Faraday’s Law

dtd B

The EMF around a closed path is equal to the rate of change of the magnetic flux inside the path.

EMF

Solenoid

Wire

Ammeter 电表The magnetic field in the solenoid increases from 0.1 T to 0.7 T in 0.2 seconds.

What current will the ammeter measure?

2solenoid cm 3A

B

Example

solenoidBAB

tBA

dtd B

solenoidEMF

5.0R

A 108.1EMF 3R

i

Solenoid

Wire

Ammeter 电表The magnetic field in the solenoid increases from 0.1 T to 0.7 T in 0.2 seconds.

What current will the ammeter measure?

2solenoid cm 3A

B

Example

0B

0EMF

dtd B

5.0R

0EMF

Ri

EMF in a coil with many turnsNCE

NCE

NCE

NCE

B

B increasing

turnone around distance

EMF turns

NC

N NC

EN

sdE

EMF in a coil with many turnsNCE

NCE

NCE

NCE

B

B increasing

turnoneEMFEMF N

EMF in a coil with many turnsNCE

NCE

NCE

NCE

B

B increasing

dtdN B

EMF

Solenoid

Wire

Ammeter 电表The ammeter measures a current of 10 A.

Now, we replace the single wire with a coil containing N = 20 turns.

What is the current now?

2solenoid cm 3A

B

Example

5.0R

A 106.3

EMF

EMF

2

turnone turnone

NiR

N

Ri

Another way to make currents with magnetic fields:

Motional EMF动生电动势

v

A conductor moves through a magnetic field. What happens to the charges in the conductor?

B

Magnetic forces polarize the conductor.

v

A conductor moves through a magnetic field. What happens to the charges in the conductor?

B

Connect it to a circuit – it acts like a battery.

i i

v

Magnetic forces acting on the moving rod create EMF in the circuit.

B

sdBvsdqFB

EMF

L i i

vB

L

BLvEMF

i i

Magnetic forces acting on the moving rod create EMF in the circuit.

v

We can also calculate the EMF using the change in magnetic flux.

B

i iL

tLvA

tv

tLvBABB

vB

i iL

tv

BLvtB

But we know this is

the same as the EMF!

We can also calculate the EMF using the change in magnetic flux.

vB

i iL

tv

dtd B

EMF

We can also calculate the EMF using the change in magnetic flux.

dtd B

EMF

This equation works, no matter if

• the magnetic field is changing (Faraday’s Law),

• the area of the circuit is changing (motional EMF),

• or both!

Magnitude of EMF = rate of change of magnetic flux

dtdAB

dtdBA

dtABd

EMF

In a uniform magnetic field:

caused by changing magnetic field

caused by motion in magnetic field

B

B

A

n̂

Example

0.20 m

0.10 m

v

What EMF is induced in the loop, as it moves to the right?

Magnetic flux is not changing, so no EMF.

0EMF

ABdtd

dtd B

30

B

T 3B

B

30

Example

0.20 m

0.10 m

n̂

2initial m 020.0m 10.0m 20.0 A

What EMF is induced in the loop, if it is stretched (被拉大了 )?

T 3B

B

30

Example

0.22 m

0.12 m

What EMF is induced in the loop, if it is stretched (被拉大了 )?

n̂

2initial m 020.0m 10.0m 20.0 A

2final m 026.0m 12.0m 22.0 A

T 3B

Stretching is done in a time of 0.1 s.

23 m 106 A

B

30

Example

0.22 m

0.12 m

What EMF is induced in the loop, if it is stretched (被拉大了 )?

n̂

dtdABAB

dtd

dtd B

EMF

T 3B

s 1.0t

23 m 106 A

B

30

Example

0.22 m

0.12 m

There will be an induced current in the loop while it is being stretched.

n̂

V 2.0s 1.0m 10630cosT 3EMF

23

T 3B

s 1.0t

23 m 106 Aii

ii

Maxwell’s Equations (so far…)

0

inside

qAdE

0 AdB

enclosed0isdB

0 sdE *Not complete

*Not complete

Maxwell’s Equations (so far…)

0

inside

qAdE

0 AdB

enclosed0isdB

AdBdtdsdE

*Not complete

Gauss’ Law for electric fields

Gauss’ Law for magnetic fields

Faraday’s Law