Electromagnetic induction - Northeastern University · PDF fileCurrent I Current I + + + + +-----B=? Clicker Question Consider the Biot-Savart Law for Magnetic Fields. Is there a Magnetic

Electromagnetic induction

rr

kQE ˆ

2=

v

20 ˆ

4 r

rvqB

×=v

v

πµ

So far we have found that

Electric charges create E-fields

Moving electric charges create B-fields

But this is not the whole story!

2


Electric Fields can also be created by a changing Magnetic Field.

Magnetic Fields can also be created by a changing Electric Field.

This is described by “Faraday’s Law”.

This will help complete the set of equations called Maxwell’s

equations of electricity and magnetism.

Electricity and Magnetism can be unified in a single theory!

The electricity and magnetic forces are manifestations of a

single “electromagnetic interaction”


Magnetic Flux

Magnetic Flux

The flux through an element of area is:

To calculate the flux through the surface, we

need to integrate:

For the simple case of a flat surface, we obtain:

B

φ

Magnetic flux

7

Clicker Question

At what angle of orientation of the surface is the

flux through the surface half of the maximum

possible flux?

A) φ = 0 degrees

B) φ = 30 degrees

C) φ = 45 degrees

D) φ = 60 degrees

E) None of the above

B

φ

Faraday’s Law of Induction

8

dt

d BΦ−=εTwo new quantities introduced.

1. Magnetic Flux = ΦB

2. Electro Motive Force (EMF) = ε

A changing (time-dependent) magnetic field “induces” (generates) an EMF ε.

The magnitude of the EMF equals minus the rate of change of the flux.

Important observations:

• Recall: EMF is not a force, but a source of voltage

capable of generating power.

• The EMF is called “motional” EMF, to distinguish

from “chemical” EMF (batteries)

• Recall: The magnetic flux through a closed surface is

zero! (Gauss’ Law for B-fields)

• The surface for Faraday’s Law is an open surface!

10

Clicker Question

We have a square imaginary loop with side of length a.

There is a Magnetic Field that is uniform throughout the

region as shown. How does the Magnetic Flux magnitude

|ΦB| change if we halve the Magnetic Field strength and

double the sides of the square loop?

A) Flux goes to ½ original value

B) Flux goes to ¼ original value

C) Flux goes to 2 times original value

D) Flux does not change

E) None of the above

X B

Changing the magnetic flux

11

1. Change the strength of the Magnetic Field

2. Change the area of the loop

3. Change the orientation of the loop (A vector) and the B-field

12

A loop of wire is moving rapidly through a uniform magnetic field as

shown. Is a non-zero EMF induced in the loop?

A) No, there in no EMF

B) Yes, there is an EMF

Clicker Question

13

A loop of wire is spinning rapidly about a stationary axis in uniform

magnetic field as shown.

True or False: There is a non-zero EMF induced in the loop.

A) False, zero EMF

B) True, there is an induced EMF

Clicker Question

and the angle φ is changing.

φcosBAF =

Direction of the EMF around a loop

Lenz’s Law

15

The (-) negative sign in Faraday’s Law is more of a reminder.

Lenz’s Law is just another way to express Faraday’s Law. Moreover,

it can be deduced from it.

dt

d BΦ−=ε

The induced EMF tends to induce a current in the

direction that opposes the changes in magnetic flux

Lenz’s Law

16

If we move the magnet closer to the loop, what is the direction of

the induced EMF and thus the direction of the current in the loop?

∫ ⋅=Φsurf

B AdBvv

1. The Magnetic

Flux is increasing.

2. Therefore 0>Φdt

d B

3. Induced current must create B-field

that fights the change!

Lenz’s Law

17

0>Φdt

d B

Induced current must create B-field that

fights the change!

An induced B-field down through the loop

will create a slight decrease in ΦB.

Thus, we have determined the direction of

the induced current and EMF.

i

B

Lenz’s Law

18

“The Magnetic Flux ΦB

Up Through the Loop is Increasing.

To Fight this Change, an Induced B-field is made that

Decreases the Flux Up Through the Loop.

This is accomplished by the induced current.”

Lenz’s law (summary)

20

A bar magnet is positioned below a loop of wire. The magnet is

pulled down, away from the loop. As viewed from above, is the

induced current in the loop clockwise, counterclockwise, or zero?

A: clockwise B: counter-clockwise

eyeball

C: zero

Answer: Counterclockwise.

As the magnet is pulled away, the flux is decreasing. To

fight the decrease, the induced B-field should add to the

original B-field.

Clicker Question

i

B

21

A loop of wire is sitting in a uniform, constant magnet field as

shown. Suddenly, the loop is bent into a smaller area loop. During

the bending of the loop, the induced current in the loop is ...

A: zero

B: clockwise

C: counterclockwise

B(in) B(in)

Answer: Clockwise. The flux into the page is decreasing as the

loop area decreases. To fight the decrease, we want the

induced B to add to the original B. By the right hand rule

(version II) , a clockwise induced current will make an induced B

into the page, adding to the original B.

Clicker Question

22

Is there an induced current in this circuit? If so, what is its

direction?

A. Yes, clockwise

B. Yes, counterclockwise

C. No

Clicker Question

* There is zero Magnetic Flux at all times through the loop.

23

A current-carrying wire is pulled

away from a conducting loop in the

direction shown. As the wire is

moving, is there a clockwise current

around the loop, a counterclockwise

current or no current?

A. There is a clockwise current around the loop.

B. There is a counterclockwise current around the loop.

C. There is no current around the loop.

Clicker Question

24

Example with numbers:

V10 cm

10 cm

B (increasing)

Suppose there are 1000 loops to the wire coil.

Area = (0.1m)2=0.01 m2

dB/dt = +0.1 Tesla/second

dt

dBABA

dt

d

dt

d B ==Φ= )(|| ε

Volts001.0)1.0)(01.0(|| ==ε

Voltdt

dN

dt

Nd BB 1)1()(

|| =Φ=Φ=ε

Generators

25

Generators use Faraday’s Law to convert mechanical energy into

electrical energy.

This is the opposite of motors which convert electrical into

mechanical energy.

The alternator

26

• Wire loop in a constant external B-field.

• Turning a crank makes the loop rotate.

• This then induces a current in the wire loop.

X X

B B

i (induced)

The alternator

27

• Wire loop in a constant external B-field.

• Turning a crank makes the loop rotate.

• This then induces a current in the wire loop.

28

Clicker Question

X

X

B

Bi

Instead of turning a crank and inducing a current, what if you hook

up the wire loop to a battery and push current through it.

What happens to the loop?

A) Nothing

B) It feels a net force

C) It feels a net torque

D) The current stops flowing

Note that the wire loop is a rectangle that is slightly rotated

from the plane of the page.

29

Generator Motor

Converts mechanical energy

into electrical energy.

Crank or otherwise

mechanically turns a wire loop

in an external B-field. This

induces an electric current in

the loop.

Converts electrical energy into

mechanical energy.

Battery creates a current in a

loop. This interacts with an

external B-field to create a

mechanical torque.

The “sliding bar generator”

30

Sliding metal bar on metal rails

L

x=vt

v

X X X X X XB

Bar is pulled by an

external agent to the

right at constant

velocity.

There is a uniform B-

field surrounding the

rails and loop

system.


31

L

x=vt

v

X X X X X XB

This creates a wire loop

whose area is growing with

time.

LvtLxA ==

Thus, the Magnetic Flux is increasing with time!

BLvLvtdt

dBBA

dt

d

dt

dB ===Φ )()(

32

L

x=vt

v

X X X X X XB

Clicker Question

What is the direction of

the induced EMF and

thus current in the wire

loop?

A) Clockwise

B) Counter Clockwise

Motional EMF induced BLv=ε


33

There is another way to see why the current in the bar is up.

Charges in the bar are moving to the right

with the bar itself.

v

X

B

There is a B-field in this region.

Thus, there is an upward force on the charges.

(up) qvBBvqF =×=v

v

v

Thus, there is an upward current in the bar!

i


34

v

X

B

i

One more consequence….

If there is an upward current in the wire and there

is an external B-field, the we get?

(left) ILBBLIFMag =×=vvv

FM

But, there is also the external agent pulling the

rod to the right.

FAgent

If the velocity is constant, what is true about the

forces? They must be equal and opposite!

A conducting loop is halfway into a magnetic field. Suppose the magnetic field begins to increase rapidly in strength. What happens to the loop?

A. The loop is pushed upward, toward the top of the page.B. The loop is pushed downward, toward the bottom of the page.C.The loop is pulled to the left, into the magnetic field.D.The loop is pushed to the right, out of the magnetic field.E. The tension is the wires increases but the loop does not move.

Clicker Question

Eddy currents

If a metal object (something in which electric charges can move) and an external source of Magnetic Field are in relative motion so that there is changing Magnetic Flux through the metal, then Faraday says there is an EMF and thus an Eddy current.

If the metal is moving, then the direction of the current is always such as to cause a force which opposes the motion (slows the metal).

Eddy currents: Applications

Eddy Current Brakes on Trains

Since there is no direct contact, there is minimal friction.Thus the brakes do not wear down!


Metal detector Portable metal detector


Induction CooktopWorks via changing B-field, inducing a current in the pot or wok (which must be metal). This current then leads to power loss as heating of the pot. The cooktop surface which is not metal and does not conduct, does not get hot.

Induced electric field

40

In the case where the loop or metal is stationary and the Magnetic

field is changing, then there is no Magnetic force on the charges in

the wire (since v=0)

BvqFM

v

v

v

×=

Thus, the only way to explain the induced current in the wire is to

admit that there is the creation of a new Electric Field!

Motional EMF revisited

41

∫ ⋅=Loop

ldEvv

εTechnically, the EMF around a closed Loop is defined as:

But, recall that Voltage is defined as:

∫ ⋅−=∆ ldEVvv

Was not this integral around a closed loop always zero?

The motional EMF is not always zero around a closed loop !

Faraday’s Law reformulated

42

dt

d BΦ−=ε

∫∫ ⋅−=⋅surfaceLoop

AdBdt

dldE

vvvv

How do the loop and the surface relate?


43

∫∫ ⋅−=⋅surfLoop

AdBdt

dldE

vvvv

Bexternal

R [ ])()2( 2RBdt

dRE ext ππ

vv

−=


44

B

Wire LoopIf B = constant,

Then Magnetic Flux = constant

dΦB/dt = 0

EMF = 0 and Voltmeter reads 0.

If B =increasing,

Then Magnetic Flux = changing

|dΦB/dt| > 0

EMF > 0 and Voltmeter reads > 0.

What if the wire loop was not there?

V


45

A changing B-field creates a new kind of E-field.

BIf B is decreasing � E-field is CCW

E

B

E

If B is increasing � E-field is Clockwise


46

The (non-Coulomb) E-field is very different from the Coulomb E-

field created by single electric charges.

qB

E

E

0≠⋅= ∫Loop

ldEvv

ε0=⋅−=∆ ∫ ldEVvv

Time dependent B-fieldsSo far we have seen that a time varying flux can be generated by moving a magnet, or by

mechanically changing the surface area…

But what about changing the magnitude of the field?

From Ampere’s Law:

Replacing in Faraday’s Law we obtain:

Videos

http://www.youtube.com/watch?v=jBrneA-wQGA

Tesla

Ampere’s Law revisited

49

Maxwell realized that Ampere’s Law must be incompletebecause he discovered a situation where it gave the wrong

answer!

?0 +=⋅∫ encildB µvv


50

Current I(t) Current I(t)

+++++

-----

1. As the currents are flowing, one has Magnetic Fields around the wires.2. Charge is also building up on the plates, creating an increasing Electric Field

between them.

B BE

Consider a charging capacitor. The charge builds up in the plates, and there is a time-dependent current flowing through the wires I(t)

51

Current I Current I

+++++

-----

B=?

Clicker Question

Consider the Biot-Savart Law for Magnetic Fields.Is there a Magnetic Field at the point labeled between the plates?A)Yes, there is a B-FieldB)No, there is zero B-field

∫∫×==2

0 ˆ

4 r

rLdIBdBtot

v

vv

πµ

52

Current I Current I

+++++

-----

B=?

Clicker Question

Now consider an Amperian Loop as drawn. According to Ampere’s Law is there a Magnetic Field at the point labeled between the plates?A)Yes, there is a B-FieldB)No, there is zero B-field

Something must be wrong with Ampere’s Law!


Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose.

Current I Current I

+++++

-----


Both surfaces have the same boundary, if Ampere’s law applies, then the result should be the same, independently of the surface we choose.

Ampere’s law: solving the paradox

The changing Electric Field inside the capacitor must act like a current enclosed for Ampere’s Law.

∫ ⎟⎠

⎞⎜⎝

⎛ Φ+=⋅dt

dIldB E

enc 00 εµvv

Where ∫ ⋅=Φ AdEE

vv

Maxwell defined a fictitious “displacement current”:

Thus, a changing Electric Field creates a Magnetic Field !

dt

dI E

D

Φ= 0ε

The displacement current

EEAEdd

ACVQ Φ==== 00

0 )( εεε

CE

D Idt

dQ

dt

dI ==Φ= 0ε

The B-field in between the plates is exactly the same as the field created by the current outside.The fictitious “displacement current” gives an idea of continuity of the current, even though there is no charge flowing in the space separating the plates

Between the plates: 0;0 =Φ= CE

D Idt

dI ε

Outside the capacitor: 0;0 ≠= CD II

Maxwell’s equations

0εencl

Surf

QAdE =⋅∫rr

Gauss’ Law for E-fields

0=⋅∫SurfAdBrr

Gauss’ Law for B-fields

⎟⎠

⎞⎜⎝

⎛ Φ+=⋅∫ dt

dIldB E

thruloop 00 εµrr

Ampere’s Law

dt

dldE B

loop

Φ−=⋅∫rr

Faraday’s Law

Documents

Electromagnetic induction - Northeastern University · PDF fileCurrent I Current I + + + + +-----B=? Clicker Question Consider the Biot-Savart Law for Magnetic Fields. Is there a Magnetic