Physics For Scientists And Engineers 6E - part 250

Problems

997

at t " 0.) Determine the emf generated in the coil as a
function of time.

38.

A bar magnet is spun at constant angular speed * around
an axis as shown in Figure P31.38. A stationary flat rectan-
gular conducting loop surrounds the magnet, and at
t " 0, the magnet is oriented as shown. Make a qualitative
graph of the induced current in the loop as a function of
time, plotting counterclockwise currents as positive and
clockwise currents as negative.

39.

A  motor  in  normal  operation  carries  a  direct  current  of
0.850 A  when  connected  to  a  120-V  power  supply.  The
resistance  of  the  motor  windings  is  11.8  (.  While  in
normal operation, (a) what is the back emf generated by
the motor? (b) At what rate is internal energy produced in
the  windings?  (c)  What  If?  Suppose  that  a  malfunction
stops the motor shaft from rotating. At what rate will inter-
nal energy be produced in the windings in this case? (Most
motors have a thermal switch that will turn off the motor
to prevent overheating when this occurs.)

40.

A semicircular conductor of radius R " 0.250 m is rotated
about the axis AC at a constant rate of 120 rev/min (Fig.
P31.40). A uniform magnetic field in all of the lower half
of the figure is directed out of the plane of rotation and
has a magnitude of 1.30 T. (a) Calculate the maximum
value of the emf induced in the conductor. (b) What is the
value of the average induced emf for each complete rota-
tion? (c) What If? How would the answers to (a) and (b)
change if

B were allowed to extend a distance R above the

axis of rotation? Sketch the emf versus time (d) when the
field is as drawn in Figure P31.40 and (e) when the field is
extended as described in (c).

41.

The rotating loop in an AC generator is a square 10.0 cm
on a side. It is rotated at 60.0 Hz in a uniform field of
0.800 T. Calculate (a) the flux through the loop as a
function of time, (b) the emf induced in the loop, (c) the

current induced in the loop for a loop resistance of
1.00 (, (d) the power delivered to the loop, and (e) the
torque that must be exerted to rotate the loop.

Section 31.6 Eddy Currents

42. Figure P31.42 represents an electromagnetic brake that

uses  eddy  currents.  An  electromagnet  hangs  from  a
railroad car near one rail. To stop the car, a large current
is sent through the coils of the electromagnet. The moving
electromagnet  induces  eddy  currents  in  the  rails,  whose
fields oppose the change in the field of the electromagnet.
The  magnetic  fields  of  the  eddy  currents  exert  force  on
the current in the electromagnet, thereby slowing the car.
The direction of the car’s motion and the direction of the
current  in  the  electromagnet  are  shown  correctly  in  the
picture. Determine which of the eddy currents shown on
the rails is correct. Explain your answer.

A conducting rectangular loop of mass M, resistance

R, and dimensions w by ! falls from rest into a magnetic
field

B as shown in Figure P31.43. During the time interval

43.

Figure P31.38

out

Figure P31.40

Figure P31.42

out

Figure P31.43

before the top edge of the loop reaches the field, the loop
approaches a terminal speed v

. (a) Show that

(b) Why is v

proportional to R? (c) Why is it inversely propor-

tional to B

Section 31.7 Maxwell’s Equations

44.

An electron moves through a uniform electric field

E " (2.50iˆ 2 5.00jˆ) V/m and a uniform magnetic field

B " (0.400kˆ)T. Determine the acceleration of the
electron when it has a velocity

v " 10.0iˆ m/s.

A proton moves through a uniform electric field given by

E " 50.0jˆ V/m and a uniform magnetic field B "
(0.200

iˆ 2 0.300jˆ 2 0.400kˆ)T. Determine the acceleration

of the proton when it has a velocity

v " 200iˆ m/s.

Additional Problems

46.

A steel guitar string vibrates (Figure 31.6). The compo-
nent of magnetic field perpendicular to the area of a
pickup coil nearby is given by

B " 50.0 mT 2 (3.20 mT) sin(2, 523 t/s)

The circular pickup coil has 30 turns and radius 2.70 mm.
Find the emf induced in the coil as a function of time.

47.

Figure P31.47 is a graph of the induced emf versus time
for a coil of N turns rotating with angular speed * in a uni-
form magnetic field directed perpendicular to the axis of
rotation of the coil. What If ? Copy this sketch (on a larger
scale), and on the same set of axes show the graph of emf
versus t (a) if the number of turns in the coil is doubled;
(b) if instead the angular speed is doubled; and (c) if the
angular speed is doubled while the number of turns in the
coil is halved.

45.

MgR

48. A technician wearing a brass bracelet enclosing area

0.005 00 m

places her hand in a solenoid whose magnetic

field is 5.00 T directed perpendicular to the plane of the
bracelet.  The  electrical  resistance  around  the  circumfer-
ence  of  the  bracelet  is  0.020 0  (.  An  unexpected  power
failure  causes  the  field  to  drop  to  1.50 T  in  a  time  of
20.0 ms. Find (a) the current induced in the bracelet and
(b)  the  power  delivered  to  the  bracelet.  Note: As  this

problem implies, you should not wear any metal objects
when working in regions of strong magnetic fields.

49.

Two infinitely long solenoids (seen in cross section) pass
through a circuit as shown in Figure P31.49. The magni-
tude of

B inside each is the same and is increasing at the

rate of 100 T/s. What is the current in each resistor?

50.

A conducting rod of length ! " 35.0 cm is free to slide on
two parallel conducting bars as shown in Figure P31.50.
Two resistors R

2.00 ( and R

5.00 ( are connected

across  the  ends  of  the  bars  to  form  a  loop.  A  constant
magnetic field B " 2.50 T is directed perpendicularly into
the page. An external agent pulls the rod to the left with a
constant  speed  of  v " 8.00 m/s.  Find  (a)  the  currents  in
both  resistors,  (b)  the  total  power  delivered  to  the  resis-
tance of the circuit, and (c) the magnitude of the applied
force  that  is  needed  to  move  the  rod  with  this  constant
velocity.

51. Suppose you wrap wire onto the core from a roll of cello-

phane tape to make a coil. Describe how you can use a bar
magnet to produce an induced voltage in the coil. What is
the order of magnitude of the emf you generate? State the
quantities you take as data and their values.

52.

A bar of mass m, length d, and resistance R slides without
friction  in  a  horizontal  plane,  moving  on  parallel  rails  as
shown  in  Figure  P31.52.  A  battery  that  maintains  a
constant  emf

is connected between the rails, and a

constant magnetic field

B is directed perpendicularly to

the plane of the page. Assuming the bar starts from rest,
show that at time t it moves with a speed

v "

(1 % e

t/mR

)

998

CHAPTE R 31 • Faraday’s Law

(mV)

t(ms)

–5

–10

Figure P31.47

0.500 m

6.00

Ω

5.00

Ω

= 0.150 m

out

= 0.100 m

3.00

Ω

× ××

Figure P31.49

×
×
×
×
×
×

5.00

Ω

×
×
×
×
×
×

2.00

Ω

Figure P31.50

B (out of page)

Figure P31.52

53.

Review problem. A particle with a mass of 2.00 + 10

and a charge of 30.0 nC starts from rest, is accelerated by a
strong electric field, and is fired from a small source inside
a region of uniform constant magnetic field 0.600 T. The
velocity of the particle is perpendicular to the field. The
circular orbit of the particle encloses a magnetic flux of
15.0 /Wb. (a) Calculate the speed of the particle. (b) Cal-
culate the potential difference through which the particle
accelerated inside the source.

54.

An  induction  furnace uses  electromagnetic  induction  to
produce eddy currents in a conductor, thereby raising the
conductor’s temperature. Commercial units operate at fre-
quencies ranging from 60 Hz to about 1 MHz and deliver
powers  from  a  few  watts  to  several  megawatts.  Induction
heating can be used for welding in a vacuum enclosure, to
avoid  oxidation  and  contamination  of  the  metal.  At  high
frequencies, induced currents occur only near the surface
of the conductor—this is the “skin effect.” By creating an
induced current for a short time at an appropriately high
frequency,  one  can  heat  a  sample  down  to  a  controlled
depth.  For  example,  the  surface  of  a  farm  tiller  can  be
tempered to make it hard and brittle for effective cutting
while keeping the interior metal soft and ductile to resist
breakage.

To explore induction heating, consider a flat conduct-

ing disk of radius R, thickness b, and resistivity 3. A sinu-
soidal magnetic field B

max

cos *t is applied perpendicular

to the disk. Assume that the frequency is so low that the
skin effect is not important. Assume the eddy currents
occur in circles concentric with the disk. (a) Calculate the
average power delivered to the disk. (b) What If? By what
factor does the power change when the amplitude of the
field doubles? (c) When the frequency doubles? (d) When
the radius of the disk doubles?

The plane of a square loop of wire with edge length
a " 0.200 m is perpendicular to the Earth’s magnetic field
at a point where B " 15.0 /T, as shown in Figure P31.55.
The total resistance of the loop and the wires connecting it
to a sensitive ammeter is 0.500 (. If the loop is suddenly
collapsed by horizontal forces as shown, what total charge
passes through the ammeter?

55.

56.

Magnetic  field  values  are  often  determined  by  using  a
device  known  as  a  search  coil.  This  technique  depends  on
the  measurement  of  the  total  charge  passing  through  a
coil  in  a  time  interval  during  which  the  magnetic  flux

linking the windings changes either because of the motion
of the coil or because of a change in the value of B.
(a) Show that as the flux through the coil changes from
!

to !

, the charge transferred through the coil will be

given by Q " N(!

% !

)/R, where R is the resistance of

the coil and a sensitive ammeter connected across it and N
is the number of turns. (b) As a specific example, calculate
B when a 100-turn coil of resistance 200 ( and cross-
sectional area 40.0 cm

produces the following results.

A total charge of 5.00 + 10

C passes through the coil

when it is rotated in a uniform field from a position where
the plane of the coil is perpendicular to the field to a posi-
tion where the coil’s plane is parallel to the field.

57.

In  Figure  P31.57,  the  rolling  axle,  1.50 m  long,  is  pushed
along  horizontal  rails  at  a  constant  speed  v " 3.00 m/s.  A
resistor R " 0.400 ( is connected to the rails at points a and
b, directly opposite each other. (The wheels make good elec-
trical contact with the rails, and so the axle, rails, and R form
a  closed-loop  circuit.  The  only  significant  resistance  in  the
circuit is R.) A uniform magnetic field B " 0.080 0 T is verti-
cally downward. (a) Find the induced current I in the resis-
tor. (b) What horizontal force F is required to keep the axle
rolling at constant speed? (c) Which end of the resistor, a or
b, is at the higher electric potential? (d) What If ? After the
axle  rolls  past  the  resistor,  does  the  current  in  R reverse
direction? Explain your answer.

58.

A conducting rod moves with a constant velocity

v in a

direction perpendicular to a long, straight wire carrying a
current I as shown in Figure P31.58. Show that the magni-
tude of the emf generated between the ends of the rod is

In this case, note that the emf decreases with increasing r,
as you might expect.

vI!

2,r

Problems

999

Figure P31.55

Figure P31.57

Figure P31.58

59.

A circular loop of wire of radius r is in a uniform magnetic
field, with the plane of the loop perpendicular to the
direction of the field (Fig. P31.59). The magnetic field
varies with time according to B(t) " a 2 bt, where a and b
are constants. (a) Calculate the magnetic flux through the
loop at t " 0. (b) Calculate the emf induced in the loop.
(c) If the resistance of the loop is R, what is the induced
current? (d) At what rate is energy being delivered to the
resistance of the loop?

60.

In Figure P31.60, a uniform magnetic field decreases at a
constant rate dB/dt " % K, where K is a positive constant.
A circular loop of wire of radius a containing a resistance
R and a capacitance C is placed with its plane normal to
the field. (a) Find the charge Q on the capacitor when it is
fully charged. (b) Which plate is at the higher potential?
(c) Discuss the force that causes the separation of charges.

A  rectangular  coil  of  60  turns,  dimensions  0.100 m  by
0.200 m  and  total  resistance  10.0  (,  rotates  with  angular
speed 30.0 rad/s about the y axis in a region where a 1.00-T
magnetic field is directed along the x axis. The rotation is
initiated  so  that  the  plane  of  the  coil  is  perpendicular  to
the  direction  of

B at t " 0. Calculate (a) the maximum

induced emf in the coil, (b) the maximum rate of change
of magnetic flux through the coil, (c) the induced emf at
t " 0.050 0 s, and (d) the torque exerted by the magnetic
field on the coil at the instant when the emf is a maximum.

62.

A small circular washer of radius 0.500 cm is held directly
below a long, straight wire carrying a current of 10.0 A.
The washer is located 0.500 m above the top of a table
(Fig. P31.62). (a) If the washer is dropped from rest, what
is the magnitude of the average induced emf in the washer
from the time it is released to the moment it hits the table-
top? Assume that the magnetic field is nearly constant over
the area of the washer, and equal to the magnetic field at
the center of the washer. (b) What is the direction of the
induced current in the washer?

61.

63.

A conducting rod of length ! moves with velocity

v parallel

to a long wire carrying a steady current I. The axis of the
rod is maintained perpendicular to the wire with the near
end a distance r away, as shown in Figure P31.63. Show
that the magnitude of the emf induced in the rod is

1 2

64.

A rectangular loop of dimensions ! and w moves with a
constant velocity

v away from a long wire that carries a

current I in the plane of the loop (Fig. P31.64). The total
resistance of the loop is R. Derive an expression that gives
the current in the loop at the instant the near side is a
distance r from the wire.

The magnetic flux through a metal ring varies with time t
according to !

3(at

) T & m

, with a " 2.00 s

and b " 6.00 s

. The resistance of the ring is 3.00 (.

Determine the maximum current induced in the ring
during the interval from t " 0 to t " 2.00 s.

66.

Review problem. The bar of mass m in Figure P31.66 is
pulled horizontally across parallel rails by a massless string
that passes over an ideal pulley and is attached to a

65.

1000

CHAPTE R 31 • Faraday’s Law

Figure P31.59

Figure P31.60

Figure P31.62

Figure P31.64

Figure P31.63

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