Chapter 6. Magnetic Fields in Matter

6.1 Magnetization• All matters are composed of atoms, each with a positive charged nucleus and a

number of orbiting electrons.

• In addition, both electrons and the nucleus of an atom rotate (spin) on their own axes with certain magnetic dipole moments.

• In the absence of an external magnetic field, the magnetic dipoles of the atoms of most materials (excepts permanent magnets) have random orientations, resulting in no net magnetic moment.

• The application of an external magnetic fields cause both an alignment of the magnetic moments of spinning electrons and an induced magnetic moment due to a change in orbital motion of electrons

e-m

nucleus because and nucleus electron nucleus electron electronm m M M

– Diamagnetic, if r 1 magnetization opposite to B• m ~ -10-5

• the orbital motion of the electrons• Copper, germanium, silver, gold

– Paramagnetic, if r 1 magnetization parallel to B• m ~ 10-5

• Magnetic dipole moments of the spinning electrons• Aluminum, magnesium, titanium and tungsten

– Ferromagnetic, if r >>1 retain magnetization even after B removed• m >> 1 ( 100~ 100,000)• Magnetized domains (strong coupling forces

between the magnetic dipole moments of the atoms• Nickel, cobalt, iron (pure), mumetal

6.1.1 Magnetic Materials: Diamagnets, Paramagnets, Ferromagnets

Magnetic Materials

6.1.2 Torques and Forces on Magnetic Dipoles

But, Torque on a rectangular current loop is not zero:The forces on the "horizontal" sides are likewise equal and opposite (so the net force on the loop is zero), but they do generate a torque:

In a uniform field B, the net force on a current loop is zero:

Magnetism is not due to magnetic monopoles.but rather to moving electric charges;

magnetic dipoles are tiny current loops.(Ampere model)

Magnetic Torques

• Small circular loop of radius b and carrying a current I in uniform B.

dl dl dl

B B B

F B B B B

,, no net force

z

r

dl Bdl

a B aF B a

1 1 1

1 2 1 2

2 2

2 2 2

0

N sin sin sin

,

N 2 sin

The total torque acting on the loop

N N 2 sin

a a

F F

a

a a

x x

x

x x

d dF b Idl B b

dF d d dl dl dl bd

d Ib B d

d Ib B d I b B

N torque , r F F Bd d d Idl

F

F

2nI bm a 2N a m Bx I b B Magnetic moment:

Magnetic Torques

• Example: The force and torque on the loop

1 1 1 3

2 2 2 4

1 2

,

sum of (1)~(4) forces, all directed toward center is zero. no torgue is produced.

Net force on the loop,

B aB a a

F B

F a a a a F

F a a a a F

F F F

z z

x x y y

x x x y y z y

y x x y y z x

BB B

dl

Ib B B Ib B

Ib B B Ib B

3 4

213 1 3 1 1 2

24 2 4 1 2

13 24 1 2

0, however, they result in a net torque.

N , 22

N ,

The total torque on the rectangular loop

N=N N N m

F

F F a a

F F a

a a

x y x y

y x

x y y x

b Ib B Ib b B

Ib b B

Ib b B B

N m a ax x y yB B

Magnetic Torques

• Direct current motors

N m B

Torques and Forces on Magnetic Dipoles

Problem 6.2 Starting from the Lorentz force law, show that the torque on any steady currentdistribution (not just a square loop) in a uniform field B is m x B.

I

rdl

0 2a

: Lorentz force law B

Magnetic Forces: Rail Gun

d

Torques and Forces on Magnetic Dipoles

But, in a nonuniform field B, F is not zero:

In a uniform field B, the net force on a current loop is zero:

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

For a circular wire of radius R

Torques and Forces on Magnetic Dipoles

Problem 6.5 A uniform current density J = J0 z fills a slab straddling the yz plane, from x = -a to x = +a. A magnetic dipole m = m0 x is situated at the origin.

(a) Find the force on the dipole

at a point x

(b) Do the same for a dipole pointing in the y direction: m = m0 y .

6.1.3 Effect of a Magnetic Field on Atomic Orbits

Electrons not only spin; they also revolve around the nucleus.For simplicity, let's assume the orbit is a circle of radius R.

The orbital motion with the period T = 2R / v look like a steady current

The orbital dipole moment:

When the atom is placed in a magnetic field, The magnetic dipole is subject to a torque (m x B).

But it's a lot harder to tilt the entire orbit than it is the spin, So the orbital contribution to paramagnetism is small.

A more significant effect on the orbital motion isThe electron speeds up or slows down, depending on the orientation of B.

(An electron circling the other way would be slowed down.)

Effect of a Magnetic Field on Atomic Orbits

When the atom is placed in a magnetic field, the electron speeds up

A change in orbital speed means a change in the dipole moment

Notice that the change in m is opposite to the direction of B.

this increments is antiparallel to the field.

This is the mechanism responsible for diamagnetism.

It is a universal phenomenon, affecting all atoms.

6.1.4 Magnetization

In the presence of a magnetic field, matter becomes magnetized,

(l) Paramagnetism the dipoles associated with the spins of unpaired electrons experience a torque tending to line them up parallel to the field)

(2) Diamagnetism the orbital speed of the electrons is altered in such a way as to change the orbital dipole moment in a direction opposite to the field).

The state of magnetic polarization Magnetization, M

M = magnetic dipole moment per unit volume

Problem 6.6 Of the following materials, which would you expect to be paramagnetic and which diamagnetic? Aluminum, copper, copper chloride (CuCl2), carbon, lead, nitrogen (N2), salt (NaCI), sodium, sulfur, water.

Aluminum, copper, copper chloride, and sodium an odd number of electrons paramagnetic. The rest (having an even number) should be diamagnetic.

Analogous to the polarization P in eIectrostatics

0

1lim A/mM mkk

Chapter 6. Magnetic Fields in Matter

6.2 The Field of a Magnetized Object6.2.1 Bound Currents

Vector potential of a single dipole m was

Total vector potential in a magnetized object is

integrating by parts

volume current density

Surface current density Magnetization current density

• Magnetization surface current density

• Magnetization volume current density

Magnetization current density

K M n A/mb

2 A/mJ Mb

P b P n

PP

: out of paperM

Kb

If is uniform (space invariant) inside, the current of the neighboring atomic dipoles will cancelled averywhere.

On the surface of the material, K n 0.No net current in the interior ( 0),or,

M

MJ

M

b

b

0 if is space invariant.J Mb

Kb

Bound Currents

Example 6.1 Find the magnetic field of a uniformly magnetized sphere.

Choosing the z axis along the direction of M,

The surface current density corresponds to a rotating spherical shell, of uniform surface charge ,

(Example 5.11)

Inside the sphere,

the internal field is uniform.

6.2.2 Physical Interpretation of Bound Currents

&

surface current: /

M

b

m at m IaI Mt

K I t M

For uniformly magnetized material,all the internal currents cancel.

0bJ M

However, at the edge there is a current.

n MbK

2 A/mJ Mb

( ) ( )x z z

zx

zb x

I MtI M y dy M y dz

MI dydzy

MJy

For a nonuniformly magnetized material, the internal currents no longer cancel.

yzb x

MMJy z

( ) ( )x y y

yx

yb x

I Mt

I M z dz M z dy

MI dzdy

zM

Jz

Physical Interpretation of Bound Currents

NOTE: Jb should obey the conservation law: 0Jb

6.2.3 The Magnetic Field Inside Matter

The effect of magnetization is to establish bound currents and

The field due to magnetization of the medium is just the field produced by these bound currents.

Now, we can use the Ampere’s law to find the Magnetic field from (bound currents) + (free currents).

J Mb K M nb