Lesson 9Dipoles and Magnets
Class 27Today 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 9Dipoles and Magnets
Section 1Force 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.
i
B
F
L
Force on a Wire in a Magnetic FieldN 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:
i
B
F
L
iLBBT
LiTF
T
Lv
T
Ne
t
qi
NevBqvBFtot
,
Force on a Wire in a Magnetic Field
More generally
BLiF
Section 2Force and Torque on Wire
Loops
Force on a Wire Loop in a Uniform Magnetic Field
The net force on a wire loop is zero.
i
B
F
F
F
F
Torque on a Wire Loop in a Uniform Magnetic Field
The net torque on a wire loop is not zero.
i
BF
F
Torque on a Wire Loop in a Uniform Magnetic Field
We define an area vector for the loop, by using the right hand rule.
B
F
A
i
F
F
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.
i
A
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!
i
A
B
Now calculate the torque about an axis through the center of the loop going into the screen.
B
A
2
a
i
F
F
Torque on a Wire Loop in a Uniform Magnetic Field
B
A
F
F
2
a
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.
B
A
F
F
sin2
a
2
a
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.
sin22 Fa
B
A
2
a
F
F
sinF
Torque on a Wire Loop in a Uniform Magnetic Field
sin22 Fa
sin
sin
sin
iAB
aiaB
aF
Torque on a Wire Loop in a Uniform Magnetic Field
Section 3Magnetic Dipoles
Define the Magnetic Dipole Moment
Since 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.
Ai
Ai
Torque on a Wire Loop in a Magnetic Field
B
B
iAB
sin
sin
In terms of the magnetic dipole moment:
Direction of the Torque
A dipole feels a torque that tends to align the dipole moment with the external field.
F
F
F
F
B
B
Potential Energy of a DipoleThe 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.
F
F
F
F
B
B
Potential Energy of a Dipole
The work done by a force is FdxW
Potential Energy of a Dipole
The work done by a force is Similarly, the work done by a torque is 12 coscossin
2
1
2
1
BdBdW
FdxW
Potential Energy of a Dipole
The work done by a force is Similarly, the work done by a torque is
Since the change in potential energy is the work it takes to rotate the dipole, we have:
12 coscossin2
1
2
1
BdBdW
CBU cos
FdxW
Potential Energy of a Dipole
We can choose the constant of integration to be anything we want.
CBU cos
Potential Energy of a Dipole
It’s simplest if we choose it to be zero.
CBU cos
BU
Potential Energy of a Dipole
Caution!!! U=0 is not the minimum energy. It is the energy when the dipole is perpendicular to the field!
BU
F
F
B
Section 4Electric Dipoles
The Electric Dipole
An electric dipole is a charge +q and a charge -- q held apart a distance apart.
E
q
q
p
F
F
sin2
The Electric Dipole Moment
An electric dipole moment is The direction of goes from the charge to the + charge.
qp
q
q
p
F
F
sin2
E
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:
Potential energy:
Ep
EpU
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.)
FF
E
p
q
q
An Electric Dipole in a Nonuniform Field
Then, the dipole feels a net force in the direction of the stronger field.
F
F
E
p
q q
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.
I F
F
F
B
A Magnetic Dipole in a Nonuniform Field
Then, they experience a net force that pulls them in the direction of the stronger field.
I F
F
F
B
Permanent Magnets
Permanent magnets have magnetic dipole moments much as current loops.
NS
Permanent Magnets
In a nonuniform external field, permanent magnets experience a torque…
NS
N
S
Permanent Magnets
…then a net force in the direction of stronger magnetic field.
NS
N
S
Permanent Magnets
…then a net force in the direction of stronger magnetic field.
NS S N
Class 28Today 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 5Paramagnetism 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 earthis a perfect sphereand it’s a magnet and it spins without stopping.
N
S
Permanent MagnetsFrom last time: Permanent magnets behave like current loops. In a nonuniform external field, permanent magnets experience a torque…
NS
N
S
Permanent Magnets
…then a net force in the direction of stronger magnetic field.
NS
N
S
Permanent Magnets
…then a net force in the direction of stronger magnetic field.
NS S N
Permanent MagnetsIf we don’t allow magnets to rotate:
S
N
S
N
S
N
S
N
F
F
F
F
Permanent MagnetsWhen dipole moments align, magnets attract. When dipole moments are opposite, magnets repel.
S
N
S
N
S
N
S
N
F
F
F
F
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?
S
N
Atoms as Magnets
If we throw a magnet really fast through a non-uniform field, what happens?
S
N
Atoms as Magnets
If we throw a magnet really fast through a non-uniform field, what happens?
S
N
Atoms as Magnets
If we throw a magnet really fast through a non-uniform field, what happens?
S
N
Atoms as Magnets
Now take Ag atoms and do the same thing: Stern- Gerlach experiment 1922.
http://phet.colorado.edu/sims/stern-gerlach/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 magnetsThe 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 5Paramagnetism 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
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
N
S
N
S
N
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.
A magnetized object has an internal magnetic field given by the relation:
VolumeM
N
ii
1
MB
0int
Getting Quantitative
The internal magnetic field can also be expressed in terms of the external magnetic field:
where is called the magnetic susceptibility.
extBB
int
Susceptibilites
35 1010 to
46 1010 to
53 1010 to
paramagnetic
diamagnetic
ferromagnetic
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.”
)(int TB
)(mTBext
Hysteresis Curve
We start with no internal
field (unaligned) and no
external field.
Hysteresis Curve
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)(mTBext
We increase the external
field, causing some of the
domains to align.
Hysteresis Curve
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)(mTBext
As the external field
increases, the internal
field eventually stops
growing. Why?
Hysteresis Curve
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)(mTBext
All the domains
eventually align.
Hysteresis Curve
)(int TB
)(mTBext
We then decrease the
external field. The domains
want to stay aligned, so
the internal field remains
large.
Hysteresis Curve
)(int TB
)(mTBext
When the external field
goes to zero, some
domains remain aligned.
Hysteresis Curve
)(int TB
)(mTBext
residual magnetization
The internal field that
remains is called the
residual magnetization.
Hysteresis Curve
)(int TB
)(mTBext
To reduce the internal
field, we must apply an
external field in the
opposite direction.
Hysteresis Curve
)(int TB
)(mTBext
coercive force
To reduce the internal
field, we must apply an
external field in the
opposite direction.
Hysteresis Curve
)(int TB
)(mTBext
coercive force
The external field needed
to bring the internal field
back to zero is called the
coercive force.
Hysteresis Curve
)(int TB
)(mTBext
As we continue to
increase the external field
in the negative direction,
domains align with the field.
Hysteresis Curve
)(int TB
)(mTBext
The process continues
just as when the external
field was in the positive
direction.
Soft Iron
)(int TB
)(mTBext
A nail made of soft iron
has domains that align
easily, but it can’t hold
the magnetization.
Soft Iron
)(int TB
)(mTBext
A nail made of soft iron
has domains that align
easily, but it can’t hold
the magnetization.
The coercive force
and the residual
magnetization of soft
iron are both small.
Good Permanent Magnet
)(int TB
)(mTBext
A good permanent magnet
has a large coercive force
and a large residual
magnetization.