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PHYS2012
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MAGNETISM AND MATERIALS
All matter is composed of atoms and atoms are composed of protons, neutrons and
electrons. The protons and neutrons are located in the atom's nucleus and the electrons
are in “constant motion” around the nucleus. Electrons carry a negative electrical charge
and produce a magnetic field as they move through space. A magnetic field is produced
whenever an electrical charge is in motion.
This may be hard to visualize on a subatomic scale but consider an electric current
flowing through a conductor. When the electrons (electric current) are flowing through
the conductor, a magnetic field forms around the conductor. The magnetic field can be
detected using a compass.
Since all matter is comprised of atoms, all materials are affected in some way by a
magnetic field. However, not all materials react the same way.
At the atomic level, the motion of an electron gives rise to current loop  magnetic
dipole moment  magnetic field (like a miniature bar magnet)
Magnetic dipole moment m  pm  mu  u [A.m2]
pm  N i A n
direction – right hand screw rule
Magnetization M
[A.m-1]
magnetic dipole moment per unit volume
M
magnetic moment
volume
The magnetic moments associated with atoms have three origins:
1
The electron orbital motion.
2
The change in orbital motion caused by an external magnetic field.
3
The spin of the electrons.
4
Nuclear spins (will ignore in this course)
When a material is placed within a magnetic field, the material's electrons will be
affected. However, materials can react quite differently to the presence of an external
magnetic field. This reaction is dependent on a number of factors such as the atomic and
molecular structure of the material, and the net magnetic field associated with the atoms.
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In most atoms, electrons occur in pairs. Each electron in a pair spins in the opposite
direction, so when electrons are paired together, their opposite spins cause there magnetic
fields to cancel each other. Therefore, no net magnetic field exists. Alternately, materials
with some unpaired electrons will have a net magnetic field and will react more to an
external field.
Most materials can be classified as ferromagnetic, diamagnetic or paramagnetic.
B, H and M fields
B  o ( H  M )
M  m H
B  o (1   m ) H  o r H   H
r  1   m
  o (1   m )
M   m H only has meaning if one can define the relative permeability of the material.
The relationship is valid for diamagnetic and paramagnetic materials but may not be
valid for ferromagnetic materials.
DIAMAGNETIC MATERIALS









m  0
m
105
r 1
Small and negative susceptibility.
Slightly repelled by a magnetic field.
Do not retain the magnetic properties when the external field is removed.
Magnetic moment – opposite direction to applied magnetic field.
Solids with all electrons in pairs - no permanent magnetic moment per atom.
Properties arise from the alignment of the electron orbits under the influence of an
external magnetic field.
Most elements in the periodic table, including copper, silver, and gold, are
diamagnetic.
m temperature independent provided no structural changes in material.
m(argon) ~ -1.010-8
m(copper) ~ -1.010-5
B
Diamagnetic material
m < 0 (small)
B = o (1+ m ) H
H
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Permeability
 = o (1+ m ) =slope of B-H line
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PARAMAGNETIC MATERIALS






 m  0 small
Small and positive susceptibility.
Slightly attracted by a magnetic field.
Material does not retain the magnetic properties when the external field is
removed.
Properties are due to the presence of some unpaired electrons and from the
alignment of the electron orbits caused by the external magnetic field.
Examples - magnesium, molybdenum, lithium, and tantalum.
m(oxygen) ~ 2.010-6
m(aluminum) ~ 2.110-5
B
Ideal magnetic material
or paramagnetic material
m > 0 (small)
B = o r H =  H
 = constant = slope of B-H curve
H
FERROMAGNETIC MATERIALS








Large and positive susceptibility.
Strong attraction to magnetic fields.
Retain their magnetic properties after the external field has been removed.
Some unpaired electrons so their atoms have a net magnetic moment.
Strong magnetic properties due to the presence of magnetic domains. In these
domains, large numbers of atomic moments (1012 to 1015) are aligned parallel so
that the magnetic force within the domain is strong. When a ferromagnetic
material is in the un-magnetized state, the domains are nearly randomly organized
and the net magnetic field for the part as a whole is zero. When a magnetizing
force is applied, the domains become aligned to produce a strong magnetic field
within the part.
Iron, nickel, and cobalt are examples of ferromagnetic materials.
B  o ( H  M ) Magnetization is not proportional to the applied field.
m(ferrite) ~ 100
m(iron) ~ 1000
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Magnetic Domains
Ferromagnetic materials get their magnetic properties not only because their atoms carry
a magnetic moment but also because the material is made up of small regions known as
magnetic domains. In each domain, all of the atomic dipoles are coupled together in a
preferential direction. This alignment develops as the material develops its crystalline
structure during solidification from the molten state. Magnetic domains can be detected
using Magnetic Force Microscopy (MFM) and images of the domains like the one shown
below can be constructed.
Magnetic Force Microscopy
(MFM) image showing the
magnetic domains in a piece
of heat treated carbon steel.
During solidification a trillion or more atomic dipole moments are aligned parallel so that
the magnetic force within the domain is strong in one direction. Ferromagnetic materials
are said to be characterized by "spontaneous magnetization" since they obtain saturation
magnetization in each of the domains without an external magnetic field being applied.
Even though the domains are magnetically saturated, the bulk material may not show any
signs of magnetism because the domains develop themselves are randomly oriented
relative to each other. Ferromagnetic materials become magnetized when the magnetic
domains within the material are aligned. This can be done my placing the material in a
strong external magnetic field or by passing an electrical current through the material.
Some or all of the domains can become aligned. The more domains that are aligned, the
stronger the magnetic field in the material. When all of the domains are aligned, the
material is said to be magnetically saturated. When a material is magnetically saturated,
no additional amount of external magnetization force will cause an increase in its internal
level of magnetization.
Unmagnetized Material
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Magnetized Material
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Domain walls move and domains grow in the alignment of the atomic magnetic dipoles.
The mobility of the domain walls is reduced by impurities and lattice imperfections. The
coercivity (how easy for a magnetic material to be de-magnetized) is determined by the
mobility of the domain walls.
M 0
M 0
H
external magnetic field
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THE HYSTERESIS LOOP AND MAGNETIC PROPERTIES
A great deal of information can be learned about the magnetic properties of a material by
studying its hysteresis loop. A hysteresis loop shows the relationship between the
induced B-field (magnetic flux density) B and the H-field (magnetizing force) H. It is
often referred to as the B-H loop. An example hysteresis loop is shown below.
[Note: B does not become saturated only M does]
The loop is generated by measuring the B-field of a ferromagnetic material while the Hfield is changed. A ferromagnetic material that has never been previously magnetized or
has been thoroughly demagnetized will follow the dashed line as H is increased. As the
line demonstrates, the greater the amount of current applied (H+), the stronger the B-field
(B+). At point "a" almost all of the magnetic domains are aligned and an additional
increase in the magnetizing force will produce very little increase in the B-field. The
material has reached the point of magnetic saturation. When H is reduced back down to
zero, the curve will move from point "a" to point "b." At this point, it can be seen that
some B-field remains in the material even though the magnetizing force is zero. This is
referred to as the point of retentivity on the graph and indicates the remanence or level
of residual magnetism in the material. Some of the magnetic domains remain aligned
but many have lost there alignment. As the magnetizing force is reversed, the curve
moves to point "c", where the B-field has been reduced to zero. This is called the point of
coercivity on the curve. The reversed H-field force has flipped enough of the domains so
that the net B-field within the material is zero. The H-field required to remove the
residual magnetism from the material, is called the coercive force or coercivity of the
material.
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As the H-field is increased in the negative direction, the material will again become
magnetically saturated but in the opposite direction (point "d"). Reducing H to zero
brings the curve to point "e." It will have a level of residual magnetism equal to that
achieved in the other direction. Increasing H back in the positive direction will return B
to zero. Notice that the curve did not return to the origin of the graph because some Hfield is required to remove the residual magnetism. The curve will take a different path
from point "f" back the saturation point where it with complete the loop.
Permeability
Permeability is a property that describes the ease with which a B-field is established in a
material. It is the ratio of the B-field to the H-field
B

H
It is clear that this equation describes the slope of the curve at any point on the hysteresis
loop. The permeability value given in papers and reference materials is usually the
maximum permeability or the maximum relative permeability. The maximum
permeability is the point where the slope of the B/H curve for unmagnetized material is
the greatest. This point is often taken as the point where a straight line from the origin is
tangent to the B/H curve.
The relative permeability r is arrived at by taking the ratio of the material's permeability
 to the permeability in free space (air) o.
r =  /


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Magnetization or B-H Curve
http://www.electronics-tutorials.ws/electromagnetism/magnetic-hysteresis.html
For a ferromagnetic material, the relative permeability is not a constant, r >> 1.
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MAGNETIZATION – macroscopic view
Consider a long solenoid with no
magnetic core of length L and crosssectional area A and N turns.
From Ampere’s Circulation Law
B-field (magnetic induction) B0  0 n I 0
H-fielod (magnetic intensity) H 0  n I 0
Now put a magnetic material inside the solenoid - it becomes magnetized. Atomic
magnetic dipoles in the material line up, producing an internal magnetic field, which may
strengthen (or oppose – diamagnetic material only) the original field. Magnetic field in
the material can be seen as the result of effective currents of the atoms' magnetic dipoles.
Magnetization M = total magnetic dipole moment per unit volume.
Internal
currents
cancel,
results in
bound
surface
current
For a paramagnetic or ferromagnetic material the B-field increases
B(total) = B(current in coil) + B(material)
B = B0 + Bm
im
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B  0 n I 0  0 n im
N
im
L
N im A
 0 H  0
L A
N pm
 0 H  0
Vol
 0 H  0 M
H  n I0
 0 H  0

B  0 H  M
M  m H
magnetic dipole moment pm  im A Vol  L A
magnetization M  dipole moment/volume 
N pm
Vol

B  o (1   m ) H  o r H   H
r  1   m
  o (1   m )
As the current (and hence applied field H) increases,
magnetization M and magnetic field B increases. If H is small or
substance weakly magnetic (paramagnetic), increase is linear.
measure from graph  r 
B/H
o
If H is large or substance strongly magnetic (e.g. ferromagnetic),
as H increases, the magnetization M (and hence B) may increase
nonlinearly - high field region where slope decreases is called
"saturation" region  max M.
measure from graph  r 
B/H
o
Since  r varies with H  could also use “differential permeability” dB/dH
The shape of the hysteresis loop tells a great deal about the material being magnetized.
The hysteresis curves of two different materials are shown in the graphs below.
“hard” magnetic materials: Hc (coercivity) is high, area of the loop is large, used for
permanent magnets.
“soft” magnetic materials: Hc is small, area of loop is small, used for transformer
cores & electromagnets.
N.B.
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very difference scales for the H-field
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soft r large
hard r
Material can be demagnetized by striking or heating it, or go
round the hysteresis loop, gradually reducing its size.
"Degaussing"
Energy dissipation – hysteresis loop
W is the energy dissipated within a unit
volume of the sample (increase in internal
energy of the sample) in the process in
taking the sample around the hysteresis
loop. Transformers must be made of
materials that have narrow hysteresis
loops.
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Consider taking the sample through one cycle of the hysteresis loop. Since there is a
changing magnetic flux, an emf is generated as the current in the coil changes.
induced emf
  N
current
i
power
P = i
dm
dB
  NA
dt
dt
HL
N
W

P dt 
cycle
energy dissipated
W


 i dt 
cycle
A L H dB  Vol
cycle
dB   H L 

  NA  
 dt
dt   N 
cycle 


H dB
cycle
So, the work/volume is equal to the area enclosed by the hysteresis loop.
permanent magnets
T < Tc
carbon steel
alnico V
platinum-cobalt
Nd2Fe (sintered)
high permeability
materials
iron
4% Si-Fe
Mu metal
Supermalloy
M282
mag03.doc
Remanence
magnetism
Br (T)
1
1.25
0.45
1
Coercivity
HC (A.m-1)
4103
4104
2105
2106
r (max)
Saturation
Bsat (T)
Coercivity
HC (A.m-1)
5103
7103
1105
8105
2.1
2.0
0.65
0.8
80
40
4
0.16
M587
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PERMANENT MAGNETS
Bar Magnets
There are no free currents - the magnet is magnetized all by itself if = 0
H
dl  0  H-field inside and the H-field outside point in opposite directions
 B  dA  0  the magnetic field lines for B must be continuous, the lines just keep
going on (there are no magnetic monopoles).
Inside the magnet: H 
1
BM
o
lines of H point in a direction opposite to M and B .
Outside the magnet: M  0
B and H have the same field pattern
HH
B
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6
S

2
1
4
N
HFe
1
H
5
2
0
Hair
2
3
Circulation loop: square side L
3
5
6
2
4
5
dl   HFe dl   Hair dl   Hair dl  
1
Bair
H air 
6
3
5
5
2
4
HFe dl  0
H Fe dl   H Fe dl    Hair dl   Hair dl
H Fe   H air
Gauss’s Law f or magnetism
Cylindrical
Gaussian
surface
Binside
Aoutside
Ainside
Boutside
 B
dA   Binside A  Boutside A  0
 Binside  Boutside
B-f ield lines –
f orm continuous loops
B
M
M 0
H
N pole
im
Bound surf ace currents i m (right hand screw rule)

mag03.doc
M
 B
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Interaction between magnetic fields
Like poles repel
Unlike poles attract
Magnetic Field of Like and Unlike Poles Together
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Why does a magnet stick to a piece of iron?
un-magnetized piece of iron
Bar magnet bought near
un-magnetized piece of iron
B
N
N
N
 Bar magnet will attract
the iron that was initially
un-magnetized
north pole attracts
south pole
Consider a magnetic field B to be non-uniform in
which the B field points in a direction orthogonal
to the plane of the current loop. There is a net force
that pulls the magnetic dipole pm towards the
region of high magnetic field.
Proof
The forces all pull the current elements
outwards. The force on each current element
is
F = BiL
Forces F3 and F4 cancel.
F1 > F2 since the B-field is non uniform, the
B-field larger on the left than the right. So
there is a net force that pulls the magnetic
dipole towards the region of larger magnetic
field.
B
B large
B small
m
F4
F1
F2
F3
i
Explain what happens in the following diagrams when a magnet is placed on a ramp.
Fe ramp
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Cu ramp
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plastic ramp
16
Uniformly magnetized sphere
B-field
H-field
B-field continuous loops (no beginning or end)
The H-field lines start where the M lines end and finish where M start.
H-field has de-magnetizing effect since H and M are in opposite direction.
M245
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M301
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Horse Shoe Magnets
A permanent iron magnet is in the form of circular disk with a radius, r and a small gap in
it of width, a. For the case when r >> a, discuss the H-field, B-field and magnetization
for this example of a horse shoe magnet.
Circulation loop f or circulation integration
used in applying Ampere’s Law
N
Use Amperes’s Law for a loop around the permanent magnetic (i = 0)
 H  dl
if
 H iron (2 r  a )  H air (a)  0
The H-field in the iron, Hiron must point in the opposite direction to the H-field in the air,
Hair.
Ampere’s Law
 B
dA  0 The B-field field is perpendicular to the plane surfaces of
the ring, and the perpendicular component of the B field is constant at an interface, so B
is constant throughout the ring, B = Bair = Biron
In the air gap H air 
B
o
or B = o Hair
N
In the iron
H iron  
a
a
H air  
B
2 r  a
(2 r  a ) o
The H-field inside the magnet is in the opposite
direction to the magnetization and has a demagnetizing effect.
Hair
Hiron
This corresponds to a points on the hysteresis loop H > 0 & B < 0 or H < 0 and B > 0.
For soft materials, the de-magnetizing effect is usually sufficient to bring the material
back to B = 0 (M = 0) i.e., an un-magnetized state. This is why a horse-shoe magnet is
stored with an iron keeper. Then, the B-field, H-field and magnetization all point in the
same direction.
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M014
M169
M014
M169
mag03.doc
M245
M282
M314
June 28, 2017
M587
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