r k;v

kÞ¼

X

N k6¼j

e k r kj

vkþðv

k�e

kjÞe

kj

2c

;(6)

wherer kj¼|rj–r k|andekj¼(rj–r k)/r kj.Althoughthereisa

correspondingHam

iltonian,it

cannotbewritten

inclosed

form

.When

theDarwin

magnetic

interactionsaretaken

into

account,theBohr-van

Leeuwen

theorem

isnolonger

valid

because

themagnetic

fieldsofthemovingcharges

willcon-

tribute

tothetotalmagnetic

energy.Thefact

that

this

inva-

lidates

theBohr-van

Leeuwen

theorem

forsuperconductors

was

stated

explicitlybyPfleiderer

17in

aletter

toNature

in1966.A

more

recentdiscussionofclassicaldiamagnetism

andtheDarwin

Ham

iltonianisgiven

byEssen.18

B.WhataboutLarm

or’stheorem?

Larmor’stheorem

states

that

aspherically

symmetricsys-

tem

ofcharged

particles

willstartto

rotate

ifan

external

magnetic

fieldisturned

on(see

Landau

andLifshitz,13x45).

Therotationofsuch

asystem

producesacirculatingcurrent

andthusamagnetic

field.Sim

ple

calculationsshow

that

this

fieldis

ofopposite

directionto

theexternal

fieldandso

the

system

isdiamagnetic.Thisidea

was

firstusedbyLangevin

toderivediamagnetism.Butfrom

thepointofview

ofthe

Bohr-van

Leeuwen

theorem

this

seem

sstrange.

InFeyn-

man’s

lectures1

0theBohr-vanLeeuwen

theorem

istherefore

notconsidered

tobevalid

forsystem

sthat

canrotate.As

notedabove,

theproblem

issimply

that

theBohr-van

Leeu-

wen

theorem

does

nottakeinto

accountthemagnetic

field

producedbytheparticles

ofthesystem

itself.When

thisin-

ternal

fieldisaccountedfor,diamagnetism

followsnaturally,

whether

forsystem

sofatomsfrom

theLarmor’stheorem

19

orforsuperconductorsfrom

theDarwin

form

alism.20

C.TheShanghaiexperim

ent—

measuringadiamagnetic

current?

Ifitis

truethat

theclassicalHam

iltonianforasystem

of

charged

particles

predictsdiamagnetism,then

whyisthephe-

nomenonso

weakandinsignificantin

most

cases?

Isthere

anyevidence

forclassicaldiamagnetism

forsystem

sother

than

superconductors?Asstressed

byMahajan,21plasm

asare

typically

diamagnetic.However,plasm

asarenotusually

inthermalequilibrium

soitisdifficultto

reachanydefinitecon-

clusionsfrom

them

.Anexperim

entonan

electrongas

inther-

malequilibrium,perform

edbyXinyongFuandZitao

Fu22in

Shanghai,istherefore

ofconsiderableinterest.

Twoelectrodes

ofAg-O

-Csside-by-sidein

avacuum

tube

emit

electronsat

room

temperature

because

oftheirlow

work

function.Ifamagnetic

fieldisim

posedonthissystem

,an

asymmetry

arises

andelectronsflow

from

oneelectrode

totheother.Forafieldstrength

ofabout4gauss

asteady

currentof�10ÿ14A

ismeasuredat

room

temperature.The

currentgrowswith

increasing

field

strength

and

changes

directionas

thepolarity

isreversed.Theauthors

interpret

thisresultas

ifthemagnetic

fieldactsas

aMaxwelldem

on

that

can

violate

the

second

law

of

thermodynam

ics.22

Inview

ofstatisticalmechanicsbased

ontheDarwin

Ham

il-

tonian,18itis

more

naturalto

interpretthis

resultas

adia-

magnetic

response

ofthesystem

.Thecurrent—

just

asthe

super-currentoftheMeissner

effect—isdueto

adiamagnetic

thermal

equilibrium.Thismeansthat

nousefulwork

canbe

extractedfrom

thesystem

.

III.

ONTHEALLEGEDIN

CONSISTENCYOF

MAGNETIC

FLUXEXPULSIO

NWIT

HCLASSIC

ALPHYSIC

S

Meissner

and

Ochsenfeld

3,23

discovered

the

Meissner

effect

in1933.Totheirsurprise

amagnetic

fieldwas

not

only

unable

topenetrate

asuperconductor,

itwas

also

expelledfrom

theinteriorofaconductoras

itwas

cooled

below

itscritical

temperature.Thefirsteffect—

idealdia-

magnetism—seem

ednaturalto

them

even

thoughitviolates

theBohr-van

Leeuwen

theorem.Asalreadydiscussed,by

violatingthistheorem

idealdiamagnetism

should

havebeen

consideredanon-classical

effect.Thefluxexpulsion,onthe

other

hand,was

explicitlyproclaimed

byMeissner

tohave

noclassicalexplanation.Meissner

does

notgiveanyargu-

mentsorreferencesto

supportthisstatem

ent,butaccording

toDahl24thetheoreticalbasis

was

Lippmann’s

theorem

on

theconservationofmagnetic

fluxthroughan

ideallycon-

ductingcurrentloop.Later,Forrest23andothershaveargued

165

Am.J.Phys.,Vol.80,No.2,February2012

H.Essen

andM.C.N.Fiolhais

165

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that

themagnetohydrodynam

ictheorem

on

frozen-in

flux

lines

also

supportsthis

notion.In

this

section,wediscuss

thesearguments

andpointoutthat

they

donotrule

outa

classicalexplanationoftheMeissner

effect.

A.Lippmann’stheorem

GabrielLippmann(1845–1921),winner

ofthe1908Nobel

Prize

inPhysics,published

atheorem

in1889statingthatthe

magnetic

fluxthroughan

ideallyconductingcurrentloopis

conserved.In

1919,when

superconductivityhad

beendis-

covered,Lippmann25againpublished

thisresultin

threedif-

ferentFrench

journals(see

Sauer

26).

Althoughtheidea

of

fluxconservationisconsidered

highly

fundam

ental,27nowa-

daysreferencesto

Lippman’s

theorem

arehardto

find.But

atthetimeLippman’s

theorem

was

quiteinfluential,and

Dahl24explainshowthiswas

oneoftheresultsthatmadethe

Meissner

effect

seem

surprising—andnon-classical—at

the

timeofitsdiscovery.

TheproofofLippmann’s

theorem

followsbynotingthat

theselfinductance

Lofaclosedcircuit,orloop,isrelatedto

themagneticfluxUfrom

thecurrentin

theloopvia

U¼

cL_ q;

(7)

wherecisthespeedoflightand_ qisthecurrentthroughthe

circuit,an

overdotdenotingatimederivative(see

Landau

andLifshitz,28Vol.8,x33).Theequationofmotionforasin-

gleloopelectriccircuitis

L€ qþR_ qþCÿ1q¼

EðtÞ;

(8)

whereRis

theresistance,C

isthecapacitance,andEðtÞis

theem

fdrivingthecurrent.If

thereis

noresistance,noca-

pacitance,andnoem

f,thisequationbecomes

L€ q¼

0:

(9)

Therefore,iftheselfinductance

Lis

constant,Eqs.(7)and

(9)tellusthatU¼constant.

B.Lippmann’stheorem

andsuperconductors

AlthoughLippmann’stheorem

iscorrect,itsrelevance

for

thepreventionoffluxexpulsionis

notclear.Forsupercon-

ductors

therearetwopoints

toconsider,theassumptionof

zero

emf,andthefact

that

constantfluxdoes

notim

ply

con-

stantmagneticfield.Weconsider

thesepointsoneatatime.

Consider

asuperconductingsphereofradiusrin

acon-

stantexternal

magnetic

fieldBe.When

thesphereexpelsthis

fieldbygenerating(surface)

currentsthat

produce

Bi¼ÿBe

initsinterior,thetotalmagneticenergyisreduced.Themag-

neticenergychangeis

DEB¼

ÿ3

4pr3 3

��

B2 e

8p;

(10)

orthreetimes

theinitialinteriormagnetic

energy.29This

energyis

thusavailable

forproducingtheem

frequired

togeneratecurrentsin

thesphere’sinterior.Theassumptionof

Lippmann’s

theorem—that

theem

fiszero—istherefore

not

fulfilled.

When

asteadycurrentflowsthroughafixed

metalwireboth

thefluxandthemagneticfielddistributionareconstant.Onthe

other

hand,when

currentflowsin

aloop

inaconducting

medium,aconstantfluxthroughtheloopdoes

notim

ply

acon-

stantmagneticfieldbecause

theloopcanchangein

size,shape,

orlocation.Norm

ally

currentloopsaresubject

toforces

that

increase

theirradius(see

e.g.,Landau

andLifshitz,28Vol.8,

x34,Problem

4,orEssen,30

Sec.4.1).

Inview

ofthis,

Lippmann’s

theorem

does

notautomatically

imply

that

the

magneticfieldmustbeconstant,even

iftheem

fiszero.

Other

authors

havereached

similar

conclusions.Mei

and

Liang31carefullyconsidered

theelectromagneticsofsuper-

conductors

in1991andwrite,“T

husMeissner’s

experim

ent

should

beviewed

throughitstimehistory

insteadofas

astrictly

dcevent.In

thatcase

classicalelectromagnetictheory

willbeconsistentwiththeMeissner

effect.”

C.Frozen-infieldlines

Thesimplest

derivationofthefrozen-infieldresultfora

conductingmedium

beginswithOhm’s

law

j¼rE.If

r!

1wemusthaveE¼0to

preventinfinitecurrent.Faraday’s

law

then

gives

@B=@t¼

ÿcr

�E

ðÞ¼

0which

tells

us

the

magnetic

field

Bis

constant(Forrest,23Alfven

and

Faltham

mar

32).

When

dealingwithaconductingfluid,theequationsof

magnetohydrodynam

icsandthelimitofinfiniteconductivity

inOhm’slawgiverise

to32

@B @t¼

r�ðv

�BÞ:

(11)

Thisresulttellsusthemagnetic

fieldconvectswiththefluid

butdoes

notdissipate.

Inaddition,Eq.(11)canbeusedto

derivethefollowingtwostatem

entsthatareoften

referred

toas

thefrozen-infieldtheorem:33

1.themagnetic

fluxthroughanyclosedcurvemovingwith

thefluid

isconstant,and

2.amagnetic

fieldlinemovingwiththefluid

remainsa

magneticfieldlineforalltime.

Thefirstoftheseisajustarestatem

entofLippmann’sthe-

orem

foracurrentloopin

afluid.

D.Magneticfieldlines

insuperconductors

Asshownin

theprevioussection,Ohm’s

law

withinfinite

conductivityim

plies

therecanbenochangein

themagnetic

fieldsince

thiswould

giverise

toan

infinitecurrentdensity.

Itis,however,notphysicallymeaningfulto

takethelimit

r!

1in

Ohm’s

law.In

amedium

ofzero

resistivityone

must

instead

use

the

equation

ofmotion

forthe

charge

carriers.Onecanthen

derive2

9,34

dj

dt¼

e2n

mEþ

e mcj�B;

(12)

forthetimerate

ofchangeofthecurrentdensity.Thetime

derivativehereis

aconvective,

ormaterial,timederivative.

Ohm’s

law

simply

does

notapply,andtherefore

itcannot

imply

thatthemagneticfielddoes

notchange.

Thesecondstatem

entofthefrozen-infieldtheorem

can

bequestioned

onthesamegrounds.Justas

withLippmann’s

theorem,theconclusionthat

themagnetic

fieldremainscon-

stantdoes

notfollow,even

ifthestatem

entisassumed

valid.

Themagnetic

fieldisdetermined

bythedensity

ofmagnetic

fieldlines

andconstancy

ofthisdensity

requires

thatthecon-

ductingfluid

isincompressible.Onecaneasily

imaginethat

166


H.Essen

andM.C.N.Fiolhais

166

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art

icle

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thefluid

ofsuperconductingelectronsis

compressible,and

that

magnetic

pressure

33pushes

thefluid

(withitsmagnetic

fieldlines)to

thesurfaceofthematerial.Corroboratingthis

pointofview,AlfvenandFaltham

mar

32(Sec.5.4.2)state

that“inlowdensity

plasm

astheconceptoffrozen-inlines

of

forceisquestionable.”

E.AclassicalderivationoftheLondonequations

In1981Edwards3

5published

amanuscriptwiththeabove

titlein

PhysicalReview

Letters.This

causedan

uproar

of

indignationandthejournal

laterpublished

threedifferent

criticisms

of

Edwards’

work.36–38

Incidentally,

all

of

Edwards’

criticsrestated

(orim

plied)thetextbookmyth

that

theMeissner

fluxexpulsiondoes

nothaveaclassicalexpla-

nation.In

addition,thejournal

Nature

published

astudyby

Taylor39pointingoutthefaultsofEdwards’

derivation.Of

course,

Edwardsisnottheonly

oneto

publish

anerroneous

derivationoftheLondonequations.A

much

earlierexam

ple

isthederivationbyMoore

40from

1976.Thefactthatvarious

derivationshavebeenwrong,ofcourse,does

notproveany-

thing—as

longas

acorrectderivationexists.

In1966Nature

published

aclassicalexplanationofthe

LondonequationsbyPfleiderer

17whichdid

notcause

any

comment.Indeedithas

notbeencitedasingle

time,

prob-

ably

because

itisverybrief

andcryptic.However,aclassical

derivationoftheLondonequationsandfluxexpulsioncan

befoundin

aclassictextbookbytheFrench

Nobel

laureate

PierreGillesdeGennes.41Because

thisderivationshould

be

more

recognized,werepeatithere.

F.deGennes’derivationoffluxexpulsion

Thetotalenergyoftherelevantelectronsin

thesupercon-

ductorisassumed

tohavethreecontributions:thecondensa-

tion

energy

associated

with

the

phase

transition

ES,the

energyofthemagneticfieldEB,andthekineticenergyofthe

movingsuperconductingelectronsEk.Thetotalenergyrele-

vantto

theproblem

isthustaken

tobe

E¼

ESþEBþEk:

(13)

Thecondensation

energy

isthen

assumed

tobeconstant

whiletheremainingtwocanvaryin

response

toexternal

fieldvariations.Thesuper-currentdensity

iswritten

as

jðrÞ

¼nðrÞevðrÞ;

(14)

wherenis

thenumber

density

ofsuperconductingelectrons

andvistheirvelocity,whichgives

akineticenergyof

Ek¼

ð

1 2nðrÞmv2ðrÞdV¼

ð

1 2

m

e2nðrÞj2ðrÞdV:

(15)

BymeansoftheMaxwellequationr

�B¼

4pj=candEq.

(3)forEB,thetotalenergy(13)becomes

E¼

ESþ

1 8p

ð

B2þk2ðr

�BÞ2

hi

dV;

(16)

wherewehaveassumed

that

nis

constantin

theregion

wherethereis

current,andtheLondonpenetrationdepth

isgiven

by

k¼

ffiffiffiffiffiffiffiffiffiffiffiffi

mc2

4pe2n

r

:(17)

Minim

izingtheenergyin

Eq.(16)withrespectto

Bthen

gives

theLondonequation

Bþk2r

�ðr

�BÞ¼

0;

(18)

inoneofitsequivalentform

s.Notice

thatthisderivationuti-

lizesnoquantum

concepts

anddoes

notcontain

Planck’s

constant.Itisthuscompletely

classical.A

similar

derivation

has

beenpublished

more

recentlybyBadıa-M

ajos.34

TheconclusionofdeGennes

isclearlystated

inhis1965

book41(emphasis

from

theoriginal):

“Thesuperconductor

findsanequilibrium

state

wherethesum

ofthekinetic

and

magnetic

energiesis

minimum,andthis

state,formacro-

scopic

samples,

correspondsto

theexpulsionofmagnetic

flux.”In

spiteofthis,most

textbookscontinueto

statethat

“fluxexpulsionhas

noclassicalexplanation”as

originally

stated

byMeissner

andOchsenfeld

3andrepeatedin

thein-

fluential

monographsbyLondon42andNobel

laureateMax

von

Laue.43

As

one

textbook

exam

ple,

Ashcroft

and

Mermin

44explain

that

“perfect

conductivityim

plies

atime-

independentmagneticfieldin

theinterior.”

G.Apurely

classicalderivationfrom

magnetostatics

Onemightobject

that

theelectronic

chargeeis

amicro-

scopic

constant,andthat

theLondonpenetrationdepth

kin

mostcasesisso

smallthat

itseem

sto

belongto

thedomain

ofmicrophysics.Soeven

ifquantum

concepts

donotenter

explicitly,theabovederivationdoes

havemicroscopic

ele-

ments.Itis

thusofinterest

that

from

apurely

macroscopic

pointofview

wecan

identify

thekinetic

energy

ofthe

conductionelectronssolely

withmagnetic

energy.45,46The

energythatshould

beminim

ized

would

then

beEq.(3)

EB¼

1 8p

ð

B2dV:

(19)

Unfortunately,this

willnotprovideanyinform

ationabout

thecurrents

that

arethesources

ofB.However,if

wecan

neglect

thecontributionfrom

fieldsat

asufficientlydistant

surface—

i.e.,when

radiationis

negligible—itis

possible

torewritethisexpressionas

E0 B¼

1 2c

ð

j�A

dV;

(20)

whereB¼

r�A.Theidea

isto

then

apply

thevariational

principleto

EjA¼

2E0 BÿEB,i.e.,theexpression

EjA¼

ð

1 cj�A

ÿ1 8p

r�A

ðÞ2

��

dV:

(21)

Heretheenergyis

written

interm

softhefieldA

andits

sourcej.TherearemanywaysofcombiningEqs.(19)and

(20)to

getan

energyexpression,butitturnsoutthatEq.(21)

istheonethatgives

thesimplestresults.

Startingfrom

Eq.(21)andaddingtheconstraintr

�j¼

0,

theintegration

issplitinto

interior,

surface,

and

exterior

regions.

Theresultis

atheorem

ofclassicalmagnetostatics

that

states,forasystem

ofperfect

conductors

themagnetic

fieldis

zero

intheinteriors

andallcurrentflowsontheir

surfaces(Fiolhaiset

al.29)This

theorem

isanalogousto

asimilar

resultin

electrostaticsforelectric

fieldsandcharge

densities

inconductors,sometim

esreferred

toas

Thomson’s

167


H.Essen

andM.C.N.Fiolhais

167

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icle

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theorem

(see

Stratton,47Sec.2.11).Readers

should

consult

thepaper

byFiolhaisetal.29fordetails.

Manyother

authors

havealso

reached

theconclusionthat

theequilibrium

stateofasuperconductorisin

fact

thestate

ofminim

um

magneticenergyofan

idealconductor.Someof

theseare,

inchronological

order,Cullwick,48Pfleiderer,17

Karlsson,49Badıa-M

ajos,34Kudinov,50andMahajan.21For

exam

ple,Cullwickwrites“T

hewell-knownMeissnereffect

inpure

superconductors

isshownto

bean

expectedrather

than

anunexpected

phenomenon…”

while

Kudinov50

explainsthat

it“isworthnotingin

this

connectionthat

the

expulsionofamagnetic

fieldto

theperiphery(theMeissner

effect)also

occurs

inaclassicalcollisionless

gas

ofcharged

particlesandthatthishappenssolely

because

thisstateisen-

ergetically

favorable.”

IV.CONCLUSIO

NS

Thereader

may

gettheim

pressionfrom

ourinvestigations

abovethat

weconsider

superconductivityto

beaclassical

phenomenon.Nothingcould

befurther

from

thetruth.As

implied

bytheGinzburg-Landautheory,51theBCStheory,52

andtheJosephsoneffect,53thephenomenonisquantum

me-

chanical

toalargeextent.Since

quantum

physics

mustlead

toclassicalphysics

insomemacroscopic

limit,itmust

be

possible

toderiveourclassicalresultfrom

aquantum

per-

spective.

Evans

and

Rickayzen54

did

indeed

derive

the

equivalence

ofzero

resistivityandtheMeissner

effect

quan-

tum

mechanically,butdid

notdiscuss

theclassicallimit.

What

wewantto

correctisthemis-statementthat

theMeiss-

ner

effect

proves

that

superconductors

are“notjust

perfect

conductors.”

Accordingto

basic

physics

andalargenumber

ofindependentinvestigators,thespecificphenomenon

of

fluxexpulsionfollowsnaturallyfrom

classicalphysics

and

thezero

resistance

property

ofthesuperconductor—

they

are

justperfectconductors.

Inconclusion,wehavecarefullyexam

ined

theevidence

fortheoftrepeatedstatem

ents

intextbooksthat

(1)thereis

noclassicaldiamagnetism

and(2)thereisnoclassicalexpla-

nationofmagnetic

fluxexpulsion.Wehavefoundthat

the

theoreticalarguments

forthesestatem

ents

arenotrigorous

andthat,forsuperconductors,theseparticularphenomena

arein

goodagreem

entwiththeclassicalphysics

ofideal

conductors.

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