6.1 Magnetization

6.2 The Field of a Magnetized Object

6.3 The Auxiliary Field

6.4 Linear and Nonlinear Media

6 Magnetization

Diamagnets, Paramagnets, Ferromagnets

• Magnetism in a material is a result of tiny dipole moments in atomic scale.

• Usually they each other due to their random orientation.• In an external magnetic field those dipoles align and the medium

becomes polarized (or magnetized)

• Paramagnets: magnetization is parallel to the applied magnetic field

• Diamagnets: magnetization is opposite to the applied magnetic field

• Ferromagnets: retain a substantial magnetization indefinitely after the

external field has been removed.

Torques and Forces on Magnetic Dipoles

A magnetic dipole experiences a torque in a magnetic field :

Torque on a rectangular current loop in a uniform magnetic field

ˆsin

ˆ ( )sin

ˆ sin

N aF i

a bIB i

mB i

θ

θ

θ

=

=

=

�

magnetic dipole moment

m abI=

N m B= � ��

F�

F�

B�

1s

•This torque lines the magnetic dipole up

parallel to �: paramagnetism.

In a uniform magnetic field, the net force is zero.

( ) ( ) 0F I dl B I dl B= × = × =∫ ∫� �� � �

� �

But in a nonuniform magnetic field, there is a net force

For an infinitesimal loop with dipole moment m in a field B

{ }

ˆ ˆˆ ˆ( ( )) (0, ,0) (0, , ) (0, , ) (0,0, )

ˆ ˆ = (0, ,0) (0, , ) (0, , ) (0,0, )

F I dl B r I dyy B y dzz B z dyy B y dzz B z

I dyy B y B y dzz B z B z

ε ε

ε ε

= × = × + × − × − ×

× − + × −

∫ ∫

∫

� � � ��� �

� � � ��

(0, ,0) (0,0, )

2

ˆ ˆ = -

ˆ ˆ ˆ ˆˆ ˆ =

ˆ ˆ

y z

yx xz

yx

B BI dyy dzz

z y

BB BB B BI z y m y x x z

y z y y z z

BBm y x

y y

ε ε

ε

∂ ∂ × + × ∂ ∂

∂ ∂ ∂∂ ∂ ∂− = − − −

∂ ∂ ∂ ∂ ∂ ∂

∂∂= −

∂ ∂

∫� �

� �

( )

ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ . 0

ˆ ˆ ˆ ( . ) ( . ) (

yx x xz z

x x x

x

BB B BB Bx z m y x z

z z y y z z

B B Bm x y z B

x y z

m B m x B mx B m

∂ ∂ ∂ ∂∂ ∂− + = − + +

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂= + + ∇ = ∂ ∂ ∂

= ∇ = ∇ = ∇ = ∇

�∵

� �. )B�

( )F m B= ∇ ⋅� ��

Effect of a Magnetic Field on Atomic Orbits

Assume the orbit of an electron a circle of radius r

e+ r�

r�

ẑ

2 2since its period 1

c

rT

v w

π π= =

In a magnetic field

( assume r = constant)

B�

22

20

1 ''

4e

e vev B m

rrπε+ =

22' ( ' )

( ' )( ' )

e

e

mev B v v

r

mv v v v

r

= −

= + −

( 2 )v v v v v′ ′∆

Bound Currents

for a single dipole

0

2ˆ

4mA

µπ

×=��r

r

m�

', m Md r rτ ′= = −��

r

0

0

0

0 0

2

'

'

ˆtotal ( ) '4

1 ( ' ) '

4

1 1 [ ( ' ) ' ' ( ) ]

4

1 1 ( ' ) ' ( )

4 4

MA r d

M d

M d M d

M d M da

µτ

π

µτ

πµ

τ τπ

µ µτ

π π

×=

= ×∇∫

= ∇ × − ∇ ×∫ ∫

= ∇ × + ×∫ ∫

∫�� �

�

� �

� � ��

r

r

r

r r

r r

2 2 21 1 ˆ1 ( ' ) '( ' ' ' )

' '

r r r r

r r r r

∧ ∧ − − ∇ = ∇ = ∇ = − = =

− −

∵ � � � �r

r r r r

1 1 1' ( ) ( ' ) ( ' )M M M

∇ × = ∇ × − × ∇

� � �∵

r r r

6.2 The Field of a Magnetized Object

magnetization M magnetic dipole moment per unit volume≡�

r

r

r ′

O

( ) problem 1.61b Ad A daτ∇ × = ×∫ ∫� � �

∵ �

Potential (and field) of a magnetized object is the same as would be produced by a

bound volume current �� �� = �′ × (��) and a surface current � �

� = (��) × ��

0 0

0 0

'1 1( ) ( ' ) ' ( )4 4

( ) ( )( ) ' '

4 4b b

V S

A r M d M da

J r K rA r d da

µ µτ

π π

µ µτ

π π

= ∇ × + ×∫ ∫

′ ′= +∫ ∫

� � �� ��

� �� � �

r r

r r

surface chargecf: in ES volume charge b Pρ = −∇ ⋅ ˆb P nσ = ⋅�

Example: Magnetic field of a

uniformly magnetized sphere

ẑ

Since is uniform:

ˆˆ0 sinb bJ M K M n M θ ϕ= ∇× = = × =� � � �

0

ˆsin

2 ˆfield produced ( )3

K v R

B zR z inside

σ σω θ ϕ

µ σ ω

= =

=

� �

�

02 ( )3

inside B M uniformµ=� �

34outside field is same as that of a dipole 3

m R Mπ=��

This is similar to the currents on a uniformly charged rotating spherical shell (Ch. 5)

Physical Interpretation of Bound Currents

For uniform

b b

m Mat

Ia I Mt

K t K M

=

= ⇒ =

= ⇒ =

��

ˆbK M n= ×� �

Internal loop currents cancel

Dipole

currents

Only the current flowing

around the boundary remains

I

Each charge moves in a small loop, but the net effect is a macroscopic

current flowing over the surface.

For non-uniform internal currents do not cancel.

A nonuniform M in y contributes to a current in x direction.

A nonuniform M in z, also contributes to

current in x direction

[ ( ) ( )]

x z z

z

I M y dy M y dz

Mdydz

y

= + −

∂=

∂

x zx

I MJ

dydz y

∂= =

∂

yx

MJ

z

∂= −

∂

( )yz

b x

MMJ

y z

∂∂∴ = −

∂ ∂

0b bJ M J⇒ = ∇ × ∇ ⋅ =� �

Current density:

6.3 The Auxiliary Field �

0

0

In Electrostatics

b fb

f

P E

D E P D

ρ ρρ

ε

ε ρ

+= −∇ ⋅ ∇ ⋅ =

= + ∇ ⋅ =

� �

� � � �

In a magnetized medium there are bound currents �� = � ×

So the total current is sum of bound currents and other (free) currents � = �� + ��

So the Ampere law: 0 0 0

1 0

or 0

( ) ( )

( )

1 where

b f f

f

f

B J J J M J

B M J

H J H B M

µ

µ µ µ

µ

∇ × = = + = ∇ × +

⇒∇ × − =

∇ × = = −

� � � � � �

� � �

� � � � �

,fH dI I enc⋅ =∫� �

�In integral form Total free current passing through an Amperian loop

0

0above below

encB d I

B B K

µ

µ

⋅ =

⇒ − =

∫� �

��ℓ�

Boundary Conditions

Since

,fH dI I enc⋅ =∫� �

�

0 above belowB da B B⊥ ⊥⋅ = ⇒ =∫

� ��

0

1 H B Mµ

= −� � �

( )

above above below below

above below above below

H M H M

H H M M

⊥ ⊥ ⊥ ⊥

⊥ ⊥ ⊥ ⊥

+ = +

− = − −

0above below fH H Kµ− =� �

linear and Nonlinear Media

6.4. Magnetic Susceptibility and Permeability

In Electrostatics

Polarization

Similar parameters can be defined for magnatostatics. In paramagnetic and diamagnetic materials magnetization is sustained by the applied magnetic field and proportional to B

As a convention magnetization is written in terms of H,

�� magnetic susceptibility

0 ( )eP E linear dielectricε χ=� �

00

( ) (1 )eE D Eρ

ε ε ε χε

∇⋅ = = = +� � �

��mM = χ H

��>0 paramagnets, ��

Example: Magnetic field of an infinite length solenoid filled with

linear material of susceptibility ��

Since B depends on bound currents, apply ampere Law for H

I

ẑ

n turns

per unit

length

0

ˆ

ˆ(1 )

enc

m

H dl I

H nIz

B nIkµ χ

⋅ =

=

= +

∫�

�

ˆˆ ˆˆ( )b m mK M n nI z n nIχ χ ϕ= × = × =� �

Bound surface currents

If the medium is paramagnetic (��>0) and field is (slightly) enhanced and if the medium diamagnetic field is reduced.

6.4.2 Ferromagnetism

Paramagnetic

Ferromagnet

no B

applied remove 0=M�

no with

B�

B�

0M =�

B�

B�

B�

6.4.2(2)

Iron is ferromagnetic,

but if T>Tcurie ( 770。 for iron )

curie point

it become a paramagnetic.

This is a phase transition.

6.4.2(3)