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Lesson 9
Dipoles and Magnets
Class 27
Today we will:
• learn the definitions of electric and magnetic
dipoles.
•find the forces, torques, and energies on dipoles
in uniform fields.
•learn what happens when we put dipoles in
nonuniform fields.
Lesson 9
Dipoles and Magnets
Section 1
Force on a Current-carrying Wire
Force on a Wire in a Magnetic Field
There is a force on charge carriers in a
current-carrying wire. This force is
transferred to the wire itself.

B
i

F
L
Force on a Wire in a Magnetic Field
N charge carriers are in length L of the wire. T is the time
it takes a charge to go the distance L. The force is:
Ftot   qvB NevB
q Ne
L
L
i

, v   F  iT B  iLB
t
T
T
T

B
i

F
L
Force on a Wire in a Magnetic Field
More generally
  
F  iL  B
Section 2
Force and Torque on Wire Loops
Force on a Wire Loop in a Uniform Magnetic Field
The net force on a wire loop is zero.

F

F

F
i

F

B
Torque on a Wire Loop in a Uniform Magnetic Field
The net torque on a wire loop is not zero.

F
i

F

B
Torque on a Wire Loop in a Uniform Magnetic Field
We define an area vector for the loop, by using
the right hand rule.

F

F

A
i

F

B
Right-hand Rule for Current Loops
The direction of an area for a current loop is
1) normal to the plane of the loop and
2) In the direction of your thumb if your fingers
loop in the direction of the current.

A
i
Right-hand Rule for Current Loops
Note that the directed area of the loop is the
same as the direction of the magnetic field
produced by the loop!

A

B
i
Torque on a Wire Loop in a Uniform Magnetic Field
Now calculate the torque about an axis through
the center of the loop going into the screen.

F

A

a
2
i

F

B
Torque on a Wire Loop in a Uniform Magnetic Field
The magnitude of the torque is the product of
the force and the moment arm.

F

A
a
2


F


B
Torque on a Wire Loop in a Uniform Magnetic Field
The magnitude of the torque is the product of
the force and the moment arm.

F

A
a
2


a
sin 
2

F

B
Torque on a Wire Loop in a Uniform Magnetic Field
a
  2 F sin 
2

F

A
F sin 


F
a
2

B
Torque on a Wire Loop in a Uniform Magnetic Field
a
  2 F sin 
2
 aF sin 
 aiaB sin 
 iAB sin 
Section 3
Magnetic Dipoles
Define the Magnetic Dipole Moment

Since iA appears in a number of formulas,
we give it a name: the magnetic dipole
moment. Note that it is a vector with the
direction given by the right-hand rule.

  iA

Torque on a Wire Loop in a Magnetic Field
In terms of the magnetic dipole moment:
  iAB sin 
 B sin 

  B


Direction of the Torque
A dipole feels a torque that tends to align
the dipole moment with the external field.

F

B

F

B





F

F
Potential Energy of a Dipole
The maximum potential energy is when the
dipole is opposite the field. The minimum
potential energy is when the dipole is in the
direction of the field.

B

F



B

F



F

F
Potential Energy of a Dipole
The work done by a force is W   Fdx
Potential Energy of a Dipole
The work done by a force is W   Fdx
Similarly, the work done by a torque is
2
2
1
1
W    d   B sin  d   Bcos  2  cos 1 
Potential Energy of a Dipole
The work done by a force is W   Fdx
Similarly, the work done by a torque is
2
2
1
1
W    d   B sin  d   Bcos  2  cos 1 
Since the change in potential energy
is the work it takes to rotate the
dipole, we have:
U   B cos   C
Potential Energy of a Dipole
U   B cos   C
We can choose the constant of
integration to be anything we want.
Potential Energy of a Dipole
U   B cos   C
It’s simplest if we choose it to be zero.
 
U    B
Potential Energy of a Dipole
 
U    B
Caution!!! U=0 is not the minimum
energy. It is the energy when the
dipole is perpendicular to the field!

B



F

F
Section 4
Electric Dipoles
The Electric Dipole
An electric dipole is a charge +q and a
charge -- q held apart a distance 
apart.

F

E
q

sin 
2


p

q

F
The Electric Dipole Moment


An electric dipole moment is p  q 
The direction of  goes from the 
charge to the + charge.

F

E
q

sin 
2


p

q

F
Torque and Potential Energy of an Electric
Dipole
Electric dipoles work just the same
way as magnetic dipoles. In uniform
fields, there is no net force on the
dipole.
 
Torque:   p  E

 
Potential energy: U   p  E
An Electric Dipole in a Nonuniform Field
First, the dipole feels a torque that
aligns the dipole with the field. ( end
toward the source of the field.)

E

F
q

p
q

F
An Electric Dipole in a Nonuniform Field
Then, the dipole feels a net force in
the direction of the stronger field.

E

F
q
q

p

F
A Magnetic Dipole in a Nonuniform Field
Magnetic dipoles behave in much the
same way. They first experience a
torque that aligns them with external

field.
F

B
I

F

F
A Magnetic Dipole in a Nonuniform Field
Then, they experience a net force that
pulls them in the direction of the
stronger field.

F

B
I

F

F
Permanent Magnets
Permanent magnets have magnetic
dipole moments much as current
loops.
S


N
Permanent Magnets
In a nonuniform external field,
permanent magnets experience a
torque…
N
S
N


S
Permanent Magnets
…then a net force in the direction of
stronger magnetic field.
S
N
S
N
Permanent Magnets
…then a net force in the direction of
stronger magnetic field.
S
N
S
N
Class 28
Today we will:
• define magnetization and magnetic susceptiblity
• learn about paramagnetic, diamagnetic, and
ferromagnetic materials
• learn about the opposing effects of domain
alignment and thermal disalignment
• learn how to understand hysteresis curves
• characterize ferromagnetic materials in terms of
residual magnetization and coercive force
Section 5
Paramagnetism and
Diamagnetism
Permanent Magnets
1) Magnetite or loadstone was known from
antiquity.
Permanent Magnets
"Magnetism" comes from the region called Magnesia,
where loadstone (magnetite) was found.
Permanent Magnets
1) Magnetite or loadstone was known from
antiquity.
2) Loadstone floating on wood rotates so one end
always points north.
Permanent Magnets
1) Magnetite or loadstone was known from
antiquity.
2) Loadstone floating on wood rotates so one end
always points north. This is the north pole.
Permanent Magnets
1) Magnetite or loadstone was known from
antiquity.
2) Loadstone floating on wood rotates so one end
always points north. This is the north pole.
3) If two magnets are placed near other, like
poles attract and unlike poles repel.
Permanent Magnets
William Gilbert in 1600
publushed De Magnete –
where he described
magnetism as the “soul of
the earth.”
Permanent Magnets
Gilbert: A perfectly
spherical magnet
spins without
stopping –
because the earth
is a perfect sphere
and it’s a magnet
and it spins without
stopping.
N
S
Permanent Magnets
From last time: Permanent magnets
behave like current loops. In a nonuniform
external field, permanent magnets
experience a torque…
N
S
N


S
Permanent Magnets
…then a net force in the direction of
stronger magnetic field.
S
N
S
N
Permanent Magnets
…then a net force in the direction of
stronger magnetic field.
S
N
S
N
Permanent Magnets
If we don’t allow magnets to rotate:

F
S
S

N


F
S


F




S
N

N

F
N
Permanent Magnets
When dipole moments align, magnets attract.
When dipole moments are opposite, magnets
repel.
F
S
S

N


F
S


F




S
N

N

F
N
Atoms as Magnets
If we throw a magnet really fast (so it doesn’t have
time to rotate) through a non-uniform field, what
happens?
N
S
Atoms as Magnets
If we throw a magnet really fast through a nonuniform field, what happens?
N
S
Atoms as Magnets
If we throw a magnet really fast through a nonuniform field, what happens?
N
S
Atoms as Magnets
If we throw a magnet really fast through a nonuniform field, what happens?
N
S
Atoms as Magnets
Now take Ag atoms and do the same thing: SternGerlach experiment 1922.
http://phet.colorado.edu/sims/sterngerlach/stern-gerlach_en.html
(PhET U. of Colo.)
Atoms as Magnets
Now take Ag atoms and do the same thing – Stern
Gerlach experiment 1922.
Conclusion: Ag atoms are magnets
The magnets seem to all be aligned either with the
field or against the field.
How could an atom be a magnet?
Atoms as Magnets
How could an atom be a magnet?
Magnetic fields are caused by moving charges, so
what’s moving?
Atoms as Magnets
Magnetic fields are caused by moving charges, so
what’s moving?
Bohr atom:
Electrons as Magnets
Electrons do the same thing (sort of). How could an
electron be a magnet?
Electrons as Magnets
An electron spins…
N
S
Electrons as Magnets
We can measure
the magnetic dipole
moment of an atom
by measuring the
force on electrons in
a nonuniform field.
N
S
Electrons as Magnets
We can model an
electron as a
spinning sphere and
determine its
radius: 2.8 fm
N
S
Electrons as Magnets
But scattering
measurements find
electrons to be
point particles or
nearly so.
N
S
Section 5
Paramagnetism and
Diamagnetism
Magnetic Properties of Magnets
In most materials, all the “little magnets” are
randomly arranged, so the material is affected
only slightly by external magnetic fields.
Materials react to external magnetic fields
in three different ways
1) Paramagnetic materials are very weakly
attracted by external magnetic fields.
Most materials are paramagnetic.
Materials react to external magnetic fields
in three different ways
1) Paramagnetic materials are very weakly
attracted by external magnetic fields.
Most materials are paramagnetic.
2) Diamagnetic materials are very weakly repelled
by external magnetic fields.
Materials react to external magnetic fields
in three different ways
1) Paramagnetic materials are very weakly
attracted by external magnetic fields.
Most materials are paramagnetic.
2) Diamagnetic materials are very weakly repelled
by external magnetic fields.
3) Ferromagnetic materials are strongly attracted
or repelled by external magnetic fields.
How do we understand paramagnetism?
Paramagnetic atoms are like little magnetic
dipoles. They experience a torque which aligns
them with the external field, then they feel a net
force that pulls them into the field.
The magnetic dipole moment results primarily
from electron spin and angular momentum.


N
S

B
How do we understand diamagnetism?
Diamagnetism is something that is not adequately
explained without resorting to quantum
mechanics.
S
N



B
How do we understand ferromagnetism?
Domain alignment: If atoms have large magnetic
dipole moments, they tend to align with each other
much as a collection of magnets tends to align.
N
N
N
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
S
S
S
How do we understand ferromagnetism?
Thermal disalignment: Heat causes atoms to
vibrate, knocking them around and disaligning the
dipoles.
How do we understand ferromagnetism?
Domains: Small regions that have aligned dipole
moments are called domains. In unmagnetized
iron, the domains are randomly oriented.
How do we understand ferromagnetism?
Domains: In a permanent magnet, the domains
tend to be aligned in a particular direction.
The Curie Point
Curie Temperature: When a ferromagnetic
material gets hot enough, the domains break
down and the material becomes paramagnetic.
Getting Quantitative
We define magnetization as the total magnetic
dipole moment per unit volume.
N

 i

M  i 1
Volume
A magnetized object has an internal magnetic field
given by the relation:


Bint  0 M
Getting Quantitative
The internal magnetic field can also be expressed
in terms of the external magnetic field:


Bint   Bext
where  is called the magnetic susceptibility.
Susceptibilites
paramagnetic
  10 5 to  10 3
diamagnetic
  10 6 to  10 4
ferromagnetic
  10 3 to  10 5
Susceptibilities for Ferromagnetic Materials
Ferromagnetic materials have a “memory.” If we
know the external field, we can’t predict the
internal field, unless we know the previous history
of the sample.
We describe the relationship between internal and
external fields by means of a “hysteresis curve.”
Hysteresis Curve
We start with no internal
field (unaligned) and no
external field.

Bint (T )

Bext (mT )
Hysteresis Curve
We increase the external

Bint (T )
field, causing some of the
domains to align.

Bext (mT )
Hysteresis Curve
As the external field

Bint (T )
increases, the internal
field eventually stops
growing. Why?

Bext (mT )
Hysteresis Curve
All the domains

Bint (T )
eventually align.

Bext (mT )
Hysteresis Curve
We then decrease the

Bint (T )
external field. The domains
want to stay aligned, so
the internal field remains
large.

Bext (mT )
Hysteresis Curve
When the external field

Bint (T )
goes to zero, some
domains remain aligned.

Bext (mT )
Hysteresis Curve
The internal field that

Bint (T )
remains is called the
residual magnetization.

Bext (mT )
residual
magnetization
Hysteresis Curve
To reduce the internal

Bint (T )
field, we must apply an
external field in the
opposite direction.

Bext (mT )
Hysteresis Curve
To reduce the internal

Bint (T )
field, we must apply an
external field in the
opposite direction.
coercive force

Bext (mT )
Hysteresis Curve
The external field needed

Bint (T )
to bring the internal field
back to zero is called the
coercive force.
coercive force

Bext (mT )
Hysteresis Curve
As we continue to

Bint (T )
increase the external field
in the negative direction,
domains align with the field.

Bext (mT )
Hysteresis Curve
The process continues

Bint (T )
just as when the external
field was in the positive
direction.

Bext (mT )
Soft Iron
A nail made of soft iron

Bint (T )
has domains that align
easily, but it can’t hold
the magnetization.

Bext (mT )
Soft Iron
A nail made of soft iron

Bint (T )
has domains that align
easily, but it can’t hold
the magnetization.

Bext (mT )
The coercive force
and the residual
magnetization of soft
iron are both small.
Good Permanent Magnet
A good permanent magnet

Bint (T )
has a large coercive force
and a large residual
magnetization.

Bext (mT )