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 FieldThere 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

iLBBTLiTF

TLv

TNe

tqi

NevBqvBFtot

,

Force on a Wire in a Magnetic FieldMore 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 LoopsThe 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 LoopsNote 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

2a

i

F

F

Torque on a Wire Loop in a Uniform Magnetic Field

B

A

F

F

2a

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

sin2a

2a

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

2a

F

F

sinF

Torque on a Wire Loop in a Uniform Magnetic Field

sin22 Fa

sinsinsin

iABaiaBaF

Torque on a Wire Loop in a Uniform Magnetic Field

Section 3Magnetic Dipoles

Define the Magnetic Dipole MomentSince 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

BiAB

sinsin

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 DipoleAn 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 MagnetsIn 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 MagnetsGilbert: 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 MagnetsAn electron spins…

N

S

Electrons as MagnetsWe can measure the magnetic dipole moment of an atom by measuring the force on electrons in a nonuniform field.

N

S

Electrons as MagnetsWe can model an electron as a spinning sphere and determine its radius: 2.8 fm

N

S

Electrons as MagnetsBut 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 CurveWe start with no internal

field (unaligned) and no

external field.

Hysteresis Curve)(int TB

)(mTBext

We increase the external

field, causing some of the

domains to align.

Hysteresis Curve)(int TB

)(mTBext

As the external field

increases, the internal

field eventually stops

growing. Why?

Hysteresis Curve)(int TB

)(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.