Class 31Today we will:• learn about EMF• learn how Faraday’s law works• learn Lenz’s Law and how to apply it

Last Time -- Induced Current

Accelerating charges produce electric fields in the opposite direction to the acceleration.

iE

E

i

Faraday’s Law

If the number of magnetic field lines through a loop is changing, we produce a looping electric field.

Induced Current

Current increases in a wire…

i

Induced Current

… so the magnetic field increases…

i

Induced Current

… so the number of magnetic field lines passing through the loop (flux) increases…

i

Induced Current

… so there is an induced EMF around the loop …

i

EMF

Induced Current

… so current flows around the loop.

i

i

EMF

Faraday’s Law of Induction

t

BE

dt

d BE

Faraday’s Law of Induction

t

BE

dt

d BE

=EMF

What is EMF?1) Any voltage, as from a battery.

2) An effective voltage produced by induced electric fields.

What is EMF?1) Any voltage, as from a battery.

2) An effective voltage produced by induced electric fields.

I usually reserve the term EMF for induced voltages.

Two Ways to Produce Induced EMF

1) Acceleration of charges – changing current in a circuit.

2)Motional EMF – charges in a conductor moving in a B field.

Motional EMF

A wire of length L moves through a magnetic field. The wire is perpendicular to B. What happens?

B

v

Motional EMF

Charges along the wire feel a Lorentz Force.

B

v

BvqF

Motional EMFCharges doesn’t increase indefinitely. Eventually a voltage develops across the wire.

B

vvBLELV

vBE

qEqvBFF EB

Motional EMFAdd three other fixed wires to make a loop. Now current will continue to flow around the loop.

B

v

Clicker Question 1

B

vv

What happens if all sides of the loop move together?A. Current flows.B. Current doesn’t flow.

What happens both ways?

B

v

iaE

E

i

What happens both ways?The magnetic flux – the number of magnetic field lines – passing through the loop changes.

B

v

iaE

E

i

Faraday’s Law of InductionFaraday’s Law of Induction

…works for both motional EMF and the EMF of accelerating charges!

What is EMF?We can think of an induced EMF as a voltage produced all along a wire segment.

B

v

What is EMF?

Let’s take a square loop with an increasing magnetic field passing through it. Assume the wire has a small resistance.

V

What is EMF?

Assume that the EMF around the loop is 40 V. What would a voltmeter read?

V

What is EMF?

The voltmeter would read zero!

V = 0V

What is EMF?

The voltmeter would read zero because the voltage drop due to resistance in the wire segment is exactly the same as the voltage increase due to induction in the wire segment.

V = 0V

What is EMF?

Another way of putting it is that the wire segment is like a lot of little batteries and resistors in series. The voltage goes up through each battery, but down by the same amount through each resistor.

V = 0V

What is EMF?

Each electron that goes around the full loop once gains 40eV of energy from the EMF and loses 40eV of energy to heat!

V =0V

What if there’s a resistor in the loop?

The total EMF is 40V.

5

What if there’s a resistor in the loop?

The total EMF is 40V. Ohm’s Law gives i = 8A.

The voltage across the resistor is 40V.

5

Where is the EMF being produced?

5

Where is the EMF being produced?

Everywhere, including through the resistor.

5

Where is the voltage dropping?

5

Where is the voltage dropping?

Primarily in the resistor – just a little in the wire. The resistor only lets a little current trickle through the wire – as compared to having no resistor.

5

Lenz’s Law

To determine the direction induced current will flow in a circuit or to determine the direction of the induced electric field, we use Lenz’s Law.

Lenz’s Law

First, we ask two questions:

1) What is the direction of the external B field?

2) Is the external B field increasing or decreasing?

Lenz’s Law

1) What is the direction of the external B field?

2) Is the external B field increasing or decreasing?3) Find the direction of the induced

magnetic field. The induced magnetic field opposes change in the external magnetic field.

Lenz’s Law

1) What is the direction of the external B field?

2) Is the external B field increasing or decreasing?3) Find the direction of the induced

magnetic field. The induced magnetic field opposes change in the external magnetic field.

4) Find the direction the induced current using the right-hand rule.

Lenz’s Law

•The external B is into the screen and increasing.

x x

x x x x

x x x x

x x

externalB

Lenz’s Law

•The external B is into the screen and increasing.•To oppose change, the induced B, must be out of the screen.

x x

x x x x

x x x x

x x

externalB

inducedB

Lenz’s Law

•The external B is into the screen and increasing.•To oppose change, the induced B, must be out of the screen.• To produce this B, the current is ccw. x x

x x x x

x x x x

x x

externalB

inducedB

i

Lenz’s Law

• The external B is into the screen and decreasing.

x x

x x x x

x x x x

x x

externalB

Lenz’s Law

• The external B is into the screen and decreasing.

• To oppose change, the induced B, must be into

the screen. x x

x x x x

x x x x

x x

externalB

inducedB

i

Lenz’s Law

• The external B is into the screen and decreasing.

• To oppose change, the induced B, must be into

the screen.• To produce this B, the current is cw. x x

x x x x

x x x x

x x

externalB

inducedB

i

Class 32Today we will:• work several Faraday’s law problems• learn about Eddy currents

Maxwell’s Equations in Integral Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0enc

E

qAdE

dt

didB E

B 00

dt

ddE B

E

0AdBB

Maxwell’s Equations in Integral Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0enc

E

qAdE

dt

didB E

B 00

dt

ddE B

E

0AdBB

Flux through a Gaussian surface

Maxwell’s Equations in Integral Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0enc

E

qAdE

dt

didB E

B 00

dt

ddE B

E

0AdBB

Flux through a Gaussian surface

Line integral around an Amperian loop

Maxwell’s Equations in Integral Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0enc

E

qAdE

dt

didB E

B 00

dt

ddE B

E

0AdBB

Flux through a Gaussian surface

Line integral around an Amperian loop

Flux through the Amperian loop

Calculating Flux through a Loop

We will assume that the magnetic field is constant over a loop. Then, the flux is:

cosBAABB

Remember that points in the direction of the normal to the loop.

A

Three Ways to Generate an EMF

1) The magnetic field changes in time.

2) The area changes in time.

3) the angle changes in time.

cosBAABB

Eddy Currents

A wire loop is moved into a region where there is a magnetic field. In what direction does current flow?

B

Eddy Currents

A wire loop is moved into a region where there is a magnetic field. In what direction does current flow?

i

B

Eddy Currents

Viewing the same thing from the side, the loop becomes a magnet that is repelled by the external field. (Remember field lines come out of the N pole.)

B

N SS SN

Eddy Currents

It takes force to push the loop into the field. Where does this energy go?

B

N SS SN

Eddy Currents

If the wire loop is moved out of the magnetic field, in what direction does current flow?

B

Eddy Currents

If the wire loop is moved out of the magnetic field, in what direction does current flow?

B

i

Eddy Currents

Viewing the same thing from the side, the loop becomes a magnet that is attracted by the external field.

B

N SN SNS

Eddy Currents

A disk is even more effective at producing induced currents. Such currents are called “eddy currents.”

B

Changing Area

Put a moveable length of wire on a fixed u-shaped wire in a uniform magnetic field.

a

x

B

R

v

Lenz’s law

Which way does the induced current flow?

a

x

R

v B

Lenz’s law

Which way does the induced current flow?

a

x

R

v

i

B

Find the Flux

a

x

B

R

v

Find the Flux

a

x

R

v

vtxas

BavtBaxB

B

Find the EMF

a

x

R

v B

Find the EMF

a

x

R

v

Bavdt

d

BavtBax

B

B

BavNot worrying about the minus sign:

B

Find the Current

a

x

R

v B

Find the Current

a

x

R

v

R

Bav

Ri

Bav

B

Find the Power Dissipated in the

Resistor

a

x

R

v B

Find the Power Dissipated in the

Resistor

R

BaviP

R

Bav

Ri

Bav

2

a

x

R

v B

Find the Force on the Moveable Wire

a

x

R

v B

Find the Force on the Moveable Wire

R

vBaF

iaBF

BLiF

2

a

x

R

v B

Find the Work Done in Moving Δx

a

x

R

v B

Find the Work Done in Moving Δx

x

R

vBaxFW

R

vBaF

2

2

a

x

R

v B

Find the Mechanical Power

a

x

R

v B

Find the Mechanical Power

R

BavP

vR

vBa

t

x

R

vBa

t

WP

xR

vBaxFW

2

22

2

a

x

R

v B

Changing the Magnetic Field

a

/0)( teBtB

b

a

b

Lenz’s law

Which way does the induced current flow? /

0)( teBtB

a

Lenz’s law

Which way does the induced current flow? /

0)( teBtB

i

b

Find the Flux

ai

b

Find the Flux

/0

/0)(

tB

t

abeBBA

eBtB

ai

b

Find the EMF

ai

b

Find the EMF

/0

/0

tB

tB

abeB

dt

d

abeB

ai

b

Where does the energy come from this time?

ai

b

Where does the energy come from this time?

ai

b

If the external field comes from a permanent magnet, the magnetic field of the loop attracts the permanent magnet, making it more difficult to move away.

Where does the energy come from this time?

ai

b

If the external field comes from an electromagnet, the interaction of the loop with the electromagnet takes some energy from the electromagnet’s circuit.

We can use Faraday’s Law to calculate the electric field as well as

the EMF in one problem only!

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces /0)( teBtB

r

R

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces

r

R

/0

/2

0

/20

22

2

)(

t

E

tB

tB

erB

rE

rE

erB

dt

d

erBAtB

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces

r

R

/0

/2

0

/20

22

2

)(

t

E

tB

tB

erB

rE

rE

erB

dt

d

erBAtB

Flux through the Amperian loop of radius r.

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces

r

R

/0

/2

0

/20

22

2

)(

t

E

tB

tB

erB

rE

rE

erB

dt

d

erBAtB

Flux through the Amperian loop of radius r.

Line integral around the Amperian loop of radius r.

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces/

0)( teBtB

r

R

A Circular Loop in the Field of an Electromagnet with Circular Pole

Faces

/2

0

/2

0

/20

22

2

)(

t

E

tB

tB

er

RB

rE

rE

eRB

dt

d

eRBAtB

r

R

A Circular Loop in the Field of an Electromagnet with Circular Pole

FacesFlux through the Amperian loop of radius r.

/2

0

/2

0

/20

22

2

)(

t

E

tB

tB

er

RB

rE

rE

eRB

dt

d

eRBAtB

r

R

A Circular Loop in the Field of an Electromagnet with Circular Pole

FacesFlux through the Amperian loop of radius r.

But the field stops at R!

/2

0

/2

0

/20

22

2

)(

t

E

tB

tB

er

RB

rE

rE

eRB

dt

d

eRBAtB

r

R

A Circular Loop in the Field of an Electromagnet with Circular Pole

FacesFlux through the Amperian loop of radius r.

Line integral around the Amperian loop of radius r.

But the field stops at R!

/2

0

/2

0

/20

22

2

)(

t

E

tB

tB

er

RB

rE

rE

eRB

dt

d

eRBAtB

r

R

Changing the Angle

Attach a handle to a circular loop of wire.

Changing the Angle

BPlace the loop in a magnetic field with the shaft

perpendicular to .

A

B

Changing the Angle

We rotate the handle with angular speed .

The flux is:

The EMF is:A

B

tBAB cos

tBAdt

d B

sin

Class 33Today we will:•learn how motors and generators work•learn about split commutators and their use in DC motors and generators

Humphrey Davy (1778-1829)

•First to isolate potassium, sodium, barium, calcium, strontium, magnesium, boron, and silicon.

•1813: Discovers Michael Faraday

Michael Faraday (1791-1867)

•1831: Discovers electromagnetic induction independently of Henry.•Develops the transformer, motor, and generator.•Discovers the Faraday Effect of light - the rotation of the plane of polarization in magnetic fields.•Develops the First and Second Laws of Electrochemistry.•Discovers paramagnetism and diamagnetism.

James Clerk Maxwellb. 1831, Edinburgh, Scotlandd. 1879, Cambridgeshire, England

•1861: Proposes "displacement current" and creates Maxwell's Equations.

•Recognizes light as electromagnetic radiation.

James Clerk Maxwell“What is done by what is

called ‘myself’ is, I feel,

done by something that is

greater than myself within

me.”

Changing the Angle

Attach a handle to a circular loop of wire.

Changing the Angle

BPlace the loop in a magnetic field with the shaft

perpendicular to .

A

B

Changing the Angle

We rotate the handle with angular speed .

The flux is:

The EMF is:A

B

tBAB cos

tBAdt

d B

sin

Engineering Considerations

We usually want to get current out of the coils as they turn, so rather than use a complete loop, we use an open loop with each end connecting to a wire.

Engineering Considerations

But fixed wires would twist and break.

-- So we use commutators and brushes.

Commutators and Brushes

These allow electrical connections to be made by pressing conductors together.

wire

wire

shaft

graphite brush spring steel clip

wire

wire

shaft

shaft

commutator

commutator

Commutators

A typical AC generator or motor uses double commutators. Each end of the loop is connected to a separate commutator.

Motors

A simple motor consists of a current-carrying coil in a uniform magnetic field.

N

S

Motors

A torque on the coil tends to align the magnetic dipole moment of the loop with the external field.

N

S

Motors

A torque on the coil tends to align the magnetic dipole moment of the loop with the external field.

N

S

+

Motors

As the dipole moment rotates past the magnetic field, however, the torque reverses.

N

S

+

Motors

We’ve created a vibrator instead of a motor.

N

S

+

Motors

But, we could keep the loop (armature) turning in the same direction, if we could reverse the magnetic dipole moment. This can be done by changing the direction of the battery.

N

S

+

Motors

It’s a little hard to keep moving the leads on the battery back and forth by hand. So we need a better way of doing it.

N

S

+

AC Current

The easiest way to do this is to use AC current instead of a battery. The direction of the current through the loop automatically changes sign periodically.

AC Current

Note that the speed of such a motor is closely tied to the frequency of the AC power supply.

The Split Commutator

Another clever solution is to use a “split commutator.” A split commutator automatically changes the end of the loop connected to the positive terminal of the battery every half cycle.

DC Motors

Thus, split commutators allow motors to be operated by DC power sources.

Generators

Generators are just motors operated in reverse.

N

S

load

AC Generators

With double commutators, we get a sinusoidal current out of a generator.

N

S

load

Quasi-DC Generators

With split commutators, we get a sinusoidal current that changes sign each half cycle.

N

S

load

DC Generators

Adding a second loop can give something that is closer to a DC output.

N

S

load