Electromagnetic Induction

Chapter 22

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 Faraday’s Law of

electromagnetic induction Understand and apply Lenz’s Law

Expectations

After this chapter, students will: Apply Faraday’s and Lenz’s 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

Motional EMF

A wire passes through a uniform magnetic field. The length of the wire, the magnetic field, and the velocity of the wire are all perpendicular to one another:

L

v

B

+

-

Motional EMF

A positive charge in the wire experiences a magnetic force, directed upward:

L

v

B

+

-

qvBqvBFm 90sin

Motional EMF

A negative charge in the wire experiences the same magnetic force, but directed downward:

These forces tend to separate the charges.

L

v

B

+

-

qvBFm

Motional EMF

The separation of the charges produces an electric field, E. It exerts an attractive force on the charges: L

v

B

+

-

EqFC E

Motional EMF

In the steady state (at equilibrium), the magnitudes of the magnetic force – separating the charges – and the Coulomb force – attracting them – are equal.

L

v

B

+

-EqqvB

E

Motional EMF

Rewrite the electric field as a potential gradient:

Substitute this result back into our earlier equation:

L

v

B

+

-

L

EMF

L

VE

E

Motional EMF

Substitute this result back into our earlier equation: L

v

B

+

-

L

EMF

L

VE

E

vLBEMF

qvBqL

EMF

qvBEq

Motional EMF

This is called motional EMF. It results from the constant velocity of the wire through the magnetic field, B.

L

v

B

+

-

E

vLBEMF

Motional EMFNow, 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

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

LBR

vBLILBF

2

Motional EMF

Right-hand rule #1 shows that this force opposes the motion 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

Motional EMF

The force’s 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

Motional EMF

Suppose that, instead of being perpendicular to the plane of the sliding-wire circuit, the magnetic field had made an angle with the perpendicular to that plane.

The perpendicular

component of B: B cos

B B cos

v

x

Motional EMF

The motional EMF:

Rewrite the velocity:

Substitute:

B B cos

v

x

cosvLBEMF

t

xv

cos

cos

LBt

xEMF

vLBEMF

Motional EMF

L x is simply the change in the loop area.

x

x

L

A = L x

t

ABEMF

LxA

Bt

LxEMF

cos

cos

Motional EMF

Define a quantity :

Then:

is called magnetic

flux.

SI units: T·m2 = Wb (Weber) x

x

L

A = L xtt

ABEMF

cos

cosAB

Magnetic Flux

Wilhelm Eduard Weber

1804 – 1891

German physicist and mathematician

Faraday’s Law

In our previous result, we said that the induced EMF was equal to the time rate of change of magnetic flux through a conducting loop. This, rewritten slightly, is called Faraday’s Law:

Why the minus sign?

tEMF

Faraday’s Law

Michael Faraday

1791 – 1867

English physicist

and mathematician

Faraday’s Law

To make Faraday’s Law complete, consider adding N conducting loops (a coil):

What can change the magnetic flux?

B could change, in magnitude or direction A could change could change (the coil could rotate)

tNEMF

Lenz’s Law

Here is where we get the minus sign in Faraday’s Law:

Lenz’s 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 Faraday’s Law reminds us of that.

tNEMF

Lenz’s Law

Heinrich Friedrich Emil Lenz

1804 – 1865

Russian physicist

Lenz’s Law

Lenz’s Law says that the direction of the induced current 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?

Lenz’s LawHow 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.

Faraday’s Law: the Generator

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

but changes

with time:

tNEMF

cosAB

t

Faraday’s Law: the Generator

So the flux also changes with time:

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

Substitute into Faraday’s Law:

tABAB coscos

tABt

sin

tNABt

NEMF sin

Faraday’s 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 max

2

nt

Back EMF in Electric Motors

An electric motor also contains a coil rotating in a magnetic field.

In accordance with Lenz’s Law, it generates a voltage, called the back EMF, that acts to oppose its motion.

Back EMF in Electric Motors

Apply Kirchhoff’s loop rule:

R

EMFVIEMFIRV B

B

0

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.

Mutual Inductance

Faraday’s Law says that the changing magnetic flux in the second coil produces a voltage in that coil.

The net flux in the

secondary:

PSS IN

Mutual Inductance

Convert to an equation, using a constant of proportionality:

PSS

PSS

MIN

IN

Mutual Inductance

The constant of proportionality is called the mutual inductance:

P

SS

PSS

I

NM

MIN

Mutual Inductance

Substitute this into Faraday’s Law:

SI units of mutual inductance: V·s / A = henry (H)

t

IM

t

MI

t

N

tNEMF PPSSS

SS

PSS MIN

Mutual Inductance

Joseph Henry

1797 – 1878

American physicist

Self-Inductance

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

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

This is called self-inductance.

Self-Inductance

The self-induced voltage is calculated from Faraday’s Law, just as we did the mutual inductance.

The result:

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

t

ILEMFself

Mutual Inductance: Transformers

A transformer is two coils wound around a common iron core.

Mutual Inductance: Transformers

The self-induced voltage in the primary is:

Through mutual induction, and EMF appears in the secondary:

Their ratio:

tNEMF PP

tNEMF SS

P

S

P

S

P

S

N

N

tN

tN

EMF

EMF

Mutual Inductance: Transformers

This transformer equation is normally written:

The principle of energy conservation requires that the power in both coils be equal (neglecting heating losses in the core).

P

S

P

S

N

N

V

V

S

P

S

P

P

S

SSPPP

N

N

V

V

I

I

IVIVP

Inductors and Stored Energy

When current flows in an inductor, work has been done 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

Inductors and Stored Energy

In general, a volume in which a magnetic field exists has an energy density (energy per unit volume) stored in the field:

0

2

2volume

energydensityenergy

B