Physics For Scientists And Engineers 6E - part 243

SECTION 31.1 • Faraday’s Law of Induction

969

Finally, the galvanometer reads zero when there is either a steady current or no
current in the primary circuit. The key to understanding what happens in this experi-
ment is to note first that when the switch is closed, the current in the primary circuit
produces a magnetic field that penetrates the secondary circuit. Furthermore, when

Active Figure 31.1 (a) When a magnet is moved toward a loop of wire connected to a

sensitive ammeter, the ammeter deflects as shown, indicating that a current is induced

in the loop. (b) When the magnet is held stationary, there is no induced current in the

loop, even when the magnet is inside the loop. (c) When the magnet is moved away

from the loop, the ammeter deflects in the opposite direction, indicating that the

induced current is opposite that shown in part (a). Changing the direction of the

magnet’s motion changes the direction of the current induced by that motion.

Ammeter

(b)

(a)

(c)

At the Active Figures link

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

can move the magnet and

observe the current in the

ammeter.

At the Active Figures link

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

can open and close the switch

and observe the current in the

ammeter.

Active Figure 31.2 Faraday’s experiment. When the switch in the primary circuit is

closed, the ammeter in the secondary circuit deflects momentarily. The emf induced in

the secondary circuit is caused by the changing magnetic field through the secondary coil.

Ammeter

Secondary

coil

Primary

coil

Iron

Switch

–

Battery

970

CHAPTE R 31 • Faraday’s Law

This statement, known as

Faraday’s law of induction, can be written

(31.1)

where !

∫ B ! dA is the magnetic flux through the circuit. (See Section 30.5.)

If the circuit is a coil consisting of N loops all of the same area and if !

is the

magnetic flux through one loop, an emf is induced in every loop. The loops are in series,
so their emfs add; thus, the total induced emf in the coil is given by the expression

(31.2)

The negative sign in Equations 31.1 and 31.2 is of important physical significance, as
discussed in Section 31.3.

Suppose that a loop enclosing an area A lies in a uniform magnetic field

B, as in

Figure 31.3. The magnetic flux through the loop is equal to BA cos #; hence, the
induced emf can be expressed as

(31.3)

From this expression, we see that an emf can be induced in the circuit in several ways:

• The magnitude of

B can change with time.

• The area enclosed by the loop can change with time.
• The angle # between

B and the normal to the loop can change with time.

• Any combination of the above can occur.

" %

(BA cos #)

" %

the switch is closed, the magnetic field produced by the current in the primary circuit
changes from zero to some value over some finite time, and this changing field induces
a current in the secondary circuit.

As a result of these observations, Faraday concluded that

an electric current can

be induced in a circuit (the secondary circuit in our setup) by a changing
magnetic field. The induced current exists for only a short time while the magnetic
field through the secondary coil is changing. Once the magnetic field reaches a steady
value, the current in the secondary coil disappears. In effect, the secondary circuit
behaves as though 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.

The experiments shown in Figures 31.1 and 31.2 have one thing in common: in

each case, an emf is induced in the circuit when the magnetic flux through the circuit
changes with time. In general,

The emf induced in a circuit is directly proportional to the time rate of change of
the magnetic flux through the circuit.

▲

PITFALL PREVENTION

31.1 Induced emf

Requires a Change

The  existence of  a  magnetic  flux
through  an  area  is  not  sufficient
to create an induced emf. There
must be a change in the magnetic
flux  in  order  for  an  emf  to  be
induced.

Faraday’s law

Loop of

area A

Figure 31.3 A conducting loop

that encloses an area A in the

presence of a uniform magnetic

field B. The angle between B and

the normal to the loop is

Quick Quiz 31.1

A circular loop of wire is held in a uniform magnetic field,

with the plane of the loop perpendicular to the field lines. Which of the following will
not cause a current to be induced in the loop? (a) crushing the loop; (b) rotating the
loop about an axis perpendicular to the field lines; (c) keeping the orientation of the
loop fixed and moving it along the field lines; (d) pulling the loop out of the field.

SECTION 31.1 • Faraday’s Law of Induction

971

Some Applications of Faraday’s Law

The ground fault interrupter (GFI) is an interesting safety device that protects users of
electrical appliances against electric shock. Its operation makes use of Faraday’s law. In
the GFI shown in Figure 31.5, wire 1 leads from the wall outlet to the appliance to be
protected,  and  wire  2  leads  from  the  appliance  back  to  the  wall  outlet.  An  iron  ring
surrounds the two wires, and a sensing coil is wrapped around part of the ring. Because
the currents in the wires are in opposite directions, the net magnetic flux through the
sensing  coil  due  to  the  currents  is  zero.  However,  if  the  return  current  in  wire  2
changes, the net magnetic flux through the sensing coil is no longer zero. (This can
happen,  for  example,  if  the  appliance  becomes  wet,  enabling  current  to  leak  to
ground.) Because household current is alternating (meaning that its direction keeps
reversing), the magnetic flux through the sensing coil changes with time, inducing an
emf in the coil. This induced emf is used to trigger a circuit breaker, which stops the
current before it is able to reach a harmful level.

Another interesting application of Faraday’s law is the production of sound in an

electric guitar (Fig. 31.6). The coil in this case, called the pickup coil, is placed near the
vibrating guitar string, which is made of a metal that can be magnetized. A permanent

Quick Quiz 31.2

Figure 31.4 shows a graphical representation of the field

magnitude versus time for a magnetic field that passes through a fixed loop and is ori-
ented perpendicular to the plane of the loop. The magnitude of the magnetic field at
any time is uniform over the area of the loop. Rank the magnitudes of the emf gener-
ated in the loop at the five instants indicated, from largest to smallest.

Quick Quiz 31.3

Suppose you would like to steal power for your home from

the electric company by placing a loop of wire near a transmission cable, so as to
induce an emf in the loop (an illegal procedure). Should you (a) place your loop so
that the transmission cable passes through your loop, or (b) simply place your loop
near the transmission cable?

Figure 31.4 (Quick Quiz 31.2) The time behavior of a magnetic field through a loop.

Figure 31.5 Essential components of a ground

fault interrupter.

Circuit

breaker

Sensing

coil

Alternating

current

Iron

ring

972

CHAPTE R 31 • Faraday’s Law

magnet inside the coil magnetizes the portion of the string nearest the coil. When the
string vibrates at some frequency, its magnetized segment produces a changing
magnetic flux through the coil. The changing flux induces an emf in the coil that is
fed to an amplifier. The output of the amplifier is sent to the loudspeakers, which
produce the sound waves we hear.

(b)

Figure 31.6 (a) In an electric guitar, a vibrating magnetized string induces an emf in a

pickup coil. (b) The pickups (the circles beneath the metallic strings) of this electric

guitar detect the vibrations of the strings and send this information through an

amplifier and into speakers. (A switch on the guitar allows the musician to select which

set of six pickups is used.)

Charles D. Winters

(a)

Magnetized

portion of

string

Guitar string

To amplifier

Magnet

Pickup

coil

Example 31.1 One Way to Induce an emf in a Coil

A coil consists of 200 turns of wire. Each turn is a square of
side 18 cm, and a uniform magnetic field directed perpendic-
ular to the plane of the coil is turned on. If the field changes
linearly from 0 to 0.50 T in 0.80 s, what is the magnitude of
the induced emf in the coil while the field is changing?

Solution The area of one turn of the coil is (0.18 m)

0.032 4 m

. The magnetic flux through the coil at t " 0 is

zero because B " 0 at that time. At t " 0.80 s, the magnetic
flux through one turn is !

BA " (0.50 T)(0.032 4 m

) "

0.016 2 T & m

. Therefore, the magnitude of the induced emf

is, from Equation 31.2,

4.1 V

4.1 T&m

/s "

200

(0.016 2 T&m

0.80 s

You should be able to show that 1 T & m

/s " 1 V.

What If?

What if you were asked to find the magnitude of

the induced current in the coil while the field is changing?
Can you answer this question?

Answer If  the  ends  of  the  coil  are  not  connected  to  a
circuit,  the  answer  to  this  question  is  easy—the  current  is
zero! (Charges will move within the wire of the coil, but they
cannot move into or out of the ends of the coil.) In order
for  a  steady  current  to  exist,  the  ends  of  the  coil  must  be
connected to an external circuit. Let us assume that the coil
is connected to a circuit and that the total resistance of the
coil and the circuit is 2.0 (. Then, the current in the coil is

I "

4.1 V

2.0 (

2.0 A

Example 31.2 An Exponentially Decaying B Field

A  loop  of  wire  enclosing  an  area  A is  placed  in  a  region
where  the  magnetic  field  is  perpendicular  to  the  plane  of
the  loop.  The  magnitude  of

B varies in time according to

the expression B " B

max

, where a is some constant. That

is, at t " 0 the field is B

max

, and for t ) 0, the field decreases

exponentially (Fig. 31.7). Find the induced emf in the loop
as a function of time.

Solution Because

B is perpendicular to the plane of the

loop, the magnetic flux through the loop at time t ) 0 is

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