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1 My Chapter 20 Lecture Outline

1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Page 1: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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MyChapter 20

LectureOutline

Page 2: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Chapter 20: Electromagnetic Induction

•Motional EMF

•Electric Generators

•Faraday’s Law

•Lenz’s Law

•Transformers

•Eddy Currents

•Induced Electric Fields

•Mutual- and Self-Inductance

•LR Circuits

Page 3: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.1 Motional EMF

Consider a conductor in a B-field moving to the right.

⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗

V

Page 4: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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An electron in the conductor experiences a force downward.

Ve-

F

The electrons in the bar will move toward the bottom of the bar.

This creates an electric field in the bar and results in a potential difference between the top and bottom of the bar.

( )BvF ×= qB

Page 5: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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What if the bar were placed across conducting rails (in red) so that there is a closed loop for the electrons to follow?

In this circuit, the electrons flow clockwise; the current is counterclockwise.

⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗

VL

Page 6: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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The motional EMF is vBL=ε

where L is the separation between the rails.

The current in the rod isR

vBL

RR

VI ==

Δ=

ε

where R is the resistance in the “wires”.

Page 7: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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The magnitude of the magnetic force on the rod is:

R

LvBLB

R

vBLILBILBF

22

90sin ===°=

The rod has a current through it. What is the direction of the magnetic force on the rod due to the external magnetic field?

( )BLF ×= I

Using the right hand rule, the force on the bar is directed to the left.

Page 8: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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To maintain a constant EMF, the rod must be towed to the right with constant speed. An external agent must do work on the bar. (Energy conservation)

Page 9: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.2 Electric Generators

A coil of wire is spun in a magnetic field. This produces an EMF and also a current; both vary with time. (AC-alternating current)

An energy source is needed to turn the wire coil. Examples include burning coal or natural gas to produce steam; falling water.

Page 10: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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The EMF produced by an AC generator is:

( ) tt ωεε sin0=

In the United States and Canada ε0 = 170 volts and f = /2 = 60 Hz.

Page 11: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.3 Faraday’s Law

Moving a conductor through a B-field will generate an EMF. Another way to generate an EMF is to place a stationary conductor in a B-field that varies with time.

Page 12: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

12⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗

The magnetic flux is proportional to the number of B-field lines that cross a given area.

Loop of wire with area A

θcosBAB =Φ The unit of magnetic flux is the weber: 1 Wb = 1 Tm2

Page 13: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Faraday’s Law:t

N B

ΔΔΦ

−=ε

An induced EMF in a “coil” of N loops is due to a changing magnetic flux.

Ways to induce an EMF:

1. Vary the magnetic field.

2. Vary the area of the coil.

3. Change the angle between B and A.

Page 14: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.4 Lenz’s Law

The direction of induced EMFs and currents always oppose the change in flux that produced them.

That is, the induced I (and thus induced B) tries to keep the total flux through the loop constant.

Page 15: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Example: Towing the bar to the right produced an induced current that was CCW. What is the direction of the induced magnetic field?

The induced B is out of the page to maintain the flux originally through the loop before the bar started to move to the right (the area of the loop is increasing).

⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗⊗

VL

Page 16: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Example (text problem 20.15): A long straight wire carrying a steady current is in the plane of a circular loop of wire. (a) If the loop of wire is moved closer to the wire, what is the direction of the induced current in the wire loop?

I

Wire loop

There is a magnetic field into the page at the location of the loop. As the loop gets closer to the wire there is an increase in flux. To negate this increase in flux, the induced B-field must point out of the page. This requires a CCW current.

Page 17: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.6 Transformers

tN B

ΔΔΦ

−= 11ε

Wrap an iron core with wire.

Primary coil

Secondary coil

Apply a varying voltage to the primary coil. This causes a changing magnetic flux in the secondary coil.

tN B

ΔΔΦ

−= 22ε

Page 18: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Since the flux through the coils is the same

2

1

2

1

N

N=

εε The “turns ratio” gives

the ratio of the EMFs.

Depending on the turns ratio, a transformer can be used to step-up or step-down a voltage.

2

1

1

2

2

1

N

N

I

I==

εε

The rate that power is supplied to both coils is the same

Page 19: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Example (text problem 20.34): The primary coil of a transformer has 250 turns and the secondary coil has 1000 turns. An AC voltage is sent through the primary. The EMF of the primary is 16.0 V. What is the EMF in the secondary?

V 64.0V 0.16250

10001

1

22

2

1

2

1

=⎟⎠

⎞⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

εε

εε

NN

NN

Page 20: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.7 Eddy Currents

If a conductor is subjected to a changing magnetic flux, a current will flow. (This includes sheets of metal, etc.)

Page 21: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Consider a metal plate that swings through a magnetic field.

An external magnetic field into the page created by a magnet. ⊗⊗⊗

⊗⊗⊗

X

pivot

Page 22: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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As the plate swings through the region of magnetic field, some regions of the plate are entering the B-field (increasing flux), and other regions of the plate are leaving the B-field (decreasing flux). There will be induced currents in the conductor called eddy currents.

The eddy currents dissipate energy (according to I2R); this results in the damping of the amplitude of the metal sheet.

Page 23: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.9 Mutual- and Self-Inductance

Coil 1 Coil 2

A variable current I1 flows in coil 1.

I1 then induces a current in coil 2.

.1212 IN ∝ΦThe flux (Φ21) through coil 2 due to coil 1 is

Page 24: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Writing this as an equality, 1212 MIN =Φ

Where M is the mutual inductance. It depends only on constants and geometrical factors. The unit of inductance is the Henry (1H = 1Vs/A).

The induced EMF in the coils will be:

t

IM

tN

t

IM

tN

ΔΔ

−=Δ

ΔΦ−=

ΔΔ

−=Δ

ΔΦ−=

21211

12122

ε

ε

Page 25: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Self-inductance occurs when a current carrying coil induces an EMF in itself.

.LIN =ΦThe definition of self-inductance (L) is

Page 26: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Example (text problem 20.51): The current in a 0.080 Henry solenoid increases from 20.0 mA to 160.0 mA in 7.0 s. Find the average EMF in the solenoid during that time interval.

( )

V 106.1

s 0.7

mA 20mA 160H 080.0

3−×−=

⎟⎠

⎞⎜⎝

⎛ −−=

Δ

Δ−=

Δ

ΔΦ−=

t

IL

tNε

Page 27: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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An inductor stores energy in its magnetic field according to:

2

2

1LIU =

The energy density in a magnetic field is:

2

02

1BuB μ

=

Page 28: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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§20.10 LR Circuits

An inductor and resistor are connected in series to a battery.

As with an RC circuit, the current in the circuit varies with time.

Page 29: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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The voltage drop across an inductor is given by .t

ILL Δ

Δ=ε

When an inductor is “charging” (the energy stored is increasing) the current in the circuit is:

( )τ/1)( tf eItI −−=

Where τ = L/R is the time constant for the circuit and If = εb/R maximum current in the circuit.

Page 30: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Applying Kirchhoff’s loop rule to the circuit gives the EMF in the inductor as:

τεεε /tbbL eIR −=−=

Page 31: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Plots of εL(t) and I(t) for this LR circuit:

Page 32: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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For a “discharging” inductor, τ/0)( teItI −=

The LR circuit time constant τ plays the same role as in an RC circuit.

where I0 is the current in the inductor when t = 0.

Page 33: 1 My Chapter 20 Lecture Outline. 2 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents

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Summary

•Motional EMF

•Faraday’s Law

•Lenz’s Law

•Transformers

•Eddy Currents

•Inductance and Inductors

•LR Circuits