CHAPTER 20 ELECTROMAGNET INDUCTION FARADAY’S LAW€¦ · 1 CHAPTER 20 ELECTROMAGNET INDUCTION...

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1 CHAPTER 20 ELECTROMAGNET INDUCTION FARADAY’S LAW Units Induced EMF Faraday’s Law of Induction; Lenz’s Law EMF Induced in a Moving Conductor Changing Magnetic Flux Produces an Electric Field Electric Generators Back EMF and Counter Torque; Eddy Currents Transformers and Transmission of Power Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI Inductance Energy Stored in a Magnetic Field LR Circuit AC Circuits and Reactance LRC Series AC Circuit Resonance in AC Circuits Michael Faraday 1791 – 1867 Great experimental scientist Invented electric motor, generator and transformers Discovered electromagnetic induction Discovered laws of electrolysis Induced EMF Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

Transcript of CHAPTER 20 ELECTROMAGNET INDUCTION FARADAY’S LAW€¦ · 1 CHAPTER 20 ELECTROMAGNET INDUCTION...

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CHAPTER 20 ELECTROMAGNET INDUCTION

FARADAY’S LAW Units

• Induced EMF • Faraday’s Law of Induction; Lenz’s Law • EMF Induced in a Moving Conductor • Changing Magnetic Flux Produces an Electric Field • Electric Generators • Back EMF and Counter Torque; Eddy Currents • Transformers and Transmission of Power • Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI • Inductance • Energy Stored in a Magnetic Field • LR Circuit • AC Circuits and Reactance • LRC Series AC Circuit • Resonance in AC Circuits

Michael Faraday

• 1791 – 1867 • Great experimental scientist • Invented electric motor, generator and transformers • Discovered electromagnetic induction • Discovered laws of electrolysis

Induced EMF

Almost 200 years ago, Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

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Faraday concluded that although a constant magnetic field produces no current in a conductor, a changing magnetic field can produce an electric current. Such a current is called an induced current. This is also called electromagnetic induction.

Faraday’s Experiment – Set Up

• A current can be produced by a changing magnetic field

– First shown in an experiment by Michael Faraday

• A primary coil is connected to a battery • A secondary coil is connected to an

ammeter

• The purpose of the secondary circuit is to detect current that might be produced by the magnetic field

• When the switch is closed, the ammeter reads a current and then returns to zero • When the switch is opened, the ammeter reads a current in the opposite direction and then

returns to zero • When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Conclusions

• An electrical current is produced by a changing magnetic field • The secondary circuit acts as if a source of emf were connected to it for a short time • It is customary to say that an induced emf is produced in the secondary circuit by the

changing magnetic field

Induced EMF

He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.

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Magnetic Flux

• The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field

• Magnetic flux is defined in a manner similar to that of electrical flux

• Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop

• You are given a loop of wire • The wire is in a uniform magnetic field • The loop has an area A • The flux is defined as

– ΦB = B

A = B A cos θ

• θ is the angle between B and the normal to the plane

• The flux can be visualized with respect to magnetic field lines – The value of the magnetic flux is proportional to the total number of lines passing

through the loop • When the area is perpendicular to the lines, the maximum number of lines pass through

the area and the flux is a maximum • When the area is parallel to the lines, no lines pass through the area and the flux is 0

Example 1: Calculate the magnetic flux A conducting circular loop of radius 0.250 m is placed in the xy-plane in a uniform magnetic field or 0.60T that points in the positive z-direction, the same direction as the normal to the plane. a) Calculate the magnetic flux through the loop.

Suppose the loop is rotated clockwise around the x-axis so the normal direction now points at a

45o angle with respect to the z-axis. Recalculate the magnetic flux through the loop.

c) What is the change in flux due to the rotation of the loop?

2 2 2(0.250 ) 0.196A r m m

2cos (0.196 )(0.360T)cos(0 )o

B AB m 20.0706T 0.0706Wbm

2cos (0.196 )(0.360T)cos(45.0 )o

B AB m 20.0499T 0.0499Wbm

2cos (0.196 )(0.360T)cos(45.0 )o

B AB m 20.0499T m 0.0499Wb

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Faraday’s Law of Induction; Lenz’s Law

• When the field is perpendicular to the plane of the loop, as in a, θ = 0 and ΦB =

ΦB, max = BA

• When the field is parallel to the plane of the loop, as in b, θ = 90° and ΦB = 0

– The flux can be negative, for example if θ = 180°

• SI units of flux are T. m² = Wb (Weber)

Electromagnetic Induction – Results of the Experiment

• A current is set up in the circuit as long as there is relative motion between the magnet and the loop

– The same experimental results are found whether the loop moves or the magnet moves

• The current is called an induced current because is it produced by an induced emf

Faraday’s Law and Electromagnetic Induction

• The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit

• If a circuit contains N tightly wound loops and the flux changes by ΔΦB during a time

interval Δt, the average emf induced is given by Faraday’s Law: Faraday’s law of induction: [1 loop]

[N loops]

Faraday’s Law and Lenz’ Law

• The change in the flux, ΔΦB, can be produced by a change in B, A or θ

– Since ΦB = B A cos θ

• The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law

– The current caused by the induced emf travels in the direction that creates a magnetic field with flux opposing the change in the original flux through the circuit

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Magnetic flux will change if the area of the loop changes:

Magnetic flux will change if the angle between the loop and the field changes:

There are many devices that operate on the basis of Faraday’s law.

An electric guitar pickup:

Applications of Faraday’s Law – Electric Guitar

• A vibrating string induces an emf in a coil • A permanent magnet inside the coil magnetizes a

portion of the string nearest the coil • As the string vibrates at some frequency, its magnetized

segment produces a changing flux through the pickup coil

• The changing flux produces an induced emf that is fed to an amplifier

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Applications of Faraday’s Law – Apnea Monitor

• The coil of wire attached to the chest carries an alternating current

• An induced emf produced by the varying field passes through a pick up coil

• When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert

Faraday’s Law of Induction

Tape recorder:

Faraday’s Law of Induction; Lenz’s Law

Problem Solving: Lenz’s Law 1. Determine whether the magnetic flux is increasing, decreasing, or unchanged. 2. The magnetic field due to the induced current points in the opposite direction to the

original field if the flux is increasing; in the same direction if it is decreasing; and is zero if the flux is not changing.

3. Use the right-hand rule to determine the direction of the current. 4. Remember that the external field and the field due to the induced current are different.

Practice with Lenz’s law. Initially, the magnetic field pointing out of the page passes through the loop. If you pull the loop out of the field, magnetic flux through the loop decreases; so the induced current will be in a direction to maintain the decreasing flux through the loop: the current will be counterclockwise to produce a magnetic field outward.

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The external field is into the page. The coil area gets smaller, so the flux will decrease; hence the induced current will be clockwise, producing its own field into the page to make up for the flux decrease.

Initially, the magnetic field pointing out of the page passes through the loop. If you pull the loop out of the field, magnetic flux through the loop decreases; so the induced current will be in a direction to maintain the decreasing flux through the loop: the current will be counterclockwise to produce a magnetic field outward.

Magnetic field lines point out from the N pole of a magnet, so as the magnet moves toward the loop, the magnet’s field points into the page and is getting stronger. The current in the loop will be induced in the counterclockwise direction in order to produce

a field B out of the page.

Initially there is no flux through the loop. When you start to rotate the loop, the external field through the loop begins increasing to the left. To counteract this change in flux, the loop will have current induced in a counterclockwise direction so as to produce its own field to the right. Example 2: Pulling a coil from a magnetic field A square coil of wire with side 5.00l cm contains 100 loops and is positioned perpendicular to a uniform 0.600T magnetic field. It is quickly pulled from the field at constant

speed (moving perpendicular to B ) to a region where B drops abruptly to zero. At t = 0, the right edge of the coil is at the edge of the field. It takes 0.100 s for the whole coil to reach the field-free region. The coil’s total resistance is100 . Find a) The rate of change in flux through the coil.

2 2 2 3 2(5.00 10 ) 2.5 10A l x m x m 3 2 3(0.600T)(2.50 10 ) 1.50 10 WbB BA x m x

320 (1.50 10 Wb)

1.50 10 Wb/s0.100

B xx

t s

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b) the emf and current induced.

2(100)( 1.50 10 Wb/s) 1.50BN x Vt

21.501.50 10 15.0

100

VI x A mA

R

By Lenz’s law, the current must be clockwise to produce more B into the page and thus oppose the decreasing flux into the page. c) How much energy is dissipated in the coil?

2 2 2 3(1.50 10 A) (100 )(0.100 ) 2.25 10 JE Pt I Rt x s x

d) What was the average force required?

3

2

2.25 100.0450

5.00 10

W x JF N

d x m

EMF Induced in a Moving Conductor

This image shows another way the magnetic flux can change:

The induced current is in a direction that tends to slow the moving bar – it will take an external force to keep it moving.

Assume that a uniform magnetic field B is perpendicular to the area bounded by the U-shaped conductor and the movable rod resting on it. If the rod is made to move at a speed v, it travels a distance x v t in a time t . Therefore, the area of the loop increases by an amount

A l x lv t in a time t . By Faraday’s law there is an induced emf whose magnitude is given by

B B A Blv tBlv

t t t

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Example 3: Does a moving airplane develop a large emf? An airplane travels 1000km/h in a region where the Earth’s

magnetic field is 55.0 10x T , and is nearly vertical. What is the potential difference induced between the wing tips that are 70m apart?

5(5.0 10 T)(70 )(280 / ) 1.0Blv x m m s V

Example 4: Electromagnetic blood-flow measurement. The rate of blood flow in our body’s vessels can be measured since blood contains charged ions. If a blood vessel is 2.0 mm in diameter, the magnetic field is 0.080T and the measured emf is 0.10mV. What is the flow velocity of the blood?

4

3

(1.0 10 )0.63 /

(0.08T)(2.0 10 )

x Vv m s

Bl x m

Mechanical Work and Electrical Energy

If the rod is to move at a constant speed, an external force must be exerted on it. This force should have equal magnitude and opposite direction to the magnetic force:

• As the bar moves to the right, the magnetic flux

through the circuit increases with time because the area of the loop increases

• The induced current must be in a direction such that it opposes the change in the external magnetic flux

Lenz’s Law This conducting rod completes the circuit. As it falls, the magnetic flux decreases, and a current is induced.

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The force due to the induced current is upward, slowing the fall.

Mechanical Work and Electrical Energy The mechanical power delivered by the external force is: Compare this to the electrical power in the light bulb: Therefore, mechanical power has been converted directly into electrical power.

Lenz’ Law – Moving Magnet Example

• A bar magnet is moved to the right toward a stationary loop of wire (a) – As the magnet moves, the magnetic flux increases with time

• The induced current produces a flux to the left, so the current is in the direction shown (b) • When applying Lenz’ Law, there are two magnetic fields to consider

– The external changing magnetic field that induces the current in the loop – The magnetic field produced by the current in the loop

Application – Tape Recorder

• A magnetic tape moves past a recording and playback

head – The tape is a plastic ribbon coated with iron oxide

or chromium oxide • To record, the sound is converted to an electrical signal

which passes to an electromagnet that magnetizes the tape in a particular pattern

• To playback, the magnetized pattern is converted back into an induced current driving a speaker

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Generators

• Alternating Current (AC) generator – Converts mechanical energy to electrical

energy – Consists of a wire loop rotated by some

external means – There are a variety of sources that can supply

the energy to rotate the loop • These may include falling water, heat

by burning coal to produce steam • Basic operation of the generator

– As the loop rotates, the magnetic flux through it changes with time

– This induces an emf and a current in the external circuit – The ends of the loop are connected to slip rings that rotate with the loop – Connections to the external circuit are made by stationary brushes in contact with

the slip rings

Electric Generators A dc generator is similar, except that it has a split-ring commutator instead of slip rings.

AC Generators

• The emf generated by the rotating loop can be found by ε =2 B ℓ v=2 B ℓ sin θ

• If the loop rotates with a constant angular speed, ω, and N turns

ε = N B A ω sin ω t • ε = εmax when loop is parallel to the field

• ε = 0 when the loop is perpendicular to the field

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Joseph Henry

• 1797 – 1878 • First director of the Smithsonian • First president of the Academy of Natural Science • First to produce an electric current with a magnetic field • Improved the design of the electro-magnetic and

constructed a motor • Discovered self-inductance

Transformers and Transmission of Power

A transformer consists of two coils, either interwoven or linked by an iron core. A changing emf in one induces an emf in the other. The ratio of the emfs is equal to the ratio of the number of turns in each coil: This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:

Transformers A transformer is used to change voltage in an alternating current from one value to another. By applying Faraday’s law of induction to both coils, we find: Here, p stands for the primary coil and s the secondary.

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Example 5: Portable radio transformer. A transformer for home use of a portable radio reduces 120-V ac to 9.0-V ac. (Such a device also contains diodes to change the 9.0-V ac to dc, to be like its 9.0-V battery.) The secondary coil contains 30 turns and the radio draws 400mA. a) Calculate the number of turns in the primary coil.

(30)(120 )

400 turns9.0 )

PP S

S

V VN N

V V

b) Calculate the current in the primary.

30

(0.40 ) 0.030400

SP S

P

NI I A A

N

c) Calculate the power transformed.

(0.40 )(9.0 ) 3.6S SP I V A V W

The power in the primary coil, P = (0.030A)(120V) =3.6W, is the same as the power in the secondary coil.

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CHAPTER 21

ELECTROMAGNETIC INDUCTION

CONCEPTS

1. Units of magnetic flux consists of: T-m2, weber, and volt-second.

2. Magnetic flux density has the same units as magnetic field.

3. A wire moves across a magnetic field. The emf produced in the wire depends on: the

field’s magnetic flux, the length of the wire, and the orientation of the wire with respect to the magnetic field vector.

4. Faraday’s law of induction states that the emf induced in a loop of wire is proportional to the time variation of the magnetic flux.

5. Doubling the number of loops of wire in a coil produces a change on the induced emf that is twice as much.

6. A coil lies flat on a table in a region where the magnetic field vector points straight up. The magnetic field vanishes suddenly. When viewed from above, the induced current flows counterclockwise.

7. A circular loop of wire is rotated at constant angular speed about an axis whose direction can be varied. In a region where a uniform magnetic field points straight down, the orientation of the loop’s axis of rotation must be vertical if the induced emf is to be zero.

8. A bar magnet is positioned inside a coil. In the following, “work” refers to any work done as a consequence of the fact that the bar is magnetic. If the bar is suddenly pulled out of the coil, more work will be done if the switch is closed.

9. A generator coil rotates through 60 revolutions each second. The frequency of the emf is 60 Hz.

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10. A lightbulb is plugged into a household outlet, and one of the wires leading to it is wound into a coil. Now a slug of iron is slid into the coil. This will cause the lamp to dim because a back emf will be produced by the coil, and this will reduce the current flowing in the coil and the lamp.

11. A transformer is a device that operates only on AC.

12. The coil of a generator rotates through one complete turn in a uniform magnetic field 50

times every second. When this generator is connected to an external load, the current through the load reverses directions 100 times every second.