Chapter 20Chapter 20Induced Voltages and Induced Voltages and

InductanceInductance

Conceptual questions: 1,2,4,6,12,13 Quick Quizzes: 1,3,5Problems: 26, 28, 34, 39,56

Induced emfInduced emf

A current can be produced by a A current can be produced by a changing magnetic fluxchanging magnetic flux

Magnetic Magnetic FluxFlux A loop of wire is in a A loop of wire is in a

uniform magnetic uniform magnetic field Bfield B

The loop has an area The loop has an area AA

The flux is defined asThe flux is defined as ΦΦBB = B = BA = B A cos θA = B A cos θ

θ is the angle between θ is the angle between B and the normal to the B and the normal to the planeplane

SI units of flux are SI units of flux are

T m² = Wb (Weber)T m² = Wb (Weber)

Magnetic Magnetic FluxFlux

The value of the magnetic flux is proportional to The value of the magnetic flux is proportional to the total number of lines passing through the the total number of lines passing through the looploop

When the area is perpendicular to the lines, the When the area is perpendicular to the lines, the maximum maximum number of lines pass through the area number of lines pass through the area and the flux is and the flux is a maximuma maximum

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

Faraday’s LawFaraday’s Law

B

B

t

Nt

For a single loop

For N tightly wound up loops

Since ΦSince ΦBB = B A cos θ, t = B A cos θ, the change in the flux, he change in the flux, ΔΦ, can be produced by a change in B, A or ΔΦ, can be produced by a change in B, A or θ. θ.

Thus, the induced electromotive force can Thus, the induced electromotive force can be produced by changing B, A or θ, or their be produced by changing B, A or θ, or their combinations. combinations.

emf induced by changing magnetic field

The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

QUICK QUIZ 20.1

Motional emf, changing AMotional emf, changing A

A straight conductor of A straight conductor of length length ℓ moves ℓ moves perpendicularly with perpendicularly with constant velocity constant velocity through a uniform fieldthrough a uniform field

The electrons in the The electrons in the conductor experience conductor experience a magnetic forcea magnetic force F = q v BF = q v B

The electrons tend to The electrons tend to move to the lower end move to the lower end of the conductorof the conductor

Motional emf, contMotional emf, cont

The potential difference between the The potential difference between the ends of the conductor can be found ends of the conductor can be found byby ΔV = B ℓ vΔV = B ℓ v

A potential difference is maintained A potential difference is maintained across the conductor as long as there across the conductor as long as there is motion through the fieldis motion through the field If the motion is reversed, the polarity of If the motion is reversed, the polarity of

the potential difference is also reversedthe potential difference is also reversed

Motional emf in a CircuitMotional emf in a Circuit

The induced, motional emf, acts like a battery in the circuitThe induced, motional emf, acts like a battery in the circuit

R

vBIandvB

You wish to move a rectangular loop of wire into a region of uniform magnetic field at a given speed so as to induce an emf in the loop. The plane of the loop must remain perpendicular to the magnetic field lines. In which orientation should you hold the loop while you move it into the region of magnetic field in order to generate the largest emf? (a) With the long dimension of the loop parallel to the velocity vector; (b) With the short dimension of the loop parallel to the velocity vector. (c) Either way—the emf is the same regardless of orientation.

QUICK QUIZ 20.3

Faraday’s Law and Lenz’ Faraday’s Law and Lenz’ LawLaw

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

produces a current whose magnetic field produces a current whose magnetic field opposes the change in magnetic flux through opposes the change in magnetic flux through the loopthe loop

The induced current tends to maintain The induced current tends to maintain the original flux through the circuitthe original flux through the circuit

Lenz’ Law Revisited – Lenz’ Law Revisited – Moving Bar ExampleMoving Bar Example

As the bar moves to As the bar moves to the right, the magnetic the right, the magnetic flux through the circuit flux through the circuit increases with time increases with time because the area of because the area of the loop increasesthe loop increases

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

Lenz’ Law, Bar ExampleLenz’ Law, Bar Example

The bar is moving The bar is moving toward the lefttoward the left

The magnetic flux The magnetic flux through the loop is through the loop is decreasing with timedecreasing with time

The induced current The induced current must be clockwise to must be clockwise to to produce its own to produce its own flux into the pageflux into the page

Lenz’ Law, Moving Magnet Lenz’ Law, Moving Magnet ExampleExample

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

with timewith time The induced current produces a flux to the left, The induced current produces a flux to the left,

so the current is in the direction shown (b)so the current is in the direction shown (b)

Lenz’ Law, Final NoteLenz’ Law, Final Note

When applying Lenz’ Law, there When applying Lenz’ Law, there are are twotwo magnetic fields to consider magnetic fields to consider The external changing magnetic field The external changing magnetic field

that induces the current in the loopthat induces the current in the loop The magnetic field produced by the The magnetic field produced by the

current in the loopcurrent in the loop

1. A circular loop is located in a uniform and constant magnetic field. Describe how an emf can be induced in the loop.

2. Does dropping a magnet down a copper tube produce a current in the tube?

12. A bar magnet is dropped toward a conducting ring lying on a floor. As the magnet falls toward the ring, does it move as a freely falling body?

4. A loop of wire is placed in a uniform magnetic field. For what orientation of the loop is the magnetic flux a maximum? For what orientation is it zero?

6. A bar moves perpendicularly to the magnetic field. Is an external force required to keep it moving with a constant velocity?

Conceptual questions

What is the direction of the current induced in the resistor at the instant the switch is closed?

I

BProblem 20.26.

I

B

Induced current

Induced B

20-28. 20-28. Find the direction of the current Find the direction of the current in R the instant the switch is closed.in R the instant the switch is closed.

Applications of Faraday’s Applications of Faraday’s Law – Electric GuitarLaw – Electric Guitar

A vibrating string induces A vibrating string induces an emf in a coilan emf in a coil

A permanent magnet A permanent magnet inside the coil inside the coil magnetizes a portion of magnetizes a portion of the string nearest the coilthe string nearest the coil

As the string vibrates at As the string vibrates at some frequency, its some frequency, its magnetized segment magnetized segment produces a changing flux produces a changing flux through the pickup coilthrough the pickup coil

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

Applications of Faraday’s Applications of Faraday’s Law – Ground Fault Law – Ground Fault InterruptersInterrupters

Tape Tape RecorderRecorder

A magnetic tape moves A magnetic tape moves past a recording and past a recording and playback headplayback head The tape is a plastic ribbon The tape is a plastic ribbon

coated with iron oxide or coated with iron oxide or chromium oxidechromium oxide

To record, the sound is To record, the sound is converted to an electrical converted to an electrical signal which passes to an signal which passes to an electromagnet that electromagnet that magnetizes the tape in a magnetizes the tape in a particular patternparticular pattern

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

RecordingRecording

Tape playingTape playing

AC AC GeneratorsGenerators As the loop rotates, the As the loop rotates, the

magnetic flux through it magnetic flux through it changes with timechanges with time

This induces an emf and This induces an emf and a current in the external a current in the external circuitcircuit

The ends of the loop are The ends of the loop are connected to slip rings connected to slip rings that rotate with the loopthat rotate with the loop

Connections to the Connections to the external circuit are external circuit are made by stationary made by stationary brushed in contact with brushed in contact with the slip ringsthe slip rings

AC AC GeneratorsGenerators If the loop rotates with If the loop rotates with

a constant angular a constant angular speed, ω, and N turnsspeed, ω, and N turnsε = N B A ω sin ω tε = N B A ω sin ω t

ε = εε = εmaxmax = NBAω when = NBAω when loop is parallel to the loop is parallel to the fieldfield

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

Problem 20.34. Problem 20.34. A coil of area 0.10 mA coil of area 0.10 m22 is rotating at is rotating at 60 rev/s with its axis of rotation perpendicular to a 60 rev/s with its axis of rotation perpendicular to a 0.20-T magnetic field. (a) If there are 1 000 turns on 0.20-T magnetic field. (a) If there are 1 000 turns on the loop, what is the maximum voltage induced in the loop, what is the maximum voltage induced in the coil? the coil? (b) When the maximum induced voltage occurs, (b) When the maximum induced voltage occurs, what is the orientation of the loop with respect to what is the orientation of the loop with respect to

the magnetic field?the magnetic field?

Using, max = NBA m2) (120 rad/s) = 7.5 103 V

Plane of the loop is parallel to the magnetic field

DC GeneratorsDC Generators

Components are Components are essentially the essentially the same as that of an same as that of an ac generatorac generator

The major The major difference is the difference is the contacts to the contacts to the rotating loop are rotating loop are made by a split made by a split ring, or commutatorring, or commutator

DC GeneratorsDC Generators

The output voltage The output voltage always has the always has the same polaritysame polarity

The current is a The current is a pulsing currentpulsing current

Self-inductanceSelf-inductance Self-inductanceSelf-inductance occurs when the changing occurs when the changing

flux through a circuit arises from the circuit flux through a circuit arises from the circuit itselfitself As the current increases, the magnetic flux As the current increases, the magnetic flux

through a loop due to this current also increasesthrough a loop due to this current also increases The increasing flux induces an emf that opposes The increasing flux induces an emf that opposes

the currentthe current As the magnitude of the current increases, the As the magnitude of the current increases, the

rate of increase lessens and the induced emf rate of increase lessens and the induced emf decreasesdecreases

This opposing emf results in a gradual increase This opposing emf results in a gradual increase of the currentof the current

Self-inductance contSelf-inductance cont The self-induced emf must be The self-induced emf must be

proportional to the time rate of proportional to the time rate of change of the currentchange of the current

L is L is inductanceinductance of the device, unit of the device, unit HenryHenry

1 H = 1 (V 1 H = 1 (V · s) / A· s) / A

t

IL

I

NB

INL B

Self inductance of a Self inductance of a solenoidsolenoid

= BA cos , when =90o

= BA

For a solenoid B = o n I = o (N/l)I,

= o A NI/l

[A is the cross-sectional area of the solenoid]

L = N /I = o A N2 /l

L depends only on geometric factors A and l, number

of turns squared, and on o

A solenoid of radius 2.5 cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75 mV.

Problem 20.39

QUICK QUIZ 20.5The switch in the circuit shown in the figure below is closed and the lightbulb glows steadily. The inductor is a simple air-core solenoid. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As the rod is inserted into the solenoid, the brightness of the lightbulb (a) increases, (b) decreases, or (c) remains the same.

Energy Stored in a Energy Stored in a Magnetic FieldMagnetic Field

PEPELL = = ½ L I½ L I22

13. If the current in an inductor is doubled, by what factor does the stored energy change?

Conceptual question

Problem 20-56Problem 20-56 A novel method of storing electrical energy has been A novel method of storing electrical energy has been

proposed. A huge underground superconducting coil, proposed. A huge underground superconducting coil, 1.00 km in diameter, would be fabricated. It would 1.00 km in diameter, would be fabricated. It would carry a maximum current of 50.0 kA through each carry a maximum current of 50.0 kA through each winding of a 150-turn Nbwinding of a 150-turn Nb33Sn solenoid. Sn solenoid.

(a)(a) If the inductance of this huge coil is 50.0 H, what is the If the inductance of this huge coil is 50.0 H, what is the total energy stored? total energy stored?

(b)(b) (b) What is the compressive force per meter length (b) What is the compressive force per meter length acting between two adjacent windings 0.250 m apart? acting between two adjacent windings 0.250 m apart? ((Hint: Hint: Because the radius of the coil is so large, the Because the radius of the coil is so large, the magnetic field created by one winding and acting on magnetic field created by one winding and acting on an adjacent turn can be considered to be that of a an adjacent turn can be considered to be that of a long, straight wire.)long, straight wire.)

Review questionsReview questions

1.1. A heavy permanent magnet is moving A heavy permanent magnet is moving toward a current carrying circular loop of toward a current carrying circular loop of wire. Which is correct?wire. Which is correct?

a.a. The coil will push or pull the magnet just as The coil will push or pull the magnet just as hard as the magnet pulls or pushes the coil.hard as the magnet pulls or pushes the coil.

b.b. The magnet pushes harder on the coil than The magnet pushes harder on the coil than the coil pushes on the magnet because the the coil pushes on the magnet because the magnet is more massive than the coil.magnet is more massive than the coil.

c.c. The magnet will push or pull on the coil, but The magnet will push or pull on the coil, but the coil will not push or pull on the magnet the coil will not push or pull on the magnet at all because the coil is not a magnet.at all because the coil is not a magnet.

2. A conducting bar is sliding at a constant velocity along two conducting horizontal rods. The rods are separated by a distance l and connected across by a resistor R. The entire apparatus is placed inside a magnetic field B directed into the page.

How will the current in the apparatus be generated?

a. sinusoidally b. clockwise

c. counterclockwise d. not enough information

v

3. A conducting coil is rotated at a constant speed in an external magnetic field. Which of the following most likely represents the current generated within the coil as a function of time?

t

i

t t

i

t

t

i

t t

i

t

a

b

c

d