Physics For Scientists And Engineers 6E - part 246

SECTION 31.4 • Induced emf and Electric Fields

981

31.4 Induced emf and Electric Fields

We  have  seen  that  a  changing  magnetic  flux  induces  an  emf  and  a  current  in  a
conducting  loop.  In  our  study  of  electricity,  we  related  a  current  to  an  electric  field
that  applies  electric  forces  on  charged  particles.  In  the  same  way,  we  can  relate  an
induced current in a conducting loop to an electric field by claiming that

an electric

field is created in the conductor as a result of the changing magnetic flux.

We also noted in our study of electricity that the existence of an electric field is

independent of the presence of any test charges. This suggests that even in the absence
of a conducting loop, a changing magnetic field would still generate an electric field in
empty space.

This induced electric field is nonconservative, unlike the electrostatic field produced

by stationary charges. We can illustrate this point by considering a conducting loop of
radius r situated in a uniform magnetic field that is perpendicular to the plane of the
loop, as in Figure 31.19. If the magnetic field changes with time, then, according to
Faraday’s law (Eq. 31.1), an emf

" %

/dt is induced in the loop. The induction

of a current in the loop implies the presence of an induced electric field

E, which must

be tangent to the loop because this is the direction in which the charges in the wire
move in response to the electric force. The work done by the electric field in moving a
test charge q once around the loop is equal to q

. Because the electric force acting on

the charge is q

E, the work done by the electric field in moving the charge once around

the loop is qE(2,r), where 2,r is the circumference of the loop. These two expressions
for the work done must be equal; therefore, we see that

Using this result, along with Equation 31.1 and the fact that !

BA " ,r

B for a

circular loop, we find that the induced electric field can be expressed as

(31.8)

If the time variation of the magnetic field is specified, we can easily calculate the in-
duced electric field from Equation 31.8.

The emf for any closed path can be expressed as the line integral of

E & ds over that

path:

&E & ds. In more general cases, E may not be constant, and the path may not

be a circle. Hence, Faraday’s law of induction,

" %

/dt, can be written in the

general form

(31.9)

The induced electric field E in Equation 31.9 is a nonconservative field that

is generated by a changing magnetic field. The field E that satisfies Equation 31.9

E&ds " %

E " %

2,r

" %

E "

2,r

" qE(2,r)

current is induced, and the induced emf is B!v. As soon as the
left side leaves the field, the emf decreases to zero.

maintain  this  motion  is  plotted  in  Figure  31.18d.  Before  the
loop enters the field, no magnetic force acts on it; hence, the
applied force must be zero if v is constant. When the right side
of  the  loop  enters  the  field,  the  applied  force  necessary  to
maintain constant speed must be equal in magnitude and op-
posite in direction to the magnetic force exerted on that side.
When  the  loop  is  entirely  in  the  field,  the  flux  through  the
loop is not changing with time. Hence, the net emf induced in

the loop is zero, and the current also is zero. Therefore, no ex-
ternal force is needed to maintain the motion. Finally, as the
right side leaves the field, the applied force must be equal in
magnitude and opposite in direction to the magnetic force
acting on the left side of the loop.

From this analysis, we conclude that power is supplied

only when the loop is either entering or leaving the field.
Furthermore, this example shows that the motional emf
induced in the loop can be zero even when there is motion
through the field! A motional emf is induced only when the
magnetic flux through the loop changes in time.

Figure 31.19 A conducting loop

of radius r in a uniform magnetic

field perpendicular to the plane of

the loop. If B changes in time, an

electric field is induced in a

direction tangent to the

circumference of the loop.

Faraday’s law in general form

▲

PITFALL PREVENTION

31.2 Induced Electric

Fields

The  changing  magnetic  field
does  not need  to  be  in  existence
at  the  location  of  the  induced
electric  field.  In  Figure  31.19,
even a loop outside the region of
magnetic  field  will  experience
an induced  electric  field.  For
another

example,

consider

Figure 31.8. The light bulbs glow
(if the switch is open) even
though the wires are outside the
region of the magnetic field.

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CHAPTE R 31 • Faraday’s Law

31.5 Generators and Motors

Electric generators take in energy by work and transfer it out by electrical transmission.
To understand how they operate, let us consider the

alternating current (AC) gener-

ator. In its simplest form, it consists of a loop of wire rotated by some external means
in a magnetic field (Fig. 31.21a).

cannot possibly be an electrostatic field because if the field were electrostatic, and
hence conservative, the line integral of

E & ds over a closed loop would be zero

(Section 25.1); this would be in contradiction to Equation 31.9.

Quick Quiz 31.9

In a region of space, the magnetic field increases at a

constant rate. This changing magnetic field induces an electric field that (a) increases in
time (b) is conservative (c) is in the direction of the magnetic field (d) has a constant
magnitude.

Example 31.8 Electric Field Induced by a Changing Magnetic Field in a Solenoid

A long solenoid of radius R has n turns of wire per unit
length and carries a time-varying current that varies
sinusoidally as I " I

max

cos *t, where I

max

is the maximum

current and * is the angular frequency of the alternating
current source (Fig. 31.20).

(A)

Determine the magnitude of the induced electric field

outside the solenoid at a distance r ) R from its long central
axis.

Solution First let us consider an external point and take
the path for our line integral to be a circle of radius r
centered on the solenoid, as illustrated in Figure 31.20. By
symmetry we see that the magnitude of

E is constant on this

path and that

E is tangent to it. The magnetic flux through

the area enclosed by this path is BA " B,R

; hence, Equa-

tion 31.9 gives

(1)

The magnetic field inside a long solenoid is given by

Equation 30.17, B " /

nI. When we substitute the expres-

sion I " I

max

cos *t into this equation for B and then substi-

tute the result into Equation (1), we find that

(2)

(for r ) R )

E "

max

sin *t

max

sin *t

E(2,r) " %,R

max

(cos *t)

E&ds " E(2,r) " %,R

E&ds " %

(B,R

) " %,R

Hence, the amplitude of the electric field outside the sole-
noid falls off as 1/r and varies sinusoidally with time.

(B)

What is the magnitude of the induced electric field

inside the solenoid, a distance r from its axis?

Solution For an interior point (r 0 R), the flux through an
integration loop is given by B,r

. Using the same proce-

dure as in part (A), we find that

(3)

(for r 0 R )

This shows that the amplitude of the electric field induced
inside the solenoid by the changing magnetic flux through
the solenoid increases linearly with r and varies sinusoidally
with time.

E "

max

r sin *t

E(2,r) " %,r

" ,r

max

* sin *t

Path of

integration

max

cos

Figure 31.20 (Example 31.8) A long solenoid carrying a time-

varying current given by I " I

max

cos

t. An electric field is

induced both inside and outside the solenoid.

In commercial power plants, the energy required to rotate the loop can be derived

from a variety of sources. For example, in a hydroelectric plant, falling water directed
against the blades of a turbine produces the rotary motion; in a coal-fired plant, the
energy released by burning coal is used to convert water to steam, and this steam is
directed against the turbine blades. As a loop rotates in a magnetic field, the magnetic
flux through the area enclosed by the loop changes with time; this induces an emf and
a current in the loop according to Faraday’s law. The ends of the loop are connected
to slip rings that rotate with the loop. Connections from these slip rings, which act as
output terminals of the generator, to the external circuit are made by stationary
brushes in contact with the slip rings.

Suppose that, instead of a single turn, the loop has N turns (a more practical

situation), all of the same area A, and rotates in a magnetic field with a constant
angular speed *. If # is the angle between the magnetic field and the normal to the
plane of the loop, as in Figure 31.22, then the magnetic flux through the loop at
any time t is

BA cos # " BA cos *t

where we have used the relationship # " *t between angular position and angular
speed (see Eq. 10.3). (We have set the clock so that t " 0 when # " 0.) Hence, the
induced emf in the coil is

(31.10)

This result shows that the emf varies sinusoidally with time, as plotted in Figure 31.21b.
From Equation 31.10 we see that the maximum emf has the value

max

NAB*

(31.11)

which occurs when *t " 90° or 270°. In other words,

max

when the magnetic

field is in the plane of the coil and the time rate of change of flux is a maximum. Fur-
thermore, the emf is zero when *

t " 0 or 180°, that is, when

B is perpendicular to the

plane of the coil and the time rate of change of flux is zero.

The frequency for commercial generators in the United States and Canada is 60 Hz,

whereas in some European countries it is 50 Hz. (Recall that * " 2,

f, where f is the

frequency in hertz.)

" %

" %NAB

(cos *t) " NAB* sin *t

SECTION 31.5 • Generators and Motors

983

(a)

Slip rings

Brushes

External

circuit

Loop

(b)

max

External

rotator

Active Figure 31.21 (a) Schematic diagram of an AC generator. An emf is induced in

a loop that rotates in a magnetic field. (b) The alternating emf induced in the loop

plotted as a function of time.

At the Active Figures link

at http://www.pse6.com, you

can adjust the speed of

rotation and the strength of the

field to see the effects on the

emf generated.

Normal

Figure 31.22 A loop enclosing an

area A and containing N turns,

rotating with constant angular

speed

in a magnetic field. The

emf induced in the loop varies

sinusoidally in time.

984

CHAPTE R 31 • Faraday’s Law

The

direct current (DC) generator is illustrated in Figure 31.23a. Such genera-

tors are used, for instance, in older cars to charge the storage batteries. The compo-
nents are essentially the same as those of the AC generator except that the contacts to
the rotating loop are made using a split ring called a commutator.

In this configuration, the output voltage always has the same polarity and pulsates

with time, as shown in Figure 31.23b. We can understand the reason for this by noting
that the contacts to the split ring reverse their roles every half cycle. At the same time,
the polarity of the induced emf reverses; hence, the polarity of the split ring (which is
the same as the polarity of the output voltage) remains the same.

A pulsating DC current is not suitable for most applications. To obtain a more

steady DC current, commercial DC generators use many coils and commutators distrib-
uted so that the sinusoidal pulses from the various coils are out of phase. When these
pulses are superimposed, the DC output is almost free of fluctuations.

Motors are devices into which energy is transferred by electrical transmission

while energy is transferred out by work. Essentially, a motor is a generator operating

Quick Quiz 31.10

In an AC generator, a coil with N turns of wire spins in a

magnetic  field.  Of  the  following  choices,  which  will  not cause  an  increase  in  the  emf
generated  in  the  coil?  (a)  replacing  the  coil  wire  with  one  of  lower  resistance
(b)  spinning  the  coil  faster  (c)  increasing  the  magnetic  field  (d)  increasing  the
number of turns of wire on the coil.

Example 31.9 emf Induced in a Generator

An AC generator consists of 8 turns of wire, each of area
A " 0.090 0 m

, and the total resistance of the wire is 12.0 (.

The loop rotates in a 0.500-T magnetic field at a constant
frequency of 60.0 Hz.

(A)

Find the maximum induced emf.

Solution First, note that * " 2,f " 2,(60.0 Hz) " 377 s

Thus, Equation 31.11 gives

max

NAB* " 8(0.090 0 m

)(0.500 T)(377 s

)

136 V

(B)

What is the maximum induced current when the output

terminals are connected to a low-resistance conductor?

Solution From Equation 27.8 and the results to part (A),
we have

11.3 A

max

136 V

12.0 (

Commutator

(a)

Brush

(b)

Armature

Active Figure 31.23 (a) Schematic diagram of a DC generator. (b) The magnitude of