electromagnetic induction - cabrillo.edumwatson/2b/spiral chapters/5. eminduction 2b.pdf ·...

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Electromagnetic Induction

Concepts and Principles

Creating Electrical Energy

When electric charges move, their electric fields vary. In the previous two chapters we

considered moving electric charges as the source of magnetic fields, but we could just as

easily have considered the variation in the electric field as the source of the magnetic field.

This leads to an interesting question: If varying electric fields can create magnetic fields, can

varying magnetic fields create electric fields? The answer is yes, and this process, termed

electromagnetic induction, is at the heart of almost all electrical power generation worldwide.

In addition to incredible technological importance, electromagnetic induction hints at a deep

inter-relationship and symmetry between electric and magnetic fields that will explored more

fully in later chapters.

Imagine a region of space with a magnetic field. Surrounding a portion of this region is a

hypothetical closed path. (Often, a real loop of wire will be the closed path of interest, but

induction occurs whether or not a real wire loop is present.)

B

First, let me define magnetic flux. The mathematical definition of magnetic flux is:

cosBA

where

B is the magnetic field within the area bounded by the closed path,

A is the area bounded by the closed path,

and is the angle between the magnetic field and the direction perpendicular to the

area. thus, if the field is perpendicular to the area, this angle is zero and the magnetic

flux is a maximum.

Conceptually, the magnetic flux is often visualized as the amount of field that “passes

through” the area enclosed by the path.

2

Second, let me define emf. (Actually, let me apologize. Emf used to stand for electromotive

force, even though it is not a force. In light of this misleading name, emf now, officially,

stands for emf. It’s not short for anything. I’m not making this up.) Emf is the name for the

electrical energy per unit charge created by changing magnetic flux. In general, any process

that generates electrical energy “creates” emf. In addition to changing magnetic flux,

chemical batteries and some solar cells create emf.

Mathematically, emf is defined as the product of the electric field that is created “around” the

closed path and the length of the path, although this definition is really all that useful in our

context. All you need to know is that emf is the energy created per unit charge. The units of

emf, joules per coulomb, are given the name volts (V). In crude language, an emf is a

“voltage”.

Let’s put this all together. The central relationship describing electromagnetic induction,

termed Faraday’s Law, claims that:

t

where

is the emf induced (the voltage created) in the closed path,

is the magnetic flux that passes through the closed path,

and the negative sign indicates that the emf’s direction in the closed path is to oppose

the change in magnetic flux. (If the closed path is a real loop of wire, the emf will

drive an induced current whose direction is such that the magnetic field produced by

this induced current is opposite to the change in magnetic field that produces the

induced current. Crystal clear?)

3

. . . . . . .. . .


Analysis Tools

Abrupt Change in Flux

A 100-turn, 1.0 cm radius coil of wire is 50 cm

from a very long straight wire carrying 2.0 A. If

the current flowing through the wire is turned

off, and drops to zero in about 0.1 s, what is the

average induced emf in the loop? What is the

direction of the induced current?

i

When the current is turned off, the magnetic flux through the small loop will decrease to zero.

By Faraday’s Law, this change in flux will create an emf in the loop. The first step toward

finding the emf involves finding the magnetic flux through the loop of interest. For this

example, the direction of the magnetic field is perpendicular to the loop’s area, so the angle in

the flux equation is 0°:

BA

BA

cos

Moreover, since the loop is far from the wire, and the loop’s diameter is small, the magnetic

field from the wire is approximately constant over the area of the loop, and the initial flux is

given by:

))(2

( 20 RNr

i

BA

initial

The N indicates the number of turns in the coil, each turn increasing the effective area of the

coil, r is the distance between the wire and the coil, and R is the radius of the coil.

28

26

1052.2

))01.0(100)()50.0(2

)2)(1026.1((

Tmx

x

initial

initial

4

This flux is reduced to zero over 0.1s. Therefore:

Vx

x

t

t

if

7

8

1052.2

1.0

)1052.20(

)(

This is the average emf induced while the current decreases to zero. Although this is a small

value, it is easily measured by precision equipment.

Since initially the magnetic field through the loop was in the +z-direction, and was then

removed, the induced emf will drive an induced current counterclockwise around the loop in

an attempt to counter the reduction in flux. In other words, the induced current will try to

maintain a constant magnetic flux through the loop.

5

Continuous Change in Flux

A 1000-turn secondary coil of radius 2.0 cm is

concentric with a 400-turn primary coil of

radius 20 cm carrying AC current at 60 Hz with

peak current 2.0 A. What is the average emf in

the secondary coil as the AC current changes

from 2A clockwise to 2A counterclockwise?

current source

i

Again, the first step toward finding the emf involves finding the magnetic flux through the

secondary loop. For this example, the direction of the magnetic field is again perpendicular to

the loop’s area, so flux reduces:

BA

Moreover, since the magnetic field is approximately constant over the relatively small area of

the secondary loop, the flux is given by:

))(2

(20

ss

p

pRN

R

iN

BA

Plugging in values yields:

23

26

1017.3

))02.0(1000)()20.0(2

)2)(1026.1)(400((

Tmx

x

When the current flows clockwise, this flux is out of the page. When it flows

counterclockwise, the flux is into the page. Calling out of the page positive, and realizing that

the current changes direction in 120th

of a second, results in:

V

xx

t

t

if

76.0

)120

1(

)1017.3)1017.3((

)(

33

Thus, as the primary current changes direction from clockwise to counterclockwise, the

secondary coil has an average induced “voltage” of 0.76V.

6

. . . . . . .. . .


Activities

7

For each of the actions below, indicate the direction of the induced current in the loop. The magnet and

loop center are in the same plane.

a.

v

N S

b.

v

S N

c.

v = 0

N S

d.

v

N S

v

e.

N S

v

8

For each of the actions below, indicate the direction of the induced current in the circular loop. The magnet

and loop center are in the same plane.

a.

N S

v = 0

b.

N S

v

c.

N

S

v

d.

N S

e.

N S

9

For each of the actions below, indicate the direction of the induced current in the left loop. The loops are

parallel and their centers are coplanar.

a.

close switch

b.

open switch

c.

v

d.

open switch

e.

v

10

For each of the actions below, indicate the direction of the induced current in the left loop. The loops are

perpendicular and their centers are coplanar.

a.

close switch

b.

open switch

c.

close switch

d.

close switch

e.

v

11

For each of the actions below, indicate the direction of the induced current in the secondary loop (the loop

without the battery). The loops are either parallel or perpendicular and their centers are coplanar.

a.

close switch

b.

v c.

close switch

d.

close switch

e.

open switch

12

For each of the actions below, indicate the direction of the induced current through the resistor.

a.

close switch b.

open switch c.

close switch d.

v

13

The device below consists of a primary coil (containing an adjustable current source) and a secondary coil.

For the given primary current (ip) vs. time graph, construct the corresponding secondary current (is) vs. time

graph. Current to the right is considered positive.

current source

i

a.

t

is

t

ip

b.

t

is

t

ip

c.

t

is

t

ip

14

The device below consists of a primary coil (containing an adjustable current source) and a secondary coil.

For the given secondary current (is) vs. time graph, construct the corresponding primary current (ip) vs. time

graph. Current to the right is considered positive.

current source

i

a.

t

is

t

ip

b.

t

is

t

ip

c.

t

is

t

ip

15

The device below consists of a primary loop (containing an adjustable current source) and a secondary

loop. For the given primary current (ip) vs. time graph, construct the corresponding secondary magnetic

flux (s) vs. time graph. Current to the right and flux out of the page are considered positive.

current source

i

a.

t

s

t

ip

b.

t

s

t

ip

c.

t

s

t

ip

16

Six identical, circular loops of wire are rotating at the same angular frequency in the uniform magnetic field

shown below. The rotation axes of A, B and C are perpendicular to the plane of the page and the rotation

axes of D, E and F are parallel to the plane of the page.

A

C

B

F E

D

B

a. At the instant shown, rank these scenarios on the basis of the flux through the loop.

Largest 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ Smallest

_____ The ranking cannot be determined based on the information provided.

b. At the instant shown, rank these scenarios on the basis of emf induced in the loop.



Explain the reason for your rankings:

17

Six new prototypes for electrical generators are tested in the laboratory. Each generator consists of N

circular loops of wire with radius R rotated at frequency f in the same uniform magnetic field.

B

N R (cm) f (Hz) A 2000 1.0 60

B 1000 1.0 120

C 1000 2.0 60

D 4000 0.5 60

E 3000 1.0 60

F 500 2.0 180

Rank these generators on the basis of the maximum emf produced.



Explain the reason for your ranking:

18

Six identical vertical metal bars start at the position shown and move at constant velocity through identical

magnetic fields. The bars make electrical contact with and move along frictionless metal rods attached to

identical light bulbs.

10 cm/s

A

10 cm/s

B

5 cm/s

C

20 cm/s

D

10 cm/s

E

0 cm/s

F

a. At the instant shown, rank these scenarios on the basis of the flux through the loop.



b. At the instant shown, rank these scenarios on the basis of the magnitude of the current in the light bulb.



Explain the reason for your rankings:

19

Six identical horizontal metal bars start at rest and fall vertically through identical magnitude magnetic

fields (except A). The bars make electrical contact with and move along frictionless metal rods attached to

identical light bulbs.

A

B

B

C

B

D

B

E

B

F

B

Rank these scenarios on the time it takes the horizontal bar to reach the bottom of the rods.




20

Six identical horizontal metal bars start at rest and fall vertically through identical magnitude magnetic

fields (except A). The bars make electrical contact with and move along frictionless metal rods attached to

identical batteries.

A

B

B

C

B

D

B

E

B

F

B

Rank these scenarios on the time it takes the horizontal bar to reach the bottom of the rods.




21

The 100 turn, 10 cm square loop at right is located in a 10 mT magnetic

field. The magnitude of the magnetic field is increased over a period of

0.50 s. The bulb's resistance is 30 .

B

Qualitative Analysis On the diagram above, indicate the direction of the induced current in the loop.

Mathematical Analysis1

To what value should the field be increased in order to supply the 10 mA necessary to light the bulb?

22

The 5.0 cm square loop at right consists of 100 turns of wire and is located

in a 0.10 T magnetic field. The polarity of the magnetic field is reversed.

The bulb's resistance is 30 .

B



Over what time interval must the reversal take place in order to supply the 10 mA necessary to light the

bulb?

23

A 200-turn loop of radius 1.0 cm is 50 cm from a very long straight wire

carrying 250 mA. The current is reduced linearly to zero in a time of 3.0 s.

i



What is the induced emf in the loop?

24

A 2000-turn secondary coil of radius 2.0 cm is concentric with a 200-turn

primary coil of radius 30 cm carrying current 300 mA. The primary

current is reduced to zero.

current source

i



Over what time interval must the primary current be reduced to zero to induce 6.0 V in the secondary coil?

25

A coil of radius 2.0 cm is concentric with a 400-turn/cm solenoid carrying

current 1800 mA. The current is linearly reduced to zero over a time

interval of 0.05 s.

current source

i

iinduced


When current flows in the solenoid, a magnetic flux exists that points to the right through the loop. If the current is reduced to zero, this flux disappears and a current will be induced in the loop to create a “replacement” flux. This current must flow down the front of the loop to create this flux.

Mathematical Analysis How many turns are needed in the secondary coil to develop an emf of 12 V?

First, I’ll determine the flux through the loop when the current flows in the solenoid. Since the field from the solenoid is uniform inside of the loop, I don’t need to integrate to find the flux.

))(( 2

0 RNni

BA

Since this flux drops abruptly to zero, the average emf induced is simply the change in flux divided by the time over which the flux disappears.

5260

)02.0()8.1)(40000)(1026.1(

)05.0)(12(

)(

)))((0(

)(

26

2

0

2

0

2

0

N

xN

Rni

tN

t

RniN

t

RNni

t

if

You’ll need about 5300 turns in the secondary coil to develop an emf of 12 V.

26

A 200-turn loop of radius 1.0 cm is 50 cm from a very long straight wire

carrying AC current at 60 Hz with peak current 250 mA. What is the

average induced emf in the loop as the current changes direction?

i


27

An n-turn secondary coil of radius r is concentric with a N-turn primary

coil of radius R. If the current source supplies AC current with maximum

current imax, what is the average induced emf in the secondary coil as the

current changes direction?

current source

i


28

A 1000-turn coil of radius 2.0 cm is concentric with a 400-turn/cm

solenoid carrying AC current at 60 Hz with peak current 2.0 A. What is

the average induced emf in the coil as the current changes direction?

current source

i


29

A car alternator consists of a 250-turn coil of radius 5.0 cm. The coil

rotates at 1000 rpm in a uniform magnetic field. In order to generate an

average induced emf of 12 V as the coil flips over, what value of magnetic

field is necessary?

B


30

A metal bar of length 5.0 cm moves at a constant speed 2.0 m/s through a

magnetic field of 2.0 T. The bar makes electrical contact with and moves

along frictionless metal rods attached to a light bulb of resistance 30 .

v

B

Qualitative Analysis On the diagram above, indicate the direction of the induced current in the bulb.


What is the current through the light bulb?

31

A metal bar of length 25 cm moves at constant speed 40 m/s through a

magnetic field 50 mT.

v

B

Qualitative Analysis On the diagram above, indicate which end of the bar is positive.


Determine the emf induced between the two ends of the bar.

32

Selected Answers

1 B = 0.16 T

2 t = 0.166 s

3 = 2.1 x 10

-9 V

4 t = 52.6 x 10

-6 s

5 average = 1.51 V

6 average = 0.127 V

7 average = 30.4 V

8 B = 0.092 T

9 i = 6.67 mA

10 = 0.50 V

electromagnetic induction - cabrillo.edumwatson/2b/spiral chapters/5. eminduction 2b.pdf ·...

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