Ch 22 Electromagnetic Induction

Chapter 22Electromagnetic Induction

AP Learning ObjectivesElectromagnetism Electromagnetic induction (including Faraday’s law and Lenz’s law)

Students should understand the concept of magnetic flux, so they can: Calculate the flux of a uniform magnetic field through a loop of

arbitrary orientation. Students should understand Faraday’s law and Lenz’s law, so they

can: Recognize situations in which changing flux through a loop will

cause an induced emf or current in the loop. Calculate the magnitude and direction of the induced emf and

current in a loop of wire or a conducting bar when the magnitude of a related quantity such as magnetic field or area of the loop is changing at a constant rate.

Table Of Contents

1. Induced Emf and Induced Current

2. Motional Emf

3. Magnetic Flux

4. Faraday's Law of Electromagnetic Induction

5. Lenz's Law

6. Applications of Electromagnetic Induction to The

Reproduction of Sound (Not AP)

7. The Electric Generator (Not AP)

8. Mutual Inductance and Self-inductance (Not AP)

9. Transformers (Not AP)

Chapter 22:Electromagnetic InductionSection 1:

Induced Emf and Induced Current

There are a number of ways a magnetic field can be used to generate an electric current.

It is the changing field that produces the current.

Induced Current

The current in the coil is called the induced current because it is broughtabout by a changing magnetic field.

Since a source emf is always needed to produce a current, the coil behavesas if it were a source of emf. This emf is known as the induced emf.

Induced EMF

An emf can be induced by changing thearea of a coil in a constant magnetic field

In each example, both an emf and a current are induced because the coil is part of a complete circuit. If the circuit were open, therewould be no induced current, but there would be an induced emf.

The phenomena of producing an induced emf with the aid of amagnetic field is called electromagnetic induction.

Ways to Induce an EMF

Chapter 22:Electromagnetic InductionSection 2:Motional Emf

Each charged particle within the conductor is moving and experiences a magnetic force

qvBF

The separated charges on theends of the conductor give riseto an induced emf, called amotional emf.

THE EMF INDUCED IN A MOVING CONDUCTOR

Motional EMF

vBL

Motional emf when v, B, and L are mutually perpendicular

Motional EMF

Example 1 Operating a Light Bulb with Motional EmfSuppose the rod is moving with a speed of 5.0m/sperpendicular to a 0.80-T magnetic field. The rod has a length of 1.6 m and a negligible electricalresistance. The rails also have a negligibleelectrical resistance. The light bulb has a resistance of 96 ohms. Find (a) the emf producedby the rod and (b) the current induced in thecircuit.

(a) vBL

RI

(b)

V 4.6 m 6.1T 80.0sm0.5

96

V 4.6A 067.0

In order to keep the rod moving at constant velocity, the force the hand exerts on the rod must balance the magnetic force onthe current:

ILBFF hand

MOTIONAL EMF AND ELECTRICAL ENERGY

The direction of the force in this figure would violate theprinciple of conservation of energy.

MOTIONAL EMF AND ELECTRICAL ENERGY

In essence, the magnetic field creates a greater amount of resistance.

Conceptual Example 3 Conservation of EnergyA conducting rod is free to slide down between two vertical copper tracks. There is no kinetic friction between the rod and the tracks. Because the only force on the rod is its weight, it falls with an acceleration equal to the acceleration of gravity. Suppose that a resistor was connected between the tops of the tracks. (a) Does the rod now fall with the acceleration of gravity? (b) How does the principle of conservation of energy apply?

a) no, it accelerates at a rate less than g. As it accelerates, a greater magnetic force is generated, causing a greater amount of resistance. It will eventually reach a critical velocity and fall at a constant v

b) As the potential energy is converted, somebecomes kinetic energy, some becomes heatas the electrons move through the resistor.

22.2.1. A coil of wire that forms a complete loop is moving with a constant speed v toward a very long current carrying wire, only a portion of which is shown. What affect, if any, does the current carrying wire have on the coil or wire?

a) Since the magnetic field increases as the coil approaches the wire, a current is induced in the coil.

b) The rectangle will be distorted as it is pulled in the direction of the current in the wire.

c) Close to the wire, a magnetic force acts on the loop that accelerates the loop away from the wire.

d) Since the magnetic field around the wire is not changing, there is no effect on the coil.

e) Since the coil and the wire are not touching, there is no effect.

22.2.2. A rectangular loop of wire is attached to a metal rod using rigid, electrically insulating rods so that the distance between the loop and metal rod is constant as the metal rod is rotated uniformly as shown. The metal rod carries a current in the direction indicated. Which of the following statements concerning an induced current in the rectangular loop as a result of the current in the metal rod is true?

a) The induced current in the loop is clockwise around the loop.

b) The induced current in the loop is counterclockwise around the loop.

c) The induced current in the loop alternates between clockwise and counterclockwise around the loop.

d) There is no induced current in the loop.

22.2.3. A rigid, conductive loop is falling through a uniform magnetic field that is perpendicular to the plane of the loop. Initially, the loop is completely within the field, but then it falls into a region where no magnetic field is present. Which one of the following quantities varies during the fall?

a) the magnetic field penetrating the loop

b) the area of the loop penetrated by the magnetic field

c) the magnetic flux

d) the current in the loop

e) all of the above

Chapter 22:Electromagnetic InductionSection 3:Magnetic Flux

Magnetic Flux Flux is a mathematical term for how much of something is

flowing through a surface. It’s actually more complicated than that, but that’s the

simple gist of it. The magnetic flux is how much of the magnetic field is

“flowing” through the area of the circuit/coil. It is equal to the dot product of magnetic field and normal

line of the area. Can be thought of as a total magnetic “effect” on a coil of

wire of a given area. SI unit is: weber (Wb)(tesla/meters2)

MOTIONAL EMF AND MAGNETIC FLUX

o

o

o

o

o

o

o

o

tt

BABAB

tt

AAB

tt

LxxLBL

tt

xxvBL

magnetic flux ABm ttt

m

o

o

A little complication…

You will almost always see the equation from the previous slide written with a minus sign.

The minus sign is introduced for the following reason: The direction of the current induced in the circuit is such

that the magnetic force on the rod to oppose its motion, thereby tending to slow down the rod

The minus sign ensures that the polarity of the induced emf sends the induced current in the proper direction so as to give rise to this opposing magnetic force.

This issue of the polarity of the induced emf will be discussed further in Section 22.5.

tm

t

m

GENERAL EXPRESSION FOR MAGNETIC FLUX

ABm

cosBAm

GRAPHICAL INTERPRETATION OF MAGNETIC FLUX

The magnetic flux is proportionalto the number of field lines that passthrough a surface.

22.3.1. A circular ring is rotated clockwise at a constant rate for an extended period of time using the apparatus shown. Which of the graphs below correctly shows the magnetic flux through the ring as a function of time? Note: At time t = 0 s, the plane of the ring is perpendicular to the direction of the magnetic field.

22.3.2. A balloon has an initial radius of 0.075 m. A circle is painted on the balloon using silver metal paint. When the paint dries, the circle is a very good electrical conductor. With the balloon oriented such that a 1.5-T magnetic field is oriented perpendicular to the plane of the circle, air is blown into the balloon so that it expands uniformly. The silver circle expands to a radius 0.125 m in 1.5 s. Determine the induced emf for this silver circle during this period of expansion.

a) 0.021 V

b) 0.031 V

c) 0.047 V

d) 0.058 V

e) 0.075 V


Faraday's Law of Electromagnetic Induction

FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION

The average emf induced in a coil of N loops is

tN

ttN

o

o

SI Unit of Induced Emf: volt (V)

Example 5 The Emf Induced by a Changing Magnetic Field

A coil of wire consists of 20 turns each of which has an area of 0.0015 m2.A magnetic field is perpendicular to the surface. Initially, the magnitude of the magnetic field is 0.050 T and 0.10s later, it has increased to 0.060 T.Find the average emf induced in the coil during this time.

tN

V 100.3 3

s 0.10

T 050.0T 060.00cosm 0015.020 2

t

BBNA o cos

t

ABBAN o

coscos

Conceptual Example 7 An Induction Stove

Two pots of water are placed on an induction stove at the same time.The stove itself is cool to the touch. The water in the ferromagneticmetal pot is boiling while that in the glass pot is not. How can sucha cool stove boil water, and why isn’t the water in the glass pot boiling?

Steel is made up mainly of iron. The stove induces a current in the iron.Since Iron has a higher resistance, the pot dissipates the energy as heatdue to the resistance. The glass is an insulator, so very little heat is created.Aluminum pots are not ferromagnetic, so there will be no induced current.

22.4.1. A permanent magnet is moved toward a 320-turn solenoid such that the magnetic field inside the solenoid increases from zero to 0.50 T in 0.75 s. The radius of the solenoid is 0.035 m. The ends of the solenoid are connected in series with a light bulb. What emf is induced during this time interval?

a) 47 V

b) 24 V

c) 5.7 V

d) 1.6 V

e) 0.051 V

Chapter 22:Electromagnetic InductionSection 5:Lenz's Law

Lenz’s Law

The induced emf resulting from a changing magnetic flux has

a polarity that leads to an induced current

The current will flow in a direction so as to oppose the

change in flux.

Use in combination with hand rule to predict current

direction.

Reasoning Strategy

1. Determine whether the magnetic flux that penetrates the coil

is increasing or decreasing.

2. Find what the direction of the induced magnetic field must be

so that it can oppose the change influx by adding or

subtracting from the original field.

3. Use RHR-2 to determine the direction of the induced current.

Conceptual Example 8 The Emf Produced by a Moving MagnetA permanent magnet is approaching a loop of wire. The external circuit consistsof a resistance. Find the direction of theinduced current and the polarity of the induced emf.

Since Field lines are more dense nearthe magnet, the magnetic flux is increasingas the magnet approaches the wire loop.

To counter this, the induced field must be in the opposite direction.

Point fingers in direction of induced field,thumb points in the direction of the current

Conceptual Example 9 The Emf Producedby a Moving Copper Ring.There is a constant magnetic field directed into the page in the shaded region. The fieldis zero outside the shaded region. A copperring slides through the region. For each of the five positions, determine whetheran induced current exists and, if so, find itsdirection.

a) none

b) Since field is into the screen and is increasing,the induced field must be opposite, out of the screen.therefore current is counter clockwise. c) none

d) Since the field is into the screen and decreasing, the induced field must also be into the screen to counterthe decrease. Therefore the current is clockwise.e) none

Motional EMF

B: magnetic field (T) L: length of bar moving through field v: speed of bar moving through field.

Bar must be “cutting through” field lines. It cannot be moving parallel to the field.

This formula is easily derivable from Faraday’s Law of Induction

BLv

t

t

BA

Assume is 0

t

BLx

BLv

22.5.1. You are looking in the direction of the magnetic field in a specific region as shown. A square, metal loop is moving from left to right at a constant speed. Which one of the graphs below shows the behavior of the current, if any, in the loop as time passes?

22.5.2. Consider the situation shown in the drawing. A conducting loop is connected to a resistor. The resistor and loop are at rest in a magnetic field that is directed toward you. Within a short period of time the magnetic field is reduced to one half of its initial value. Which one of the following statements concerning an induced current, if any, in the loop is true?

a) During the time the magnetic field is decreasing, a current is induced that is directed counterclockwise around the loop.

b) During the time the magnetic field is decreasing, a current is induced that is directed clockwise around the loop.

c) No current is induced in the loop at any time.

d) A current is induced that is directed clockwise around the loop, which also continues after the magnetic field attains a constant value.

e) A current is induced that is directed counterclockwise around the loop, which also continues after the magnetic field attains a constant value.

22.5.3. A permanent magnet is moved toward and away from a solenoid with a frequency of 60 Hz. The ends of the solenoid are connected in series with a light bulb. Which one of the following statements concerning an induced current, if any, in the loop is true?

a) An alternating current will be induced in the circuit that is clockwise when the magnet moves to the right and counterclockwise when the magnet moves to the left.

b) An alternating current will be induced in the circuit that is clockwise when the magnet moves to the left and counterclockwise when the magnet moves to the right.

c) An direct current will be induced in the circuit that is clockwise as the magnet oscillates to the left and to the right.

d) An direct current will be induced in the circuit that is counterclockwise as the magnet oscillates to the left and to the right.

e) No current will be induced in the circuit by using this method.

22.5.4. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is entering the magnetic field, what will be the direction of any induced current present in the loop?

a) clockwise

b) counterclockwise

c) No current is induced.

22.5.5. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is exiting the magnetic field, what will be the direction of any induced current present in the loop?

a) clockwise

b) counterclockwise


22.5.6. A rigid, circular metal loop begins at rest in a uniform magnetic field directed away from you as shown. The loop is then pulled through the field toward the right, but does not exit the field. What is the direction of any induced current within the loop?

a) clockwise

b) counterclockwise


Chapter 22:Electromagnetic InductionSection 6:Applications of Electromagnetic Induction to The Reproduction of Sound

Applications of Electromagnetic Induction to the Reproduction of Sound

Chapter 22:Electromagnetic InductionSection 7:The Electric Generator

HOW A GENERATOR PRODUCES AND EMF

sinBLvBLv E

2

Wrv

ttNAB o sinsin

f 2

Emf induced in a rotating planar coil

to sin

THE ELECTRICAL ENERGY DELIVERED BY A GENERATORAND THE COUNTERTORQUE

When the generator is delivering current, there is a magnetic forceacting on the coils.

The magnetic force gives rise to acountertorque that opposes the rotational motion.

THE BACK EMF GENERATED BY AN ELECTRIC MOTOR

When a motor is operating, two sources of emf are present: (1) theapplied emf V that provides current to drive the motor, and (2) the emf induced by the generator-like action of the rotating coil.

R

VI

Consistent with Lenz’s law, the induced emf acts to oppose theapplied emf and is called back emf.

Example 12 Operating a Motor

The coil of an ac motor has a resistance of 4.1 ohms. The motor is plugged into an outlet where the voltage is 120.0 volts (rms), and the coildevelops a back emf of 118.0 volts (rms) when rotating at normal speed. The motor is turning a wheel. Find (a) the current when themotor first starts up and (b) the current when the motor is operating atnormal speed.

A 294.1

V 0V 120

R

VI

(a)

A 49.04.1

V 0.118V 120

R

VI

(b)

22.7.1. The drawing shows one type of electric generator. An aluminum disc is rotated with a portion near the edge in between the poles of a powerful C-shaped magnet. As the disc rotates, two metal brushes contact it, one near the outer radius, and one near the center as shown. A voltmeter measures the potential difference between the two brushes. How does this generator produce this potential difference?

a) A current appears in the disc to oppose the magnetic field of the magnet.

b) The rubbing of the brushes releases electrons from the aluminum and collect near the outer radius of the disc.

c) Because the disc is rotating, free electrons in the plate experience a magnetic force that causes them to move toward the outer radius, thus creating a potential difference.

d) The magnet exerts a force on free electrons in the aluminum that causes them to move toward the center of the disc, thus creating a potential difference.

22.7.2. The drawing illustrates an electric generator in which there are two independent loops that are oriented perpendicular to one another. As the generator rotates, the loops in turn make contact with the circuit shown for nearly one fourth of a revolution and the light bulb is illuminated. Which one of the graphs shown below best represents the voltage generated by this generator as a function of time?

Chapter 22:Electromagnetic InductionSection 8:Mutual Inductance and Self-inductance

MUTUAL INDUCTANCE

The changing current in the primary coil creates a changingmagnetic flux through the secondary coil, which leads to aninduced emf in the secondary coil.

The effect in calledmutual induction.

tN

Emf due to mutual induction

t

IM

ps

mutual inductance

SI Unit of mutual inductance: (Henry) H 1As1V

SELF INDUCTANCE

The effect in which a changing current in a circuit induces and emfin the same circuit is referred to as self induction.

SI Unit of self inductance: (Henry) H 1As1V

t

IL

self inductance

THE ENERGY STORED IN AN INDUCTOR

Energy stored in an inductor

Energy density

221Energy LI

2

2

1densityEnergy B

o

22.8.2. A potential difference exists between the ends of an inductor. At the instant shown, the upper end is at a higher potential than the lower end. Which one of the following statements accurately describes the current in the inductor?

a) The current has a constant value and is directed from the top of the inductor toward the bottom of the inductor.

b) The current is increasing and is directed from the top of the inductor toward the bottom of the inductor.

c) The current is decreasing and is directed from the top of the inductor toward the bottom of the inductor.

d) The current is increasing and is directed from the bottom of the inductor toward the top of the inductor.

e) The current may be increasing and directed from the top toward the bottom or decreasing and directed from the bottom toward the top of the inductor.

22.8.3. A circuit contains a battery, a switch, an inductor, and a resistor connected in series. Initially, the switch is open. In which one of the following intervals does the energy stored in the inductor have the largest value?

a) before the switch is closed

b) immediately after the switch is closed when the current in the circuit is increasing

c) a long time after the switch is closed

22.8.4. Two solenoids, A and B, have the same length and cross-sectional area. Solenoid B has three times the number of turns per unit length. What is the ratio of the self-inductance of solenoid B to that of solenoid A?

a) 1/3

b)

c) 3

d) 6

e) 9

3

22.8.5. In each of the three cases shown, a time-varying current is flowing through the larger coil that produces a magnetic field. Rank the mutual inductance for the three cases shown from smallest to largest.

a) MA, MB, MC

b) MC, MB, MA

c) MC, MA, MB

d) MB, MC, MA

e) MB, MA, MC


Transformers

A transformer is a device for increasing or decreasing an ac voltage.

tN

sst

N

pp

p

s

p

s

N

N

Transformers

p

s

p

s

N

N

V

VTransformer

equation

Transformers

s

p

s

p

p

s

N

N

V

V

I

I

A transformer that steps up the voltage simultaneously stepsdown the current, and a transformer that steps down the voltagesteps up the current.

Transformers

Transformers

22.8.1. Circuit A contains a battery, a switch, and a resistor connected in series. Circuit B contains a battery, a switch, an inductor, and a resistor connected in series. Initially, the switch is closed in both circuits. How does the behavior of the current in circuit B compare with that in circuit A as both switches are opened at the same time?

a) The current in both circuits decreases at the same rate because inductors do not affect the current in a circuit.

b) The current in circuit B decreases more slowly than that for circuit A since the inductor acts to maintain the current in the circuit.

c) The current in circuit B decreases more quickly than that for circuit A since the inductor increases the current in the circuit as its stored energy is released.

d) The behavior of the current in the circuit depends on the inductance of the inductor. If the inductance is small, the current will decrease rapidly; and if the inductance is large, the current will increase for a short time before decreasing.

22.9.1. A transformer has two coils. Coil A has 75 turns and coil B has 100 turns. In the first instance, coil A is used as the primary coil, which is connected to an ac current source. As a result, the peak voltage of the secondary coil, B, has a value VP. Then, coil B is used as the primary coil and coil A becomes the secondary coil. The ac current source is then adjusted to deliver the same current to coil B as it did to coil A in the first instance. For this case, how does the peak voltage of the secondary coil, A, compare with VP.

a) The peak voltage of A is VP.

b) The peak voltage of A is 1.25VP.

c) The peak voltage of A is 0.75VP.

d) The peak voltage of A is 1.33VP.

e) The peak voltage of A is 0.80VP.

22.9.2. The ac adapter for a laptop computer contains a transformer. The input of the adapter is the 120 volts from the ac wall outlet. The output from the transformer is 20 volts. What is the turns ratio of the transformer?

a) 0.17

b) 6

c) 100

d) This cannot be determined without knowing how many turns one of the coils in the transformer has.

22.9.3. Transformer A has a primary coil with 400 turns and a secondary coil with 200 turns. Transformer B has a primary coil with 400 turns and a secondary coil with 800 turns. The same current and voltage are delivered to the primary coil of both transformers. The secondary coils are connected to identical circuits. How does the power delivered by secondary coil A compare to the power delivered by secondary coil B?

a) Secondary coil A delivers one fourth the power delivered by secondary coil B

b) Secondary coil A delivers one half the power delivered by secondary coil B

c) Secondary coil A delivers the same power as delivered by secondary coil B

d) Secondary coil A delivers twice the power delivered by secondary coil B

e) Secondary coil A delivers four times the power delivered by secondary coil B

Ch 22 Electromagnetic Induction

Education

Transcript of Ch 22 Electromagnetic Induction