Magnetic Materials
Reading: Chapter 14 in Kong & Shen
Outline
- Magnetization and Magnetic Susceptibility
- Ferromagnetism, Paramagnetism, Diamagnetism
- ‘Magnetic Charges’
1
True or False?
1. For a solenoid, if we
increase the number of loops
N by 5% and increase the
height h by 5%, then the Hfield inside will not change.
2. For a toroid, if we increase
the number of loops N, but do
not increase the radius, then
the H-field inside stays the
same.
NI
h
NI
2πr
B
3. Lenz’s Rule states that the induced E-field
drives the current in the direction to try to
keep the magnetic flux constant.
2
Maxwells Equations
Electric Fields
Magnetic Fields
0 E · dA =
S
B · dA = 0
ρdV
V
S
MQS
EQS
E · dl = 0
C
3
Magnetic Fields
· dl =
H
J · dA
S
= Ienclosed
· dA
=0
B
Magnetic
Moment
magnetic
monopoles
do not exist
m=ia
4
Microscopic Magnets
Magnetic moment of an atom due to
electron orbit…
Magnetic moment of an atom due to
electron/nuclear spin…
matom
2
A
·
m
≈ 10−23
atom
The magnetization or net magnetic dipole
moment per unit volume is given by
= Nm
M
[A/m]
5
Number of
dipoles per unit
volume [m-3]
average
magnetic
dipole moment
[A m2]
Generating Strong Magnetic Fields
Superposition with current loops…
d = 0.012 meters
h = 0.019 meters
B = 0.5 Tesla
−7
μo = 4π × 10
T
A/m
Image in the
Public Domain
6
Jm
0.5 T Electromagnets
h = 0.019meters
d = 0.012meters
B = μo H = μo ni = 0.5T
Ni
0.5
ni =
=
h
4π × 10−7
Bh
i=
μo N
0.5 Tesla with 100 turn solenoid…
0.5 Tesla with current loop…
n = 100
n=1
i
i0.5 T ≈ 7600 Amps
i0.5T ≈ 7.6Amps
...but the wires are microscopic !
7
Generating Strong Magnetic Fields
Will it be “easier” to generate a 0.5-T magnetic flux
density with a permanent magnet or an electromagnet ?
Image in the Public Domain
−23
matom ≈ 10
iatom
∼ 3Å
2
A·m
atom
2
−19 2
˚
m
and aatom ≈ 9A ≈ 10
A
−4
≈
3000
K
≈ 10 A and
atom
cm
8
Induced Magnetization
For some materials, the net magnetic dipole
moment per unit volume is proportional to
the H field
= χm H
M
MAGNETIC
SUSCEPTIBILITY
(dimensionless)
units of
both M and H
are A/m.
The effect of an applied magnetic field on a magnetic material
is to create a net magnetic dipole moment per unit volume M
9
Type of
Magnetism
Suscept
ibility
Example /
Susceptibility
Atomic / Magnetic Behaviour
M
Diamagnetism
Paramagnetism
Ferromagnetism
Antiferromagnet
Small
negative
Small
positive
Large
positive
Small
positive
Atoms have
no magnetic
moment
Au
Cu
-2.74x10-6
-0.77x10-6
β-Sn
Pt
Mn
0.19x10-6
21.04x10-6
66.10x10-6
Fe
~100,000
Cr
3.6x10-6
H
M
Randomly
oriented
magnetic
moments
H
Parallel
aligned
magnetic
moments
M
Parallel and
anti-parallel
aligned
magnetic
moments
M
H
H
10
Diamagnetic Materials
Diamagnetism is the property of an object
which causes it to create a magnetic field
in opposition of an externally applied
magnetic field, thus causing a repulsive
effect. It is a form of magnetism that is
only exhibited by a substance in the
presence of an externally applied
magnetic field. Diamagnetism is generally
a quite weak effect in most materials,
although superconductors exhibit a strong
effect.
On Right: A small (~6mm) piece of
pyrolytic graphite levitating over a
permanent neodymium magnet array
(5mm cubes on a piece of steel).
Note that the poles of the magnets
are aligned vertically and alternate
(two with north facing up, and two
with south facing up, diagonally)
from Wikipedia
Image in the Public Domain
11
Ferromagnetic Materials
Impact that the imposition of a magnetic field H on a ferromagnetic material has on the resulting
magnetic flux density B. The field causes the magnetic moments in each of the domains to begin to
align. When the magnetizing force H is eliminated, the domains relax, but don’t return to their
original random orientation, leaving a remanent flux Br ; that is, the material becomes a “permanent
magnet.” One way to demagnetize the material is to heat it to a high enough temperature (called
the Curie temperature) that the domains once again take on their random orientation. For iron, the
Curie temperature is 770ºC.
r
B
C
−H
s
B
B,J
0
C
H
H
Initial
Magnetization
Curve
H>0
r
−B
hysteresis
s
−B
0
Slope =
μ
H
BEHAVIOR OF AN INITIALLY UNMAGNETIZED MATERIAL
Domain configuration during
several stages of magnetization
•  Why is there a B even when H is zero ??
•  Why doesn’t the magnetization change sign until HC ??
Magnetic Materials
Which has lower energy ?
Orbital 1
Orbital 2
Orbital 1
Orbital 2
Atomic Number
Element
Electronic Structure of 3d
Moment (μB)
21
Sc
1
22
Ti
2
23
V
3
24
Cr
5
25
Mn
5
26
Fe
4
27
Co
3
28
Ni
2
29
Cu
0
= electron spin orientation
13
Magnetic Materials
magnetic susceptibility
χm
3
4
5
6
7
8
9
:
;
32
33
34
35
36
37
38
39
3:
.-4,7/
.-3,3/
46 -45
-3; -44 .-8,5/ 9,;
.-6,2/
! ! %
:,3 7,9
43 -5,6 -45 -34 .-44/ .-33/
% # $ * # & % %
7,9 43 486 3:3 5:5 489 :4: 4,38 3,98 2,83 -;,9 -34 -45 -9,5 -7,6 -3: -38 .-38/
% % $ * # # 6,6 58 343 32; 458 33; 595 88 392 9:5 -47 -3; -:,4 4,6 -89 -46 -44 .-46/
& & % ' * ! $
'
#
7,5 8,9 85
93 397 9:
;8
37
59 486 -56 -4: -58 -38 -375
"#(
%"#(
%%$"#(
All values given for a temperature of 300 K
In case of ferromagnetic materials: saturation
polarization
numbers without (): ·10−6
numbers with(): ·10−9
14
Magnetic refrigeration
N
Heat Irradiation
Magnetic Field
S
Gadolinium
Gd
3 to 4K per Tesla
Image in the Public Domain
One of the most notable examples of the magnetocaloric effect is in the
chemical element gadolinium and some of its alloys. Gadolinium's
temperature is observed to increase when it enters certain magnetic fields.
When it leaves the magnetic field, the temperature drops. Praseodymium
alloyed with nickel (PrNi5) has such a strong magnetocaloric effect that it
has allowed scientists to approach within one thousandth of a degree of
absolute zero.
Image by Jurii http://commons.wikimedia.org/
wiki/File:Gadolinium-2.jpg on Wikimedia Commons
15
Source: Wikipedia
Today’s Culture Moment
Magnetic
Levitation
Image in the Public Domain
Image in the Public Domain
Shanghai Maglev Train goes up to 431km/h
S
N
S
N
N
S
N
N
S
N
Maglev Propulsion
16
N
S
S
N
S
S
Magnetic Materials
weak magnetism
strong magnetism
diamagnetism
ferromagnetism
Induced magnetic dipoles directed reverse
to the external magnetic field.
Spontaneous alignment of all permanent
magnetic dipoles within a domain in the crystal
lattice (only metals)
M
M
H
H
What kind of Magnetism is present in these materials ?
= Ferromagnetism
=
17
Diamagnetism
Maxwell’s Equations for Magnetic Materials
Magnetic Fields
· dl =
H
J · dA
S
= Ienclosed
· dA
=0
B
Image in the Public Domain
How do we incorporate magnetic materials in Maxwell’s Equation ?
18
Analogy Between Magnetic and Electric Dipoles
Magnetic Fields
Electric Field
Image in the Public Domain
Magnetic
Moment
m=ia
Electric Dipole
Moment
m=qd
The H-field lines for a magnet looks
like the E-field lines for a electric dipole!
19
Superposition of Magnetic Moments
N
S
The field outside looks the same as a ‘magnetic charge’ dipole
20
Equivalent Magnetic Charges
Going from region of high
magnetization to low magnetization
it appears as if there are ‘magnetic
charges (monopoles)’ within the
material !!
μo M
ρM = −∇
S(-)
S(-)
These ‘magnetic charges’ obey a
Gauss
Law…
=
· dA
μo H
ρM dV
V
S
= − μo M
· dA
· dA
μo H
S S
+M
· dA
=0
μo H
21
S
Magnetic Flux
Φ [Wb] (Webers)
due to macroscopic
& microscopic
Magnetic Flux Density
B [Wb/m2] = T (Teslas)
Magnetic Field Intensity
H [Amp-turn/m]
due to macroscopic
currents
· dA
Φ= B
= μo H
+M
= μo H
+ χm H
= μo μr H
B
Magnetization M
is due to material’s
microscopic response
to magnetic field H
22
Magnetic Susceptibility and Permeability
= μo H
+M
= μo H
+ χm H
= μo μr H
B
Image by MIT OpenCourseWare.
μr μ=r =1 1+
+ χχm is the relative permeability of the material
m
μ=
μ =μμ
oμ
rr is the permeability of the material
ομ
23
Magnetic Storage
Ring Inductive
Write Head
Shield
GMR
Read
Head
Magnetic
flux
R
Recording Medium
Electric
current
24
Horizontal
Magnetized Bits
Ferromagnetic Materials
r
B
C
−H
s
B
B,J
0
C
H
H
Initial
Magnetization
Curve
H>0
r
−B
hysteresis
s
−B
0
H
Slope = μ
i
Behavior of an initially unmagnetized
C
Hr : coercive magnetic field strength material.
Domain configuration during several stages
B S : remanence flux density
of magnetization.
B : saturation flux density
Choosing Magnetic Storage
Which material is most attractive for the storage media ?
M
Retains a large
fraction of the
saturation field
when driving
field removed.
Saturation
Magnetization
M
H
H
Bo
Bo
M
H
Bo
Narrow hysteresis loop implies
A small amount of dissipated
Energy in repeatedly reversing
The magnetization.
26
Thin Film Write Head
Recording Current
Magnetic Head Coil
Magnetic Head Core
Recording Magnetic
Field
Close-up of a hard disk
head resting on a disk
platter. A reflection of
the head and its
suspension is visible on
the mirror-like disk.
27
Practical Issues with Scaling
As the magnetic domain size
shrinks, the read/write head
must move closer to the hard
drive surface…
read/
write
head
human hair
smoke
particle
Image in the Public Domain
dust
particle
fingerprint
aluminum platter
Altitude: 1.5 mm
magnetic oxide
coating
Lubricant: 0.15 mm
Carbon overcoat: 0.5 mm
Scaled up disk
structure
28
Summary
Magnetic
Moment
m=ia
= Nm
M
= χm H
M
+M
= μo H
B
+ χm H
= μo H
r
B
C
−H
= μ o μr H
s
B
0
C
H
r
−B
s
−B
29
H
MIT OpenCourseWare
http://ocw.mit.edu
6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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