Electromagnetic Induction - FCPS 29 notes.pdfinduced current or emf. Lenz’s law states: The...

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures forUniversity Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 29

Electromagnetic Induction


Goals for Chapter 29

•

To consider Faraday’s Law

•

To consider Lenz’s Law

•

To study motional emf

•

To explore induced electric fields

•

To summarize Maxwell’s equations and see their application to displacement current


How to achieve these goals:Read this PowerPoint.

Read the Chapter.

Do the Homework:

5, 11, 17, 23, 29, 33, 35, 37


Introduction

• The motion of a magnet can induce current in practical ways. If a credit card has a magnet strip on its back, “swiping”

the card can generate

tiny currents that send information to cash registers.

• A coil of wire and magnets set into motion around each other will generate currents in the wire. A source of mechanical energy can drive the rotation and make a waterfall into an electrical power station.


Induced current• Joseph Henry worked in the United States and Michael Faraday

worked in England to discern the details of current generated in wire and permanent magnets in motion

relative to each other.

• See Figure 29.1 below.


Induced Current ObservationsWhen there is no current in the electromagnet, so that ,

the

galvanometer shows no current.When the electromagnet is turned on, there is a momentary current through the meter as increases.When levels off at a steady value, the current drops to zero, no matter how large is.With the coil in a horizontal plane, we squeeze it so as to decrease the cross-sectional area of the coil. The meter detects current only during

the deformation, not before or after. When we increase the

area to return the coil to its original shape, there is current in the opposite direction, but only while the area of the coil is changing.If we rotate the coil a few degrees about a horizontal axis, the

meter

detects current during the rotation, in the same direction as when we decreased the area. When we rotate the coil back, there is a

current in the opposite direction during this rotation.

0=B

B

BB


Induced Current Observations ContinuedIf we jerk the coil out of the magnetic field, there is a current during the motion, in the same direction as when we decreased the area.If we decrease the number of turns in the coil by unwinding one or more turns, there is a current during the unwinding, in the same

direction as when we decreased the area. If we wind more turns onto the coil, there is a current in the opposite direction during the winding.When the magnet is turned off, there is a momentary current in the direction opposite to the current when it was turned on.The faster we carry out any of these changes, the greater the current.If all these experiments are repeated with a coil that has the same shape but different material and different resistance, the current in each case is inversely proportional to the total circuit resistance. This shows that the induced emfs

that are causing the current to not depend

on the material of the coil but only on its shape and the magnetic field.


Knowing the magnetic fluxThe common element in each situation is a changing magnetic flux. The induced emf is proportional to the rate of change of magnetic flux, while the direction depends on the increase or decrease of flux.For an infinitesimal area element in a magnetic field, the magnetic flux through the area is:

The total magnetic flux through a finite area is the integral of the above expression over the area:

If the magnetic field is uniform over a flat area:

φcosBdAdABAdBd B ==⋅=Φ ⊥

φcosBAAdBB =⋅=Φ

∫∫ =⋅=Φ φcosBdAAdBB


CAUTIONIn equations 29.1 and 29.2 we have to be careful to define the direction of the vector area dA or A unambiguously. There are always two directions perpendicular to any given area, and the sign of the magnetic flux through the area depends on which one we choose to be positive. For example, in Figure 29.3 we chose dA to point upward so φ

is less than 90o

and B · dA is positive. We could have

chosen instead to have dA point downward, in which case φ

would have been more than 90o

and B · dA would have

been negative. Either choice is equally good, but once we make a choice we must stick with it.


Knowing the magnetic fluxFaraday’s law of induction states:The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.

Try example 29.1.dt

d BΦ−=ε


Direction of Induced EMFUse equation 29.3 with the following rules to get the direction of the induced emf.Define a positive direction for the vector area.From the directions of and the magnetic field, determine the

sign

of the magnetic flux and its rate of change.Determine the sign of the induced emf or current. If the flux is increasing, so dΦB

/dt

is positive, then the induced emf or current is negative; if the flux is decreasing, dΦB

/dt

is negative and the induced emf or current is positive.Finally, determine the direction of the induced emf or current using your right hand. Curl the fingers of your right hand around thevector , with your right thumb in the direction of . If the

induced

emf or current in the circuit if positive, it is in the same direction as your curled fingers; if the induced emf or current is negative, it is in the opposite direction.

A

AA


CAUTIONSince magnetic flux plays a central role in Faraday’s law, it’s tempting to think that flux

is the cause of

induced emf and that an induced emf will appear in a circuit whenever there is a magnetic field in the region bordered by the circuit. But equation 29.3 shows that only a change

in the flux through a circuit, not flux

itself, can induce an emf in a circuit. If the flux through a circuit has a constant value, whether positive, negative, or zero, there is no induced emf.


A Coil with N number of turnsIf we have a coil with N identical turns, and the flux varies at the same rate through each turn, the total rate of change through all the turns is:

dtdN BΦ

−=ε


Faraday’s Law•

Follow Problem-Solving Strategy on 1112.

•

Refer to Example 29.2.

•

Figure 29.6 illustrates Example 29.2.


The search coil, generator example I•

Follow Conceptual Example 29.3.

•

Consider Example 29.4 and Figures 29.8 and 29.7.


Generator example II

•

The example considers a DC generator and back emf

in a motor.


Generator example III

•

Follow Example 29.6.

•



Work and power in a slide wire generator

•


•



29.1 & 29.2 Summary and HomeworkFaraday’s law states that the induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. This relationship is valid whether the flux change is caused by a changing magnetic field, motion of the loop, or both.

On page 1138:

5 & 11


Lenz’s LawLenz’s law is a convenient alternative to determine the direction of an induced current or emf. Lenz’s law states:The direction of any induction effect is such as to oppose the cause of the effect.The induced current or emf produces a magnetic field opposite to

the

magnetic field that is changing.Try examples 29.8 and 29.9.


If the cause of the current is opposed . . .If the induced current or emf opposes the cause, how can the flux change at all?The direction is opposite to the cause, but the magnitude depends on the resistance of the circuit. The larger the resistance, the less induced current.Lenz’s law states that an induced current or emf always tends to oppose or cancel out the change that caused it. Lenz’s law can be derived from Faraday’s law and is often easier to use.

On page 1039: 17


A conducting rod moving in a uniform magnetic field If an isolated rod moves in a uniform magnetic field, the magnetic forces will eventually be canceled out by the resulting electric field from the separated charges.When the rod is connected to a U shape wire, an induced emf and current

will flow in the direction indicated, with magnitude: ε

= vBL. The current has an associated magnetic field that points out of the page on the inside of the loop.


EMF for any closed loopWe can generalize an expression for the emf induced in any shape loop, if the magnetic field does not vary with time. For an element dl of a conductor, the contribution to the emf is the magnitude dl multiplied by the component of v x B parallel to dl:

For any closed loop the total emf is:

This expression is equivalent to our previous version of emf. Either can be used for moving conductors, but for stationary conductors in a changing magnetic field, equation 29.3 must be used.

( ) ldBvd ⋅×=ε( )∫ ⋅×= ldBvε


The Faraday Disk dynamo

Follow Example 29.11.Figure 29.15 illustrates Example 29.11.


Induced electric fieldsThe magnetic field in the center of the solenoid B

= μo

nI. Neglecting the small field outside the coils and take A to be parallel to B, the flux is ΦB

= BA = μo

nIA.

When I changes with time, and emf is induced in the galvanometer loop:

If the resistance in the loop is R, the loop current I’

= ε/R.

What force causes the charges to move around the loop?

The conductor isn’t moving in a magnetic field, and in fact isn’t even in a magnetic field.

dtdInA

dtd

oB με −=

Φ−=


Induced electric fieldsWe are forced to conclude that there is an induced electric field in the conductor, caused by the changing magnetic flux.

This electric field is not conservative. The line integral around a closed path is not zero, in fact the line integral representing the work done by the induced electric field per unit charge is equal to the induced emf:

This is only true if the path of integration is stationary.

dtdldE

ldE

BΦ−=⋅

=⋅

∫

∫ ε


Induced electric fieldsLets say the galvanometer loop has a radius

r. The line integral becomes the

electric field magnetude

multiplied by the circumferance, 2πrE

and equation

29.10 gives:

The direction of the electric field must be opposite to that of the change in magnetic flux, to oppose the effects of the change in flux.

This electric field is a nonelectrostatic field, it is not conservative and the

concept of potential has no meaning.

The force on any charge is still qE.

dtd

rE BΦ=

π21


Induced electric fields II •

Refer to Figure 29.18 below.

•



29.5 Summary and HomeworkWhen an emf is induced by a changing magnetic flux through a stationary conductor, there is an induced electric field E of nonelectrostatic origin. This field is nonconservative and cannot be associated with a potential.

1140: 23 & 29


Displacement current and Maxwell’s equationsConsider Ampere’s law. What is the result for the line integral for the plane surface? What about the bulging surface? It cannot be both.Consider equation 29.12 on 1129. Taking the time derivative of both sides yields 29.13.The displacement current is a virtual (fictitious) current used to describe the region between the plates.


Generalized Ampere’s law

( )enclDCo iildB +=⋅∫ μThis equation is valid no matter which surface we use. For the plane surface, iD

is zero, for the bulging surface iC

is zero. It is also true that iC

= iD

.We have pulled this concept out of thin air, like Maxwell did in

1865,

but we will see that it saves Ampere’s law for these situations.


Displacement current and Maxwell’s equationsNot only does this fake current save Ampere’s law, it solidifies Kirchhoff's junction rule when dealing with capacitors, because we can apply a current through

a capacitor.

Lets use this idea of a displacement current to calculate the magnetic field between the plates of a capacitor.Consider figure 29.22. We apply Ampere’s law to a circle of radius r

passing through points a

and b. The total current enclosed by the circle is jD

times its area, or (iD

/πR2)(πr2) = iD

r2/R2

Remember that iD

= iC

, so Ampere’s law becomes:

Co

Co

iRrB

iRrrBldB

2

2

2

2

2

πμ

μπ

=

==⋅∫

Experimental varification

of this result comfirms

the role of displacement current as

the source of magnetic field.


Maxwell’s Equation of ElectromagnetismMaxwell did not discover all of these equations single handedly,

but he did put them together and recognize there significance.

Gauss’s law for E

Gauss’s law for B

Ampere’s law

Faraday’s law

( )

dtdldE

iildB

AdB

QAdE

B

enclDCo

o

encl

Φ−=⋅

+=⋅

=⋅

=⋅

∫

∫∫

∫

μ

ε

0


Electrostatic and nonelectrostatic fieldsThe total electric field at a point in space can be the superposition of an electrostatic field,Ec , caused by a distribution of charges at rest and a magnetically induced, nonelectrostatic

field, En .

E = Ec + EnThe electrostatic part is always conservitive, so the line integral is equal to zero. This does not contribute to the integral in Faraday’s law, so we can take E to be the total electric field. Similarly, the nonconservative

part does not contribute to the

integral in Gauss’s law, because this is not caused by static charges, so E dot dA

is always zero.

In all the Maxwell equations, E is the total electric field.


29.7 Summary and HomeworkA time-varying electric field generates a displacement current iD

which acts as a source of magnetic field in exactly the same way as conduction current. The relationships between electric and magnetic fields and their sources can be stated compactly in four equations, called Maxwell’s equations. Together they form a complete basis for the relationship of E and B fields to their sources.

1140; 33, 35, 37


Chapter Review Follows


A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field is directed into the plane of the figure. If the magnetic field magnitude is constant,

Q29.1

A. the induced emf

is clockwise.

B. the induced emf

is counterclockwise.

C. the induced emf

is zero.

D. The answer depends on the strength of the field.


A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field is directed into the plane of the figure. If the magnetic field magnitude is constant,

A29.1

A. the induced emf

is clockwise.

B. the induced emf


C. the induced emf

is zero.



A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field is directed into the plane of the figure. If the magnetic field magnitude is decreasing,

Q29.2

A. the induced emf

is clockwise.

B. the induced emf


C. the induced emf

is zero.



A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field is directed into the plane of the figure. If the magnetic field magnitude is decreasing,

A29.2

A. the induced emf

is clockwise.

B. the induced emf


C. the induced emf

is zero.



A circular loop of wire is placed next to a long straight wire. The current I

in the long straight

wire is increasing. What current does this induce in the circular loop?

Q29.3

A. a clockwise current

B. a counterclockwise current

C. zero current

D. not enough information given to decide


A circular loop of wire is placed next to a long straight wire. The current I

in the long straight

wire is increasing. What current does this induce in the circular loop?

A29.3

A. a clockwise current

B. a counterclockwise current

C. zero current

D. not enough information given to decide


Q29.4

A. flows downward through resistor R

and is proportional to B.

B. flows upward through resistor R


C. flows downward through resistor R

and is proportional to B2.

D. flows upward through resistor R


E. none of the above

A flexible loop of wire lies in a uniform magnetic field of magnitude B

directed into the

plane of the picture. The loop is pulled as shown, reducing its area. The induced current


A29.4

A. flows downward through resistor R


B. flows upward through resistor R


C. flows downward through resistor R


D. flows upward through resistor R


E. none of the above

A flexible loop of wire lies in a uniform magnetic field of magnitude B

directed into the

plane of the picture. The loop is pulled as shown, reducing its area. The induced current


The rectangular loop of wire is being moved to the right at constant velocity. A constant current I

flows

in the long straight wire in the direction shown. The current induced in the loop is

Q29.5

A. clockwise and proportional to I.

B. counterclockwise and proportional to I.

C. clockwise and proportional to I2.

D. counterclockwise and proportional to I2.

E. zero.



flows

in the long straight wire in the direction shown. The current induced in the loop is

A29.5





E. zero.


The loop of wire is being moved to the right at constant velocity. A constant current I

flows in the long

straight wire in the direction shown. The current induced in the loop is

Q29.6





E. zero.


The loop of wire is being moved to the right at constant velocity. A constant current I

flows in the long

straight wire in the direction shown. The current induced in the loop is

A29.6





E. zero.


Q29.7

A. L: to the left; R: to the left

B. L: to the left; R: to the right

C. L: to the right; R: to the left

D. L: to the right; R: to the right


flows in

the long wire in the direction shown. What are the directions of the magnetic forces on the left-hand (L) and right-hand (R) sides of the loop?



flows in

the long wire in the direction shown. What are the directions of the magnetic forces on the left-hand (L) and right-hand (R) sides of the loop?

A29.7

A. L: to the left; R: to the left

B. L: to the left; R: to the right

C. L: to the right; R: to the left

D. L: to the right; R: to the right


The drawing shows the uniform magnetic field inside a long, straight solenoid. The field is directed into the plane of the drawing, and is increasing.

What is the direction of the electric force on a positive point charge placed

at point a?

Q29.8

A. to the left

B. to the right

C. straight up

D. straight down

E. misleading question —

the electric force at this point is zero




at point a?

A29.8

A. to the left

B. to the right

C. straight up

D. straight down






at point b?

Q29.9

A. to the left

B. to the right

C. straight up

D. straight down






at point b?

A29.9

A. to the left

B. to the right

C. straight up

D. straight down






at point c

(at the center of the solenoid)?

Q29.10

A. to the left

B. to the right

C. straight up

D. straight down






at point c

(at the center of the solenoid)?

A29.10

A. to the left

B. to the right

C. straight up

D. straight down



Electromagnetic Induction - FCPS 29 notes.pdfinduced current or emf. Lenz’s law states: The...

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