Lecture Notes 19: Magnetic Fields in Matter I; Dia-/Para-/Ferro ...

UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede 

LECTURE NOTES 19 

MAGNETIC FIELDS IN MATTER 

THE MACROSCOPIC MAGNETIZATION, Μ 

There exist many types of materials which, when placed in an external magnetic field 

 

Bext 

( r ) become magnetized — i.e. at the microscopic level ∃ internal atomic/molecular magnetic 

dipole moments m atom 

or m 

molecular 

, which, in the presence of the external aligning magnetic field 

 

Bext 

( r ) produce magnetic torques τ ( r ) = m ( r ) × B 

ext ( r 

) which act on the individual 

atomic/molecular dipole moments, thereby causing a net alignment of the atomic/molecular 

 

magnetic dipole moments matom mmolecular 

which in turn results in a net, macroscopic magnetic 

polarization, also known as the magnetization, Μ 

( r ) . This is analogous to the situation 

associated with dielectric materials where electrostatic torques τ ( r ) = p ( r ) × E 

ext ( r 

) act on 

 

individual atomic/molecular electric dipole moments patom pmolecular 

in an external electric 

 

field Eext 

( r) 

resulting in a net, macroscopic electric polarization, Ρ 

( r ) . 

 

In the absence of an external applied magnetic field (i.e. Bext 

( r ) = 0) the macroscopic 

 

alignment of the atomic/molecular magnetic dipole moments matom mmolecular 

(in many, but not all 

magnetic materials) is random, due to fluctuations in the internal thermal energy of the material 

at finite temperature (e.g. room temperature). Thus, no net macroscopic magnetization 

 

M ( r ) exists in many such materials for Bext 

( r ) = 0 at finite (absolute) temperature, T. 

 

We define the macroscopic magnetic polarization (a.k.a. magnetization) M ( r ) of a magnetic 

material in complete analogy to that associated with the macroscopic electric polarization 

 

P( r ) of a dielectric material: 

 

Macroscopic Electric Polarization P( r ) : 

⎛electric dipole moment ⎞ 

P ( r) 

= ⎜ ⎟ at point r SI Units of P : Coulombs/m 2 

⎝ unit volume ⎠ 

 

N 

 

N 

pmol ( r) ( ) ( ) 

( ) 

i i Qd 

i i 

ri 

P r = nmol 

pmol 

r ≡ ∑ 

= 

Volume, V ∑ Volume, V 

n 

mol 

= 

# atoms/molecules 

unit volume 

i= 1 i= 

1 

 

Macroscopic Magnetic Polarization/Magnetization M ( r ) : 

⎛magnetic dipole moment ⎞ 

M ( r) 

= ⎜ ⎟ at point r SI Units of M : Amperes/meter 

⎝ unit volume ⎠ 

N 

 

N 

mmol ( r) ( ) ( ) 

( ) 

i i Ia 

i i 

ri 

M r = nmol 

mmol 

r ≡ ∑ 

= 

Volume, V ∑ Volume, V 

i= 1 i= 

1 

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 

2005-2008. All Rights Reserved. 

1


Note that the magnetization ( r ) 

 

K( r) 

(Amperes/meter), whereas the electric polarization P( r ) 

surface charge density, σ ( r 

) (Coulombs/m 2 ). 

There are (at least) four kinds of magnetism: 

1.) DIAMAGNETISM: 

Μ 

has SI units the same as that for a surface current density, 

has SI units the same as that for a 

The induced macroscopic magnetization Μ 

( r ) is antiparallel to B ( r ) 

dia 

 

ext 

. Due to the physics 

origin of diamagnetism at the microscopic scale – i.e. at the atomic/molecular scale, ALL 

substances are diamagnetic! However, diamagnetism is very a weak phenomenon – other kinds 

of magnetism (see below) can “over-ride”/mask out the diamagnetic behavior of a material. 

Diamagnetism results from changes induced in the orbits of electrons in the atoms/molecules 

of a substance, due to the applied/external magnetic field. The direction of the change in orbital 

motion of the electrons is such that it to opposes the change in applied magnetic flux (this is 

nothing more than Lenz’s Law acting at the microscopic/atomic/molecular scale!). 

Superconductors are examples of strong diamagnets – they are in fact perfect diamagnets, 

 

Bext 

r (* if no flux-pinning 

Μ 

dia 

r vanishes when 

 

B r = . 

completely* screening out the applied external magnetic field ( ) 

defects are present in the superconducting material). Note that ( ) 

ext 

( ) 0 

2.) PARAMAGNETISM: 

The induced macroscopic magnetization, Μ 

para ( r ) is parallel to Bext 

( r ) . Atoms or molecules 

that have a net orbital and/or intrinsic spin magnetic dipole moment m (e.g. atoms/molecules 

with unpaired electrons – such as A 

, Ba, Ca, Na, Sr, U, 

… and also metals – due to the 

magnetic dipole moments m associated with intrinsic spins of the conduction electrons) are 

 

paramagnetic materials. The external applied magnetic field Bext 

( r ) exerts a torque on these 

atomic/molecular magnetic dipole moments m which tends to (partially) align them, giving rise 

to a net Μ 

para ( r ) which is parallel to Bext 

( r ) . The energy of alignment UM 

( r ) =−m ( r ) i Bext 

( r ) 

is a minimum when m 

is parallel to Bext 

( r ) . This is analogous to the net induced electric 

polarization Ρ 

( r ) which is parallel to Eext 

( r ) in dielectric materials, the energy of alignment 

 

UE( r) =−p( r) iEext( r) 

when p 

is parallel to Eext 

( r) 

. Note that Μ 

para ( r ) also vanishes 

 

when B ( r ) = 0. 

ext 

Μ 

Μ 

dia 

para 

 

( r ) 

 

( r ) 

 

B 

 

B 

ext 

ext 

 

( r ) 

 

( r ) 

2 


2005-2008. All Rights Reserved.


Μ 

depends on the (entire) 

!! There exists a non-linear hysteresis-type relation between 

3.) FERROMAGNETISM: The macroscopic magnetization, 

ferro ( r ) 

 

past history of exposure to Bext 

( r ) 

Μ 

( r ) and B ( r ) 

 

ferro 

ext . Iron and other ferromagnetic materials have a “macroscopic” crystalline 

domain structure (a typical scale length involves many thousands of atoms), within a domain 

(nearly) all of the atomic/molecular magnetic dipole moments m are aligned parallel to each 

other ⇒Μ 

domain ( r ) can be very large. However, the orientation of Μ 

domain 

over many domains is 

 

≈ random, unless B ( r) ≠ 0 . However, ferromagnetic materials have a critical temperature 

ext 

(known as the Curie Temperature T 

C 

) below which the domains can spontaneously align − a 

phase transition occurs in the material at this temperature! In the presence of an external applied 

magnetic field B 

ext 

the alignment of ferromagnetic domains tends to be parallel to B 

ext 

, but it is in 

fact (more) complicated than this, because it it history dependent!!! The alignment arises from 

quantum mechanics – intrinsic spin and the Pauli exclusion principle. Thus, Μ 

ferro ( r ) does not 

 

vanish when B ( r ) = 0!!! 

ext 

History-Dependence / Hysteresis Relation Between Μ 

ferro 

and B 

ext 

for Ferromagnetic Materials 

for T < T (= Curie Temperature): 

C 

Ferromagnetic behavior vanishes for T > TC 

The material then becomes paramagnetic. 

The arrows indicate the path taken for Μ 

ferro 

: B 

ext 

starts at B 

ext 

= 0 , then goes to B 

max 

, then 

through 0, going to B min 

, then through 0 again and then going to B 

max 

, etc…. 

4.) ANTI-FERROMAGNETISM (a.k.a. FERRIMAGNETISM) 

In some magnetically-ordered materials ∃ an anti-parallel alignment of intrinsic spins, due to 

two (or more) inter-penetrating crystalline structures, such that no spontaneous magnetization in 

the bulk material occurs. Ferrimagnetism/antiferromagnetism occurs for temperatures T < T Ne ' el 

. 

Materials exhibiting antiferromagnetic properties are relatively uncommon – e.g. URu 2 Si 2 . 



3


FORCES & TORQUES ON MAGNETIC DIPOLES 

When a magnetic dipole with magnetic dipole moment m is placed in an external magnetic 

field B 

ext 

a torque on the magnetic dipole τ 

M 

= m× B 

 

ext will occur, just as we saw for the case of 

an electric dipole with electric dipole moment p when it is placed in an external electric field 

E 

giving rise to a torque on the electric dipoleτ E 

= p× 

E ext . 

ext 

As we also learned for the case of an electric dipole in a uniform external electric field, 

similarly, for a magnetic dipole placed in a uniform external magnetic field, there is no net force 

acting on the magnetic dipole. 

For a magnetic dipole with magnetic dipole moment m (e.g. arising from a current loop) 

placed in an uniform external magnetic field B 

ext 

the net force on the is zero: 

F 

net 

I d 

( r 

) B 

m ext( r 

) I ( d 

( r 

)) B 

= ∫ 

′ ′ × ′ = ′ ′ × 

ext( r 

 

 

′) 

= 0 

C′ ∫ 

C′ 

 

cf w/ that for an electric dipole placed in a uniform external electric field E ext 

: 

 

net 

F 

= F r + F r = qE r − qE r = q E r − E r = 

≡ 0 

( ) 0 

( ) ( ) ( ) ( ) ( ) ( ) 

p + + − − ext + ext − ext + ext − 

 

The nature of the magnetic (electric) torque τ 

M 

= m× B 

 

ext ( τ 

E 

= p × Eext) 

is such that it tends 

 

to align m( p) 

with (i.e. parallel to) the applied/external B ext ( E 

ext ) respectively. 

 

⇒ The effect(s) of magnetic torque explains paramagnetism, with Μ 

para 

Bext 

. One might be 

tempted to believe that paramagnetism should be a universal phenomenon, common to all 

materials. However, paramagnetism is connected to the intrinsic magnetic dipole moment of an 

unpaired electron and/or its orbital magnetic dipole moment. Because of the Pauli exclusion 

principle (identical fermions, here, electrons) cannot be in the exact same quantum state, hence 

pairs of electrons can only be in the same quantum state with one of them spin-up, and the other 

spin down. Thus, torques on paired magnetic dipole moments (or more correctly, the B -fields 

associated with the paired electron magnetic dipole moments m ) cancel. 

⇒ Paramagnetism only arises in atoms/molecules with an odd number of electrons – the 

outermost electron is unpaired ⇒ hence it (alone) is subject to magnetic torque(s). 

As we saw in the case for an electric dipole with electric dipole moment p in a non-uniform 

external electric field E 

ext 

, a non-zero force acts on the electric dipole. Similarly, for a magnetic 

dipole, with magnetic dipole moment m in a non-uniform external magnetic 

field B 

ext 

experiences a non-zero force: 

 

F 

m 

r m r Bext r m r Bext 

r 

 

F 

r p r E r p r E r 

( ) ( ( ) ) ( ) 

( ) ( ( ) ) ( ) 

( ) =∇ ( ) i ( ) = i ∇ {last step valid iff m( r) 

 

( ) =∇ 

 

( ) i 

 

( ) = 

 

i ∇ 

 

{last step valid iff p ( r ) 

p ext ext 

e 

= constant vector} 

= constant vector} 

4 




Similarly, work (= potential energy) of a magnetic (electric) dipole moment in an external 

 

magnetic (electric) field, Bext 

( Eext 

) are (respectively) given by: 

 

 

W 

= PE . . =−m i B vs. W 

= PE . . =−p i E 

m m ext 

p p ext 

The Physics of Diamagnetism 

Atomic electrons orbit/revolve around the nucleus of the atom at some mean / average / 

characteristic radius, R. Atomic electrons bound to the nucleus of an atom no longer behave like 

point-like particles, but as quantum-mechanical matter waves. However, an orbiting atomic 

electron “wave” still constitutes a circulating current: 

I 

QM 

eve eve eve 

~ = = λe C = 2π 

R for ground state 

λ C 2πR 

e 

Gnd State 

Gnd State 

Conventional 

Current, I 

ẑ 

 

B 

ext 

= Bzˆ 

o 

R 

 

m 

e 

 

v 

e 

=−m zˆ 

e 

e − 

= vϕˆ 

e 

Classically, a circulating point electric charge has: 

I 

Class 

e 

= with τ 2 

orbit 

= C = π R 

τ 

ve 

v 

⇒ 

e 

orbit 

I 

Class 

eve 

= = I 

2π R 

QM 

⎛ ev 

Then: m= Ia =− e 

⎞ 

⎜ ⎟ 

⎝2π 

R ⎠ 

π 2 1 

R zˆ 

=− ( ev ) ˆ 

eR z 

2 

due to e − charge 

 

With no external magnetic field applied B 

ext 

= 0, thus the forces acting on the atomic electron are: 

 

Felectrostatic 

= Fcentripetal 

2 

2 

2 

2 

1 Ze 

ve 

1 Ze v 

− ˆ 

ˆ 

2 r =− m 

e a 

centipetal 

=− m 

e r ⇒ Equation A: 

2 r 

e 

ˆ= 

m ˆ 

e r 

4πε 

R 

R 

4πε 

R R 

o 

e 

m = mass of electron 

Z = nuclear electric charge # {+Ze = nuclear charge} 

o 



5


 

With an external magnetic field present Bext 

≠ 0, thus the forces acting on the atomic electron are: 

 

net 

FEM = Felectrostatic + FB = F′ 

centripetal 

 

2 

net 

1 Ze 

F ˆ 

EM 

= Felectrostatic + FB =− r− e 

2 ( ve × Bext 

) 

4πε 

o 

R 

 

Suppose ˆ 

Bext 

= Bz 

0 

and v ˆ 

e 

= vϕ 

e 

(as shown in above pix) 

ˆ ϕ × zˆ = ˆ ϕ× cosθrˆ 

− sinθθˆ 

 

θ = 90 

rˆ 

× ˆ θ = ˆ ϕ 

ˆ θ× ˆ ϕ = rˆ 

ˆ ϕ × rˆ 

= ˆ θ 

Then: 

Then: ( ) 

( ˆr ) 

=− ˆ ϕ× ˆ θ =− − = + ˆr 

2 

net 1 Ze 

 

v′ 

2 

e 

ˆ 

EM 

=− −ev′ 

2 eBor 

= ′ 

centripetal 

=− 

e 

4πε 

o 

R 

R 

F 

F m rˆ 

 

sinθ 

= sin 90 = 1 

 

cosθ 

= cos90 = 0 

 

2 

2 

1 Ze 

v′ 

e 

Then for Bext 

≠ 0 we have Equation B: + ev′ 

2 eBo = me 

4πε 

o 

R 

R 

Note that since we have an additional term on LHS of Equation B, then we see that: 

 

 

v′ B ≠0 ≠ v B = 0 . 

( ) ( ) 

e ext e ext 

Subtract Equation A from Equation B: 

e 2 2 

ev′ 

eB0 

= m ( v′ 

e 

−ve 

) 

0 

R 

for 

 

⇒ v′ 

ˆ 

e 

> ve Bext 

=+ B z 

> 0 

If the change in , 

e 

But: ve ve ve 

ˆ θ × rˆ 

=−ˆ 

ϕ 

ˆ ϕ× ˆ θ =−rˆ 

rˆ 

× ˆ ϕ =−ˆ 

θ 

> 0 

( v′ for ˆ 

e 

< ve Bext 

=−B0 z) 

v Δv ≡( v′ 

− v ) is small, then: v′ 2 − v 2 = ( v′ − v )( v′ + v ) =Δ v ( v′ 

+ v ) 

e e e 

′ = +Δ (since v ( v′ 

v ) 

Δ ≡ − ) 

e e e 

2 2 

∴ v′ e 

− ve =Δ ve( ( ve +Δ ve) 

+ ve) =Δ ve( ve +Δ ve + ve) =Δ ve( 2ve +Δ ve) 

= 2v Δ v +Δv 2v Δv 

e e 

2 

e e e 

neglect 

m 

R 

e 

∴ ev′ B = e( v +Δv ) B ( v Δv 

) 

e 0 e e 0 

2 

e e 

2 v e 

me 

ve 

ev e 

B0 

Δ 

R 

eBo 

R 

or: Δve 

 

2m 

e 

e e e e e e e e e 

 

6 




But if: 

eve 

ev′ 

e 

I = and I′ = and v′ 

e 

> v 

2πR 

2πR 

Then: 

e( v′ e 

− ve) 

eΔve 

Δ I = I′ 

− I = = but: 

2π 

R 2π 

R 

∴ 

2 

2 

eBo 

R eBo 

Δ I = = 

4 π me 

R 4π me 

Δ I = 

2 

eB0 

π m 

Then: 

4 

e 

m Ia Iπ 

R 

2 

= = and 

′ ′ 

m′ Ia ′ I′π 

R 

e 

eBo 

R 

Δve 

 

2m 

2 

2 

= = ( a= 

π R ) 

2 

Thus: Δ m= m − m= ( I − I) a=Δ Ia=Δ 

Iπ 

R 

2 

2 

⎛ eB 

∴ Δ m=Δ Iπ 

R = o 

⎜ 

⎝4 

π me 

 

But recall that m=−mzˆ 

⎞ 

⎟ π R 

⎠ 

2 

eBR 

= 

4m 

2 2 

o 

e 

e 

Therefore: 

i.e. m points down. 

2 2 

eBR 

o 

Δ m=− zˆ, 

B ˆ 

ext 

= Bo 

z 

4m 

e 

Or: 

2 2 

⎛eR 

⎞ 

Δ m=−⎜ ⎟B 

⎝ 4me 

⎠ 

ext 

The point is, that for diamagnetic materials, the change in the magnetic dipole moment m , Δm 

 

is opposite to the direction of B 

ext 

- i.e. if B ˆ 

ext 

= Bz 

o 

increases, then m also increases, but in the 

opposite direction to try to cancel/buck the external/applied magnetic field, B 

ext 

. This is a 

simply a manifestation of Lenz’s Law at the atomic scale!!! 

This is what phenomenon of diamagnetism is due to, at least from a ≈ semi-classical perspective. 

The induced dipole moments in diamagnetic materials (essentially every material) point in the 

direction opposite to the applied magnetic field. The macroscopic magnetization Μ resulting 

from diamagnetism is relatively speaking very small. Diamagnetism (except in superconductors) 

is extremely weak. 



7


 

THE MAGNETIC VECTOR POTENTIAL Ar ( ) , THE MAGNETIC FIELD B( r) 

=∇× A( r) 

OF A MAGNETIZED OBJECT WITH MAGNETIZATION Μ 

( r ) 

 

Recall that the magnetic vector potential Ar ( ) of a magnetic dipole with magnetic dipole 

2 

moment ( Amp-m ) 

m 

is: 

 

⎛ μo 

⎞m× 

rˆ 

Adipole 

( r) = ⎜ ⎟ 2 

⎝4π 

⎠ r 

{SI Units: Tesla-meters = Newtons/Ampere = F/I !!!} 

Thus, in a magnetized object with macroscopic magnetization 

(magnetic dipole moment per unit volume) Μ 

( r′ ) , each volume 

element dτ ′within the volume v′ has a magnetic dipole moment 

 

m r′ =Μ 

r′ dτ ′. 

associated with it of: ( ) ( ) 

Thus, the infinitesimal contribution to the magnetic vector 

 

Ar 

m r′ 

 

potential ( ) due to the magnetic dipole moment ( ) 

associated with the macroscopic magnetization Μ 

( r′ ) in the 

infinitesimal volume element dτ ′is: 

 

 

 

 

⎛ μ m( r ) ˆ ( r ) d ˆ 

o ⎞ ′ × r ⎛ μ Μ ′ τ ′ × 

o ⎞ r 

dA( r ) = ⎜ ⎟ = 

2 ⎜ ⎟ 2 

⎝4π 

⎠ r ⎝4π 

⎠ r 

 

with r = r − r′ 

 

Then the total magnetic vector potential Ar ( ) is obtained by integrating this expression over the 

entire volume v′ of the magnetized material: 

 

 

 

⎛ μ ( r ) d ˆ 

o ⎞ Μ ′ τ ′ × r 

Ar ( ) = ∫ dAr ( ) = 

v′ ⎜ ⎟ 

2 

4π 

∫v′ 

⎝ ⎠ r 

⎛1⎞ 

1 rˆ 

Now again: ∇ ′ ⎜ ⎟=∇ ′ = 

2 

⎝ r ⎠ r − r′ 

r 

⎛ μ ⎞ ⎡ ⎛ 1 ⎞⎤ 

Ar = ⎜ Μ r′ × ∇′ dτ 

′ 

4π ⎟ ∫v′ 

⎢ ⎜ ⎟⎥ 

⎝ ⎠ ⎣ ⎝ r ⎠⎦ 

 

∇× fA = f ∇× A − A× ∇f 

o 

Thus: ( ) ( ) 

Integrating by parts, and using ( ) ( ) ( ) 

 

μ ⎧ 

 

⎛ ⎞⎪ 1 ⎡Μ 

( r′ 

) ⎤ ⎫⎪ 

Ar = ⎜ ⎨ ∇ ′ ×Μ r′ ⎤dτ 

′ − ∇ ′ × ⎢ ⎥dτ 

′ ⎬ 

4π ⎟ ∫v′ ⎣ ⎦ ∫v′ 

⎝ ⎠⎪ ⎩ 

r 

⎣ r ⎦ ⎪⎭ 

o 

Then: ( ) ⎡ ( ) 

∫ 

Then using: V ( r) dτ 

V ( r) da V( r) 

v 

 

∇× =− ∫ × = 

S 

(See Griffiths Problem 1.60 (b), page 56) 

: 

( Arbitrary Vector Point Function) 

8 




Ar 

μ 1 1 

= ⎛ ⎞⎧ ⎡∇ ⎨ ′ ×Μ 4π r 

′ ⎤ d ′ + ⎡Μ 

r 

′ × da 

⎜ ⎟ ′ ⎤⎬ 

⎫ 

v′ ⎣ ⎦ 

s′ 

⎩ r 

 

r ⎣ ⎦ 

⎭ 

o 

Thus: ( ) ( ) τ ( ) 

But: 

 

da′ = nda ˆ ′ 

⎝ ⎠ ∫ ∫ 

μ 1 

o 

μo 

1 

Then: Ar ( ) ⎡ 

 

 

= ( r) 

dτ 

( r) 

nˆ 

da ABound 

( r) 

4π∫ 

∇ ′ ×Μ ′ ⎤ ′ + ⎡Μ ′ × ⎤ ′ = 

v′ r ⎣ ⎦ 4π 

∫S′ 

⎣ ⎦ 

 

r 

 

 

≡ JBound( r′ ) 

≡ KBound( r′ 

) 

 

 

 

 

 

μ JBound 

( r′ o 

) μ K 

o Bound ( r′ 

) 

Or: ABound 

( r) 

= dτ 

′ + 

da′ 

4π∫v′ r 4π∫ 

with r = r −r′ 

S′ 

r 

 

Compare this result to that which we obtained for the magnetic vector potential Ar ( ) associated 

 

with a free volume current density J 

free ( r′ ) and a free surface/sheet current density K 

free ( r′ ) 

(see P435 Lecture Notes 16, page 6): 

 

 

μ J 

free ( r′ ) K 

free ( r ) 

o 

μ ′ 

o 

 

Afree 

( r) 

= dτ 

′ + 

da′ 

4π∫v′ r 4π∫ 

with r = r −r′ 

S′ 

r 

Thus for a magnetized material with macroscopic magnetization (magnetic dipole moment per 

unit volume) Μ 

( r′ ) contained within in the enclosing source volume v′ bounded by the surface 

arising from the sum total of 

S′ , the magnetic vector potential at the field/observation point Ar ( ) 

the macroscopic magnetization ( r′ ) 

 

contributions from an equivalent bound volume current density JBound 

( r′ ) ≡∇ ′ ×Μ( r′ 

) 

 

equivalent bound surface current density K ( r′ ) ≡Μ ( r′ 

) × nˆ 

Μ present in the material can be equivalently represented by 

Bound 

normal at the surface of the magnetized material. 

surface 

and an 

where ˆn = outward unit 

On the interior of the magnetized material: 

 

JBound 

( r′ ) ≡∇ ′ ×Μ( r′ 

) = equivalent bound volume current density, SI units = Amps/m2 

 

2 

Amps / m 

1/ m 

Amps / m 

On the surface(s) of the magnetized material: 

 

K ( ) ( ) ˆ 

Bound 

r′ ≡Μ r′ 

× n = equivalent bound surface current density, SI units = Amps/m 

 

Then: ( ) 

Amps / m Amps / m 

 

 

 

 

 

μ J 

o Bound ( r′ ) μ K 

o Bound ( r′ 

) 

ABound 

r = dτ 

′ + 

da′ 

4π∫v′ r 4π∫ 

with r = r − r′ 

S′ 

r 



9


MAGNETIC MATERIALS 

 

JBound 

( r′ ) ≡∇×Μ( r′ 

) 

 

K r′ ≡Μ r′ 

× n 

Bound 

( ) ( ) ˆ 

surface 

DIELECTRIC MATERIALS 

 

ρ 

Bound 

r′ =−∇Ρ i r′ 

 

σ r′ =Ρ 

r′ 

i n 

⇔ ( ) ( ) 

⇔ ( ) ( ) ˆ 

Bound 

surface 

So again, instead of integrating over the macroscopic magnetization Μ 

( r′ ) (or polarization 

Ρ 

( r′ ) ) arising from the direct contributions from the infinitesimal magnetic (and/or electric) 

 

 

dipoles m( r′ ) (and/or p ( r′ )), we replace these by macroscopic bound volume and surface 

 

 

current distributions JBound 

( r′ ) and KBound 

( r′ 

) (and/or ρ 

Bound ( r′ 

) and σ 

Bound ( r 

′) 

); we can then 

 

obtain Ar ( ) (and/or V ( r 

)). Once Ar ( ) (and/or V ( r ) ) is known, we can then obtain B ( r ) 

B 

( r) =∇× 

A 

( r) 

(and/or E( r) 

from ( 

E r) 

=−∇V 

( r) 

) ! 

Note that for a magnetized material with macroscopic magnetization Μ 

( r′ ) we can also 

obtain the equivalent bound current, IBound 

from: 

 

 

I = J r′ da′ + K r′ d 

′ 

∫ 

( ) ( ) 

∫ 

Bound Bound Bound surface 

S⊥′ ⊥ 

⊥ 

C⊥′ 

surface 

from 

Consider the equivalent bound surface current K 

Bound 

associated with a thin slab of 

 

magnetized material that has been placed in uniform magnetic field B ˆ 

ext 

= Bz 

o 

, in turn producing 

a uniform macroscopic magnetization (magnetic dipole per unit volume) Μ 

=Μ z ˆ 

o 

. At the 

microscopic level, atoms and/or molecules will tend to have their induced and/or permanent 

magnetic dipole moments lined up parallel/anti-parallel to B 

ext 

for paramagnetic / diamagnetic 

materials, respectively. Suppose that the material is paramagnetic, as shown in the figure below: 

 

B ˆ 

ext 

= Bz 

o 

produces 

m= Ia = Iazˆ 

uniform magnetization 

Μ 

=Μ z ˆ 

o 

It can be seen from the above figure that on the interior of the uniformly magnetized material 

the atomic/molecular microscopic currents will cancel each other (for uniform magnetization, 

Μ=Μ 

z ˆ 

o 

) except on the periphery (i.e. the surface) of the magnetic material. 

For uniformly magnetized material(s), e.g. Μ 

=Μ z 

 

ˆ J r ≡ ∇×Μ r =∇× Μ zˆ = 0 

o 

. 

: ( ) ( ) ( ) 

Bound 

o 

10 




I 

K r 

= ∫ 

′ d ′ for uniform magnetization, e.g. Μ=Μ 

z ˆ 

o 

. 

Then: ( ) 

Bound Bound surface 

C⊥′ 

⊥ 

surface 

Example: Consider a cylindrical rod of radius a and length of magnetized material immersed 

 

in a uniform B ˆ 

ext 

= Bz 

o 

as shown in the figure below. Then the magnetization is uniform, e.g. 

Μ=Μ 

 

z 

 

ˆ 

o 

. Thus, no equivalent bound volume current density JBound 

( r) 

exists, because 

 

J r =∇×Μ r =∇× Μ zˆ = 0 for uniform magnetization, Μ 

=Μ z ˆ 

o 

. 

Bound 

 

Μ=n m 

( ) ( ) ( ) 

mol 

mol 

Uniform 

o 

Μ=Μ 

z ˆ 

o 

=Μ z ˆ 

a 

o 

 

Μ= 

 

Μ= 

 

m 

 

m 

Tot 

Tot 

Volume 

π a 

2 

ẑ 

bound 

 

B 

ext 

= Bzˆ 

produces uniform Μ=Μ 

z ˆ 

o 

K ( ) 

o 

I 

K r 

= ∫ 

′ d ′ 

Bound 

C surface 

Bound ⊥ surface 

⊥′ 

ŷ = K ˆ 

bound 

ϕ 

 

ˆϕ K = Μ× nˆ 

with nˆ 

= ˆ ρ 

 

m 

Tot 

= total dipole moment ˆx ˆρ =Μ ( ˆ ˆ) 

ˆ ˆ 

o 

z× ρ =Μ 

oϕ = Koϕ 

 

of magnetized ∴ I ˆ 

Bound 

= Kboundϕ =Μo 

ˆ, ϕ 

 

cylinder i.e. K =Μ ˆ ϕ = K ˆ ϕ 

Bound 

surface 

Bound o o 

surface 

 

 

If Bext 

≠ uniform magnetic field, will result in a non-uniform magnetization, i.e. Μ≠uniform, 

 

which in turn also implies that the equivalent bound volume current density J 

Bound 

=∇×Μ≠0 

. 

This means that at microscopic level the atomic/molecular current loops no longer 

 

cancel each 

other (completely) in the interior region of the magnetized material. Hence for Μ≠uniform: 

 

Volume 

 

JBound ( r′ ) =∇×Μ( r′ ) ≠0 

⇒ IBound =∫ JBound 

( r′ 

) da 

′ 

⊥ 

Similarly, we also expect for non-uniform Μ that K ( r ) = Μ ( r ) × nˆ ≠0 

 

S⊥ 

Bound 

 

′ ′ 

surface 

and thus we 

will also have an equivalent bound surface current: 

then I Surface 

K 

Bound Bound ( r 

= ∫ 

′) 

d ′ 

′ 

⊥ surface (for magnetized cylinder in above figure: d ⊥′ = dz) 

C⊥ 

surface 

Then using the principle of linear superposition: 

 

Tot Volume Surface 

 

I = I + I = J r′ da + K r′ d 

′ 

∫ 

( ) ( ) 

Bound Bound Bound Bound Bound surface 

S⊥′ ⊥ 

⊥ 

C⊥′ 

surface 

Note that these equivalent bound currents are flowing in different places in/on the magnetized 

material – one is flowing inside the material, the other is flowing on the surface of the material. 

∫ 



11


 

 

( ) 0 

Note also that: ∇ i JBound 

( r) =∇× ∇×Μ ( r) 

= always in magnetostatics, because i J ( r) 

is the LHS of the Continuity Equation for equivalent bound currents (i.e. conservation of bound 

charge): 

 

 

∂ρBound 

( rt , ) 

 

∇ i JBound 

( r, t) 

=− = 0 if ρBound 

( rt , ) ≠ fcn(t). 

∂t 

Note also that (here): ( ) 

 

∇× ∇×Μ =∇ ∇ Μ −∇ Μ = 0 

2 

( r ) ( ( r) 

) ( r) 

i i.e.: ∇∇Μ i ( r) 

{We will come back to this relation in the near future…} 

∇ 

Bound 

 

2 

( ) =∇Μ( r) 

 

Griffiths Example 6.1: 

 

Determine the magnetic field B ( r ) associated with a uniformly magnetized sphere of radius R 

with uniform magnetization Μ=Μ 

z ˆ 

o 

as show in the figure below. Choose the local originϑ 

to be at the center of the magnetized sphere: 

Μ=Μ 

 

z ˆ 

o 

ϑ 

ẑ 

ϕ 

θ 

ˆϕ 

ˆr 

ˆ θ 

P( r ) Field/Observation Point 

ŷ 

ˆϕ 

ˆx 

 

ˆ 

Bound 

= ∇×Μ =∇× Μ 

o 

= 0 . 

 

K r r n z r θ ϕ where: nˆ 

= rˆ 

Since the magnetization of the sphere is uniform, then: J ( r) ( r) ( z) 

However: ( ) =Μ ( ) × ˆ =Μ ( ˆ× ˆ) =Μ sin ˆ 

Bound 

surface 

o o 

Note: z rˆ ( rˆ ) rˆ ( rˆ) 

surface 

ˆ × = cosθ − sinθθˆ × = 0 − sinθ ˆ θ × =+ sin θ ˆ ϕ since: rˆ× rˆ= 0, ˆ θ × rˆ=− 

ˆ ϕ 

Now recall that we learned in Griffiths Example 5.11 (p. 236-7)/P435 Lecture Note 16 p. 18-19 

 

(the charged spinning hollow sphere) that: K = σ v = σω × r′ 

= σωRsin 

ϕ ˆ ϕ 

free 

Uniformly Magnetized Sphere: 

 

K sin ˆ 

Bound 

=Μo 

θ ϕ 

 

2 2 

Binside ( r < R) 

= μoΜ ozˆ 

= μoΜ 

3 3 

 

⎛ μo 

⎞ m 

B ( ) 

3 ( 2cos ˆ sin ˆ 

outside 

r > R = ⎜ ⎟ θr+ 

θθ ) 

⎝4π 

⎠r 

4 

3 4 3 

m= πR Μ = πR Μ ˆ 

oz 

3 3 

vs. 

Charged Spinning Hollow Sphere: 

 

K sin ˆ 

free 

= σωR 

θϕ ⇒ Μ o 

= σωR 

 

2 

Binside 

r < R = μo 

σωR zˆ 

3 

 

⎛ μo 

⎞ m 

B 2cos ˆ sin ˆ 

outside 

r > R = ⎜ ⎟ θr+ 

θθ 

3 

⎝4π 

⎠r 

4 4 

m= π R 3 σωR zˆ 

= π R 4 σωzˆ 

3 3 

⇐ ( ) ( ) 

⇐ ( ) ( ) 

vs. ( ) 

12 




For magnetic media, we have obtained the following relations: 

 

 

∂ρBound 

rt , 

Bound current continuity equation: ∇ i JBound 

( r, 

t) 

=− 

∂t 

 

( ) 

Equivalent bound volume current density JBound 

( r ) =∇×Μ( r 

) , where ( r ) 

 

magnetic dipole moment per unit volume) Μ ( ) = ( ) = ( ) 

 

Μ = magnetization (a.k.a. 

r nmo 

mmol r mTot 

r volumeand the 

 

KBound 

r = Μ r × n ≠ with corresponding relations 

surface 

I Surface 

K r 

d 

Bound 

= ∫ 

Bound 

 

⊥ surface 

equivalent bound surface current density ( ) ( ) ˆ 0 

 

Volume 

 

I 

Bound 

= ∫ JBound 

( r) 

da 

S⊥ 

⊥ 

and ( ) 

C⊥ 

surface 

Using the principle of linear superposition: total current density = free current + bound current 

density: 

 

Jtot ( r) = J 

free ( r) + JBound 

( r) 

 

K r = K r + K r 

( ) ( ) ( ) 

tot free Bound 

Ampere’s Circuital Law becomes (in differential form) for the magnetic field B ( r) 

 

∇× B ( r) = μ J ( r) = μ J ( r) + μ J ( r) 

o Tot o free o Bound 

Note that this is the analog of Gauss’ Law (in differential form) for the electric field E( r) 

∇ 

1 1 

E r = 

Tot 

r 

free 

r 

Bound 

r 

ε 

= 

ε 

+ 

i 

( ) 

( ) ρ ( ) ρ ( ) ρ ( ) 

o 

o 

 

: 

 

: 

 

Now: J ( r) ≡∇×Μ( r) 

Bound 

 

 

( ) 

∴ ∇× B ( r) = μ J ( r) + μ ∇×Μ( r) 

o free o 

 

∇× B r − ∇×Μ r = J r 

or: ( ) μ ( ) 

 

 

( ) μ ( ) 

o o free 

1 

μ 

or: ∇× B ( r) −∇×Μ ( r) = J ( r) 

o 

⎧ 1 ⎫ 

⎨ 

⎬ 

⎩μo 

⎭ 

free 

or: ∇× B ( r) −Μ ( r) = J ( r) 

free 

 

 

1 

μ 

We now define the auxiliary field: H( r) ≡ B( r) −Μ( r) 

o 

SI Units of H = Amperes/meter 

– the same as that for Μ !!! 

 

We could call H( r) 

the magnetic displacement, in analogy to the electric displacement: 

But usually we just call H “the H -field”. 

 

D r E r r 

( ) = ε ( ) +Ρ( ) 

o 



13


Ampere’s Law for the H -field (in differential form) then becomes: 

 

 

∇× H( r) = J 

free ( r) 

where: ( ) 1 

H r ≡ B( r) −Μ( r) 

μ 

o 

Ampere’s Law for the H -field is the analog of Gauss’ Law for the D -field, in differential form: 

 

∇ D r = ρ r 

D 

r ≡ ε E 

r +Ρ 

 

r 

i ( ) ( ) where: ( ) ( ) ( ) 

 

free 

 

n.b. both H( r 

) and D( r 

) are auxiliary fields, B ( r ) and E ( r ) 

In integral form, these relations become: 

H 

 

r d 

i I 

C 

enclosed 

∫ ( ) = free 

( ) 

∫ D r 

da 

Q 

S 

o 

 

 

 

 

are fundamental fields. 

1 enclosed enclosed enclosed 

∫ B r d = I Tot 

= I I 

free 

+ 

Bound 

C 

0 

μ 

 

i 

enclosed 

( ) i = free 

( ) 

ε ∫ E r 

da 

i = Q = Q + Q 

S 

enclosed enclosed enclosed 

o ToT free Bound 

We also have the relations: 

 

JBound 

( r′ ) =∇×Μ( r′ 

) 

 

K ( ) ( ) ˆ 

Bound 

r′ =Μ r′ 

× n 

 

Volume 

 

I 

Bound 

= ∫ JBound 

( r′ ) da′ 

⊥ 

S⊥ 

 

surface 

 

I 

Bound 

= ∫ KBound 

( r′ ) d′ 

C⊥ 

 

Volume 

I 

free 

= ∫ J 

free ( r′ ) da′ 

S 

⊥ 

⊥ 

 

Surface 

I = K r′ d′ 

free 

∫ 

C⊥ 

free 

( ) 

surface 

⊥ 

⊥ 

ρ 

σ 

Bound 

Bound 

Surface 

Bound 

 

 

 

( r′ ) =−∇Ρ i ( r′ 

) 

 

( r′ ) =Ρ( r′ 

) i nˆ 

S′ 

Bound 

( ) 

surface 

Volume 

QBound 

= ∫ ρBound 

( r′ ) dτ 

′ 

v′ 

 

Q = σ r′ da′ 

Surface 

free 

∫ 

Volume 

Qfree 

= ∫ ρ 

free ( r′ ) dτ 

′ 

v′ 

 

Q = σ r′ da′ 

∫ 

S′ 

free 

( ) 

And the-time dependent Continuity Equations – separate conservation of bound and free charge: 

∂ρ 

∇ =− 

( r, 

t) 

( rt , ) 

free 

i J 

free 

⇐ Free charge is conserved. 

∂ρ 

∇ =− 

( r, 

t) 

∂t 

 

( rt , ) 

Bound 

i JBound 

⇐ Bound charge is conserved. 

∂t 

 

n.b. There are actually two 

separate bound charge continuity 

equations here, because we have 

bound charges in dielectric media 

and effective bound currents in 

magnetic media! 

Then using the principle of linear superposition: 

 

JTot ( r, t) = J 

free ( r, t) + JBound 

( r, 

t) 

 

 

 

∂ρ free ∂ρBound ∂ρTot 

⇒ ∇ iJTot ( r, t) =∇ iJ free ( r, t) +∇ i JBound 

( r, 

t) 

=− − =− 

∂t ∂t ∂t 

 

 

∂ρTot 

( rt , ) 

⇒ ∇ i JTot 

( r, 

t) 

=− 

⇐ Total charge is conserved. 

∂t 

( rt , ) ( rt , ) ( rt , ) 

14 




Griffiths Example 6.2: A long copper rod of radius R carries a steady, uniformly-distributed 

 

 

 

free current I ˆ 

free 

= I 

freez 

with J ˆ 

free 

= Jo free 

z as shown in the figure below. Determine H ( ρ ) 

inside and outside the copper rod. Note that copper is weakly diamagnetic, so at the microscopic 

level the magnetic dipoles of the copper atoms will align opposite/antiparallel to the magnetic 

 

field B ~ ˆ ϕ , resulting in a bound volume current I 

running antiparallel to the free current 

Volume 

free 

Volume 

Bound 

I . All currents are longitudinal ( ie . . in the zˆ 

direction) 

2 

2 

enclosed 

⎛ρ 

⎞ 

o 

= 

free free 

π with I 

free ( ρ ≤ R) 

= I 

free ⎜ ⎟ 

J I R 

⎝R 

⎠ 

± . 

where 

Use Ampere’s Circuital Law for the H 

-field: zI ˆ, 

free, 

J 

H 

 

enclosed 

( r 

) d 

∫ i = I 

C 

free 

R 

 

inside 1 ρ 

H ( ρ ≤ R) = I ˆ 

2 free 

ϕ 

2π 

R 

 

outside 1 

H ( ρ ≥ R) 

= I free 

ˆ ϕ 

2πρ 

Note that: 

 

 

inside 

outside 

H ρ = R = H ρ = R 

( ) ( ) 

ϑ 

ϕ 

2 2 

ρ = x + y (in cylindrical coords.) 

free 

 

H( r) 

ŷ 

ˆϕ 

~ ρ 

ρ = R 

 

H 

= I 

max 

( ρ = R) 

free 

2π 

R 

~1 ρ 

ρ 

ˆx 

ˆρ 

1 

thus: B( r) = μoH( r) +Μ( r) 

μo 

 

 

outside outside μ 

 

o 

B ρ > R = μoH ρ > R = I 

freeϕ 

Because Μ outside 

( ρ > R) 

≡0 

2πρ 

outside 

= same as B 

for non-magnetized wire! 


B ρ ≤ R ? 

 

 

 

inside inside inside 

B ρ ≤ R = μ H ρ ≤ R + μ Μ ρ ≤ R = μ H ρ ≤ R +Μ ρ ≤ R 

Now: H( r) ≡ B( r) −Μ( r) 

Then: ( ) ( ) ˆ 

( ) 

What is ( ) 

( ) ( ) ( ) ( ) ( ) 

0 0 0 

We don’t (yet) have the “tools” in hand to know/determine Μ ( ρ ≤ R) 


when we have these, we can then determine B ( ρ ≤ R) 

. 

( ) 

 

- but we will, shortly…. 



15


H 

r d 

i I 

C 

Note that from Ampere’s Circuital Law for the H enclosed 

-field: ( ) = 

enclosed 

This relation says that if we measure I 

free 

then we can compute H . This makes H more 

useful e.g. than D (the electric displacement). 

In the “old” days (e.g. 1800’s), it was easier to reliably measure a free current I ( in Amps) than 

voltage V ( in Volts). Reliably measuring a current required the use of galvanometer (an early 

type of ammeter – which is a very low input impedance device – ideally zero Ohms), whereas 

reliably measuring the voltage V (with respect to a local ground) required the use of a voltmeter 

with a very high input impedance (ideally infinite Ohms), which was very difficult to achieve 

back then! In the “old” days, a good galvanometer was easy to make a good ammeter, but a good 

voltmeter was very difficult to make. These days, garden-variety/“vanilla” digital voltmeters 

typically have input impedances of ~ 10 Meg-Ohms. 

Measuring a current thus enabled the monitoring of the H -field, e.g. for an electro-magnet 

(big fields ⇒ big magnet coils ⇒ lots of current!) 

∫ 

free 

The Magnetic Permeability μ and Magnetic Susceptibility χ of Linear Magnetic Materials 

Recall that for linear dielectric materials in electrostatics, that: 

D = ε 

oE 

+Ρ= ε E 

 

where ε is the electric permittivity of the material and ε = εo( 1+ χe) 

. 

 

= εo( 1+ 

χe) 

E where χ 

e 

is the electric susceptibility of the dielectric material. 

= ε E 

+ ε χ E 

⇒ Ρ= 

ε χ E 

 

. o e 

o o e 

It would seem reasonable/logical/rational for linear magnetic materials in magnetostatics, that 

we could define a magnetic permeability μ and related magnetic susceptibility χ m 

in a manner 

similar to that for howε and χ 

e 

were defined for linear dielectric materials in electrostatics, i.e.: 

1 1 

H ≡ B−Μ = B with μ = μo 

( 1 + χm 

) . 

μo 

μ 

 

However, the 1 μ factor really messes things up!!! For if H = B μ and we want to have 

1 

μ= μo( 1+ χm) 

then H = B and mathematically there is no rigorous way to separate 

μo( 1+ 

χm) 

the RHS of this relation into two separate pieces that would enable us to relate the magnetization 

Μ directly to the magnetic field B , analogous to obtaining the relation Ρ= 

εχE 

 

o e 

for linear 

dielectric media. 

If χm 

1 then: 

1 

1 

m 

1+ χ ≈ −χ 

Thus, we see that for χm 

1 , that 

m 

and then: ( 1 χ ) 

m 

1 1 1 1 

H ≈ − 

m 

B= B− χmB= B−Μ 

μo μo μo μo 

1 

Μ χmB 

in analogy to o e 

μ 

Ρ= 

ε χ E 

 

. 

o 

16 




o 

= 1+ m 

1 i.e. χm 

1 so 

therefore we cannot use this approximation!!! We must do “something” else!!! 

However, there are many linear magnetic materials where μ μ ( χ ) 

−7 

We could e.g. re-define the magnetic permeability of free space, μo 

= 4π 

× 10 Henrys/meter 

(=N/A 2 1 1 7 

) in terms of its inverse, e.g. define: ξ ≡ o 

10 

μ 

= o 

4π 

× meters/Henry (= A 2 /N), 

1 

Then H ≡ B−Μ 

⇒ H ≡ξoB 

−Μ = ξB 

 

, whereξ is a “new” inverse magnetic permeability 

μo 

* 

defined such that ξ = ξo( 1− χm) 

with “new” magnetic susceptibility χ * m 

such that 

 

* * 

* 

H ≡ξoB−Μ = ξB= ξo( 1− χm) 

B= ξoB−ξoχmB 

and thus Μ= 

ξ χ B 

 

o m in analogy to Ρ= 

ε χ E 

 

o e 

for linear dielectrics. But note that since ξ ~ 1/ μ , then as (old) μ increases, the 

(new)ξ decreases (and vice-versa)! So this approach has some troubles also… see Appendix at 

end of this lecture note for a bit more info on this…. 

However, we shouldn’t get too hung-up on this, because e.g. we have already seen that there 

is a vast difference between the nature of the electric field E vs. the nature of the magnetic field 

B in terms of how they are specified by their respective divergences and curls, and thus we have 

absolutely every reason to believe that there is also a vast difference between the nature of the 

two auxiliary fields D and H in terms of how they are specified by their respective divergences 

and curls. Hence insisting on (or wanting) “symmetry” between relations associated with E vs. 

those for B is illusory. In fact, only the macroscopic matter fields Ρ (the electric polarization / 

electric dipole moment per unit volume) and Μ ( the magnetic polarization/magnetic dipole 

moment per unit volume) are analogous/similar fields (by deliberate construction on our part)! 

What people (Maxwell, et al.) actually did was to start with the auxiliary relation: 

Multiply both sides by μ : μoH 

= B−μoΜ 

o 

 

 

, then rearrange: B= μ ( ) 

o 

H +Μ 

1 

H ≡ B−Μ 

μ 

n.b. This latter relation erroneously causes people to (wrongly) think that the H -field is the 

fundamental field and therefore that the B -field is the auxiliary field. ⇒ WRONG !!! ⇐ 

For linear magnetic materials, the magnetic permeability μ can then be defined such that μ 

connects H to B 

 

via the relation H = B μ (the magnetic analog of D= ε E ). 

The magnetic susceptibility can then be defined as: μ μo( 1 χm) 

linear dielectrics: ε ≡ ε ( 1+ 

χ ) 

≡ + paralleling that done for 

o e 

 

= μ = μo + χ m 

= μ o 

+ μoχ 

but: ( 

) 

 

m 

B μo H μoH 

μo 

 

Then we see that: B H ( 1 ) H H H = +Μ = + Μ 

Then “viola”: Μ= 

χ H 

 

 

m , which is not analogous to: Ρ=ε χ E 

o e because we don’t have a direct 

relationship between Μ and (the fundamental field) B . 

o 



17


 

However, since H 

 

= B μ 

 

 

χmH χmB μ ⎣χm χm ⎤⎦B 

μo 

. 

then Μ= = = ⎡ ( 1+ 

) 

χ , then the factor ⎡⎣χ m ( 1+ χm) 

⎤⎦ ≈ χm, and ( 1 ) 

Now if 

m 

1 

 

Thus: Μ= χmH 

≈χmB 

μo 

for χm 

1. 

μ = μ + χ ≈ μ . 

o m o 

The following table lists the magnetic susceptibilities for a few typical types of diamagnetic and 

paramagnetic materials. Note that systematically, χm 

1for both types of magnetic materials 

(except for gadolinium). 

In this course, we want to “hang on to” the following: 

 

E( r) and B( r) 

are fundamental fields. 

 

D( r) and H( r) 

are auxiliary fields associated with the E & M properties of matter. 

 

⎧D( r) = ε E( r) 

⎫ 

⎪ 

⎪ 

For linear dielectrics and linear magnetic materials: ⎨ 1 ⎬ 

⎪H( r) = B( r) 

μ 

⎪ 

⎩ 

⎭ 

ε 

ε = electric permittivity of matter = Keε K 

o 

e 

= ε 

rel 

≡ = ( 1+ 

χe) 

ε 

o 

dielectric “constant” electric susceptibility 

(a.k.a. relative electric permittivity) 

μ = magnetic permeability of matter Kmμo 

= K = μ ≡ = ( 1+ 

χ ) 

μ 

m rel m 

μo 

relative magnetic permeability 

magnetic susceptibility 

18 




For Diamagnetic Materials: 

dia 

dia 

dia dia dia μ 

dia 

χ 

m 

< 0 ⇒ μ = μo( 1+ χm ) < μo, Km = = ( 1+ χm 

) < 1 

μo 

For Paramagnetic Materials: 

para 

para 

para para para μ 

para 

χ 

μ 

> 0 ⇒ μ = μo( 1+ χm ) > μo, Km = = ( 1+ χm 

) > 1 

μ 

For Ferromagnetic Materials: 

ferro 

χm 

0 but is in fact dependent on past magnetic history of material !!! 

A non-linear hystesis-type relation exists between Μ vs. H (and/or Μ vs. B ) for ferromagnetic 

materials. 

 

Magnetic materials that obey the relation B= μH = μo( 1+ χm) 

H = μoH 

+ μoΜ 

⇒ Μ= 

χ H 

 

m 

are known as linear magnetic materials, i.e. μ = the magnetic permeability of the magnetic 

 

material and μ = constant of proportionality between B and H, 

whereas χ 

m 

= magnetic 

 

susceptibility of the magnetic material and χ = constant of proportionality between Μ and H. 

If 

m 

B 

ext 

becomes extremely large, then the relation between B and H, 

and Μ and H can/does 

 

 

 

2 

2 

B = μ 1+ c μ+ c μ + H Μ= χ 1+ a χ + a χ + … H 

become non-linear, e.g. ( 2 3 

…) 

and 

m( 2 m 3 m ) 

 

Note that various crystalline magnetic materials are anisotropic, hence: B = μH 

and 

o 

Μ= 

χ 

m H 

⎛μxx μxy μ ⎞ 

xz 

⎜ ⎟ 

μ = ⎜μyx μyy μyz 

⎟ 

⎜ ⎟ 

⎝μzx μzy μzz 

⎠ 

m m m 

⎛χxx χxy χ ⎞ 

xxz 

⎜ ⎟ 

m m m 

χm = ⎜χyx χyy χyz 

⎟ 

⎜ m m m 

χzx χzy χ ⎟ 

⎝ 

zz ⎠ 

magnetic 

permeability 

tensor 

magnetic 

susceptibility 

tensor 

Note also that: μij = μ 

ji 

and μxx + μyy + μzz 

= 0 and likewise: 

χ 

m 

ij 

= χ and χ m + χ m + χ 

m = 0 . 

m 

ji 

xx yy zz 



19


 

B r 

We have the Maxwell relation: ∇ i ( ) = 0 (no magnetic charges/no magnetic monopoles) 

 

and the constitutive relation: H( r ) 1 

= B( r 

) −Μ( r 

 

) . Then: ( ) 1 

∇ H r = ∇ B ( 

r ) 

−∇ Μ ( 

i i i r ) 

μo 

μ 

o 

= 0 

 

i i ⇐ These divergences do not necessarily vanish!!! Often they don’t! 

(especially on the surfaces of magnetized materials) 

 

 

∇Μ i r = , does ∇ i H( r) = 0 (and not vice-versa!!!) 

or: ∇ H( r) =−∇ Μ( r) 

Only when ( ) 0 

Consider a bar magnet (permanent magnet) with uniform Μ ( r ) ≠ 0 

or outside: 

S 

Μ 

N 

Consider Ampere’s Circuital Law for H : ( ) 

enclosed 

∫ 

 

H 

r d 

i = I 

C 

free 

 

 

inside, thus B( r) ≠ 0 

 

H r H r 


outside 

But: ∃ no free current(s) in a bar magnet – does this mean that ( ) = ( ) = 0 

 

Μ =Μ 

 

( r) zˆ 

B ( r ) 

o 

inside the bar magnet. 

for a cylindrical bar magnet = B ( r ) 

!!! NONSENSE !!! 

for a short solenoid (w/ no pitch angle). 


!!!??? 

Lines of B : 


B is in the same direction as Μ=Μ z ˆ 

o 

 

Lines of H : 

Outside: 

 

H 

out 

1 

= B 

μ 

0 

out 

in 

Inside: H 

is in the opposite 

direction to Μ !!! 

 

Compare these pix to that for ED , and Ρ for the bar electret – see P435 Lecture Notes 10, p. 33. 

20 




 

Note that since J ( r) =∇×Μ( r) 

Bound 

 

However: ∇× H( r) = J ( r) 

free 

 

 

 

 

and Μ ( r) = χ H( r) 

⇒ J ( r) = χ J ( r) 

m 

. 

Bound m free 

, 

then: J ( r) = χ ∇× H( r) 

Bound 

m 

This relation says that unless a free current actually flows through a linear magnetic material of 

 

susceptibility χ m 

, with free volume current density J 

free ( r) 

, then (and only then) will there be / 

 

arise a corresponding non-zero equivalent bound volume current density JBound 

( r) 

which is 

related to J 

free ( r 

 

) via ( ) 

( 

J ) 

Bound 

r = χmJ free 

r . This is analogous to the relationship that we found 

 

between ρ Bound ( r ) and ρ free ( r 

) for linear dielectric materials: ( ) ⎛ 1 ⎞ 

1 ( 

ρ 

) 

Bound 

r =−⎜ 

− ⎟ρfree 

r 

⎝ Ke 

⎠ 

 

{See P435 Lecture Notes 10, page 21}. If J 

free ( r ) = 0 inside a magnetic material, then 

 

Jbound 

( r ) = 0 inside the magnetic material, also. In this situation, any/all non-zero effective 

bound currents can only exist on the surfaces of the magnetic material!!! 

MAGNETOSTATIC BOUNDARY CONDITIONS FOR MAGNETIC MEDIA 

⊥ = normal (i.e. perpendicular) component 

relative to plane of interface 

= parallel component relative to plane of 

interface (tangential component) 

 

From ∇ i B( r) = 0 (no magnetic charges/no monopoles) in integral form: ∫ B ( r 

) i nda ˆ = 0 . 

S 

Use a Gaussian pillbox for the enclosing surface S, vertically centered on the interface between 

the two magnetic media. We then shrink the height of pillbox to infinitesimally above/below the 

interface – then only the top/bottom portions of the surface integral will contribute anything. 

 

We thus obtain a condition on the perpendicular components of B ( r ) above/below interface: 

⊥ 

above 

⊥ 

above 

B r = B r 

( ) ( ) 

2 

surface 

1 

 

We also have the constitutive relation: B ( r) μ 

o 

= H( r) +Μ( r) 

and thus ∇ B( r) = 0 

 

 

 

∇ iH( r) =−∇i Μ( r) 

. In integral form this relation becomes: ( ) iˆ 

=− Μ( ) 

surface 

 

i ⇒ 

∫ 

H r nda r nda ˆ 

S ∫ i . 

S 

Using the same Gaussian pillbox, we obtain the following condition on the perpendicular 

 

H r Μ r above/below interface: 

 

components of ( ) and ( ) 



21


⎡H ( r ) − H ( r ) ⎤ =−⎡Μ ( r ⎣ ⎦ ⎣ ) −Μ ( r 

) ⎤ 

⎦ 

⊥ above ⊥ above ⊥ above ⊥ below 

2 1 

surface 

2 1 

 

 

Ampere’s Law for B ( r ) is: ∇× B ( r ) = μoJTot 

( r 

) , with ( ) ( ) ( 

J ) 

Tot 

r = J 

free 

r + JBound 

r 

 

enclosed 

in integral form becomes: ( ) i = μ 

enclosed enclosed enclosed 

I I I 

∫ 

B r d I 

C 

o Tot 

, with 

surface 

 

Tot free Bound 

which 

= + . We can take 

a (rectangular) contour vertically centered above/below the interface between the two magnetic 

media; we then shrink the height of this contour to be infinitesimally above/below the interface, 

thus only the tangential portions of the line integral above/below the interface will contribute. 

We obtain the following condition on the tangential components of B ( r ) 

 

interface, where K ( r) = K ( r) + K ( r) 

Tot free Bound 

1 

above below 

⎣ 

2 1 

o 

B r B r KTot 

r 

μ ⎡ 

⎦surface 

( ) − ( ) ⎤ = ( ) 

We can write this more compactly/succinctly in vector form as: 

 

 

=∇× 

Since B ( r) A( r) 

 

above / below the 

and ˆn is shown in the figure above: 

surface 

1 above 

below 

B2 ( r) B1 

( r) K ( ) ˆ 

Tot 

r n 

μ ⎡ − 

⎣ 

⎤ ⎦ 

= 

× 

surface 

o 

surface 

, this relation can also be equivalently written as: 

 

above 

below 

1 ⎡∂A2 ( r) ∂A1 

( r) 

⎤ 

⎢ − ⎥ =−K 

μo 

⎢⎣ 

∂n 

∂n 

⎥⎦ 

surface 

 

Similarly, Ampere’s Law for H ( r 

) is: ( ) ( 

∇× H r = J ) free 

r 

 

enclosed 

( ) i = 

Tot 

 

( r) 

surface 

which in integral form becomes: 

∫ 

free . Again, we can take a (rectangular) contour vertically centered above / 

H r d I 

C 

below the interface between the two magnetic media; we then shrink the height of this contour to 

be infinitesimally above/below the interface, thus only the tangential portions of the line integral 

above/below the interface will contribute. We obtain the following condition on the tangential 

components of H( r) 

above/below the interface: ⎡ 

above 

( ) below 

H r − H ( r ) ⎤ = K ( r 

) 

⎣ 

2 1 

⎦ 

free 

surface 

which can also be written compactly/succinctly in vector form as: 

 

above 

below 

⎡H2 ( r) H1 ( r) ⎤ 

⎣ 

− 

⎦ 

= K ( ) ˆ 

free 

r × n 

surface 

surface 

 

Since B ( r) = μH( r) 

or: H( r) = B( r) 

μ and B 

( r) =∇× 

A 

( r) 

, this relation can also be 

equivalently written as: 

 

above 

below 

⎡⎛ 1 ⎞∂A2 ( r) ⎛ 1 ⎞∂A1 

( r) 

⎤ 

⎢⎜ ⎟ − ⎜ ⎟ ⎥ =−K 

free ( r) 

μ 

surface 

⎢⎣⎝ 2 ⎠ ∂n 

⎝μ1⎠ 

∂n 

⎥⎦ 

surface 

surface 

22 




Appendix: 

If we wanted to/needed to define the macroscopic magnetization Μ (auxiliary macroscopic 

matter field) in terms of the fundamental field B 

, in analogy to Ρ ( r) =εχ 

o eE( r) 

. We have seen 

(above) that given the constitutive relation H 

( r) = B 

( r) μo 

−Μ 

 

( r) 

we are unable to do so. 

The problem here actually focuses squarely on μ 

o 

, the magnetic permeability of free space: 

Note that μ 

o 

is actually derived/defined from: 

c 

1 

= ⇒ μo 

≡ 

2 

ε μ ε 

2 1 

o 

o 

o c 

−12 

The electric permittivity of free space is ε o 

= 8.85× 10 Farads/meter. The Farad is the SI unit 

of capacitance, C (in electrostatics) – the ability of something (in this case, the vacuum) to store 

energy in the electric field of that something. Note thatε o 

has the dimensions of capacitance/unit 

length – Farads/meter. 

The numerical value of the magnetic permittivity of free space μo 

is defined from the 

8 

experimental measurement of c= 3× 10 m s (speed of light in free space) and the electric 

−12 

permittivity of free spaceε o 

= 8.85× 10 Farads/meter, thus: 

−7 

μo 

≡ 4π× 10 Newtons/Ampere 2 = (kg-meter/sec 2 )/Ampere 2 = Henrys/meter 

{1 Newton/ Ampere 2 = 1 Henry = 1 Tesla-m 2 /Ampere = 1 Weber/Ampere} 

The Henry is the SI unit of inductance, L (in magnetostatics) – the ability of something (in this 

case, the vacuum) to store energy in the magnetic field of that something. Note that μo 

has the 

dimensions of inductance/unit length – Henrys/meter. 

7 

However, if we alternatively define 1 10 

ξo 

≡ = Amperes 

μ 

2 /Newton = meters/Henry, 

o 4π 

Thenξ o 

= inverse magnetic permeability (magnetic “reluctance”??) of free space. 

2 ξo 

2 

Then: c = ε 

or: ξo 

= c ε 

o 

o 

1 

Then the magnetic constitutive relation becomes: H = B− 

M ⇒ H = ξoB −M 

in analogy 

μo 

to D 

= ε 

oE 

+Ρ 

and we also have (for linear materials) H = ξ B in analogy to D= ε E . 

* 

ξ ξ 1 χ 

μ ≡ μ 1+ χ . 

However, here we will define o( m) 

Then: ( 1 ) 

≡ − in contrast to ( ) 

* * 

H ≡ξoB 

−Μ = ξB = ξo − χm B = ξoB −ξoχmB 

 

and thus 

Ρ= 

 

εχ 

for linear dielectrics. 

o 

e E 

 

 

χmH χmB μ χm χm ⎤⎦B 

μo 

Since: Μ= = = ⎡⎣ 

( 1+ 

) 

* 

or: χ = χ ( 1+ χ ). 

m m m 

o 

o 

m 

Μ= 

 

ξ χ 

in analogy to 

* 

B m 

* 

we see that = ( 1+ 

) 

ξχ χ χ μ 

o m m m o 



23

Lecture Notes 19: Magnetic Fields in Matter I; Dia-/Para-/Ferro ...

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