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Electromagnetic Induction

Chapter 22

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Expectations

After this chapter, students will:

Calculate the EMF resulting from the motion of

conductors in a magnetic field

Understand the concept of magnetic flux, and

calculate the value of a magnetic flux

Understand and apply Faradays Law of

electromagnetic induction

Understand and apply Lenzs Law

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Expectations

After this chapter, students will:

Apply Faradays and Lenzs Laws to some

particular devices:

Electric generators

Electrical transformers

Calculate the mutual inductance of two

conducting coils

Calculate the self-inductance of a conducting coil

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Motional EMF

A wire passes through

a uniform magnetic

field. The length of

the wire, themagnetic field, and

the velocity of the

wire are allperpendicular to

one another:

L

v

B

+

-

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Motional EMF

A positive charge in

the wire

experiences a

magnetic force,directed upward: L

v

B

+

-

qvBqvBFm 90sin

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Motional EMF

A negative charge in the

wire experiences the

same magnetic force,

but directeddownward:

These forces tend toseparate the charges.

L

v

B

+

-

qvBFm

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Motional EMF

The separation of the

charges produces an

electric field,E. It

exerts an attractiveforce on the charges: L

v

B

+

-

EqFC E

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Motional EMF

In the steady state (at

equilibrium), the

magnitudes of the

magnetic forceseparating the

chargesand the

Coulomb force

attracting themareequal.

L

v

B

+

-EqqvB

E

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Motional EMF

Rewrite the electric

field as a potential

gradient:

Substitute this result

back into our earlierequation:

L

v

B

+

-

L

EMF

L

VE

E

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Motional EMF

Substitute this result

back into our earlierequation: L

v

B

+

-

L

EMF

L

VE

E

vLBEMF

qvBqL

EMF

qvBEq

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Motional EMF

This is calledmotional

EMF. It results fromthe constant velocity

of the wire through

the magnetic field,B.

L

v

B

+

-

E

vLBEMF

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Motional EMF

Now, our moving wire slides over two other wires,

forming a circuit. A current will flow, and power

is dissipated in the resistive load:

L

v

B

+

-

R

I

R

vBLP

R

vBLvBLVIP

R

vBL

R

VI

vBLVEMF

2

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Motional EMF

Where does this power come from?

Consider the magnetic

force acting on the

current in the sliding

wire:L

v

B

+

-

R

I

R

LBvF

LB

R

vBLILBF

2

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Motional EMF

Right-hand rule #1 shows that this force opposes themotion of the wire. To move the wire at constant

velocity requires an equal and opposite force.

That force does work:

The power:

L

v

B

+

-

R

I

FvtFxW

Fvt

Fvt

t

WP

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Motional EMF

The forces magnitude was calculated as:

Substituting:

which is the same as the

power dissipated electrically.

L

v

B

+

-

R

I

R

vBLv

R

BLvFvP

22

R

BLvF

2

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Motional EMF

Suppose that, instead of being perpendicular to theplane of the sliding-wire circuit, the magnetic field

had made an angle fwith the perpendicular to that

plane.

The perpendicular

component ofB:B cos f

BB cos f

f

v

x

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Motional EMF

The motional EMF:

Rewrite the velocity:

Substitute:

BB cos f

f

v

x

fcosvLBEMF

t

xv

f

f

cos

cos

LBt

xEMF

vLBEMF

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Motional EMF

L x is simply the change in the loop area.

x

x

L

A = L x

t

ABEMF

LxA

B

t

LxEMF

f

f

cos

cos

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Motional EMF

Define a quantityF:

Then:

F is calledmagnetic

flux.

SI units: Tm2 = Wb (Weber) x

x

L

A = L xtt

ABEMF

F

fcos

fcosABF

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Magnetic Flux

Wilhelm Eduard Weber18041891

German physicist and

mathematician

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Faradays Law

In our previous result, we said that the induced EMFwas equal to the time rate of change of magnetic

flux through a conducting loop. This, rewritten

slightly, is called Faradays Law:

Why the minus sign?

tEMF

F

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Faradays Law

Michael Faraday

17911867

English physicist

and mathematician

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Faradays Law

To make Faradays Law complete, consider addingNconducting loops (a coil):

What can change the magnetic flux?

B could change, in magnitude or direction

A could change

fcould change (the coil could rotate)

t

NEMF

F

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Lenzs Law

Here is where we get the minus sign in FaradaysLaw:

Lenzs Law says that the direction of the induced

current is always such as to oppose the change in

magnetic flux that produced it.

The minus sign in Faradays Law reminds us of that.

tNEMF

F

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Lenzs Law

Heinrich Friedrich Emil Lenz

18041865

Russian physicist

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Lenzs Law

Lenzs Law says that the direction of the inducedcurrent is always such as to oppose the change in

magnetic flux that produced it.

What does that mean?

How can an induced current oppose a change in

magnetic flux?

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Lenzs Law

How can an induced current oppose a change in

magnetic flux?

A changing magnetic flux induces a current.

The induced current produces a magnetic field.

The direction of the induced current determines

the direction of the magnetic field it produces.

The current will flow in the direction (remember

right-hand rule #2) that produces a magnetic field

that works against the original change in magnetic

flux.

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Faradays Law: the Generator

A coil rotates with a constant angular speed in amagnetic field.

but fchanges

with time:

tNEMF

F

fcosABF

tf

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Faradays Law: the Generator

So the flux also changes with time:

Get the time rate of change (a calculus problem):

Substitute into Faradays Law:

tABAB f coscos F

tABt

sinF

tNABt

NEMF sinF

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Faradays Law: the Generator

The maximum voltage occurs when :

What makes the voltage larger? more turns in the coil

a larger coil area

a stronger magnetic field a larger angular speed

NABEMF max2

n

t

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Back EMF in Electric Motors

Apply Kirchhoffs loop rule:

REMFVIEMFIRV BB 0

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Mutual Inductance

A current in a coil produces a magnetic field.

If the current changes, the magnetic field changes.

Suppose another coil is nearby. Part of the magnetic

field produced by the first coil occupies the inside

of the second coil.

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Mutual Inductance

Faradays Law says that the changing magnetic fluxin the second coil produces a voltage in that coil.

The net flux in thesecondary:

PSS IN F

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Mutual Inductance

Convert to an equation, using a constant ofproportionality:

PSS

PSS

MIN

IN

F

F

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Mutual Inductance

The constant of proportionality is called themutualinductance:

P

SS

PSS

I

NM

MIN

F

F

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Mutual Inductance

Substitute this into Faradays Law:

SI units of mutual inductance: Vs / A = henry (H)

t

IM

t

MI

t

N

tNEMF PPSSSSS

F

F

PSS MIN F

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Mutual Inductance

Joseph Henry

17971878

American physicist

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Self-Inductance

Changing current in a primary coil induces a voltagein a secondary coil.

Changing current in a coil also induces a voltage inthat same coil.

This is calledself-inductance.

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Self-Inductance

The self-induced voltage is calculated fromFaradays Law, just as we did the mutual

inductance.

The result:

The self-inductance,L, of a coil is also measured inhenries. It is usually just called the inductance.

t

ILEMFself

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Mutual Inductance: Transformers

The self-induced voltage in the primary is:

Through mutual induction, and EMF appears in thesecondary:

Their ratio:

tNEMF PP

F

tNEMF SS

F

P

S

P

S

P

S

N

N

tN

tN

EMF

EMF

F

F

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Inductors and Stored Energy

When current flows in an inductor, work has beendone to create the magnetic field in the coil. As

long as the current flows, energy is stored in that

field, according to

2

2

1LIE

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Inductors and Stored Energy

In general, a volume in which a magnetic field existshas an energy density (energy per unit volume)

stored in the field:

0

2

2volume

energydensityenergy

B