Physics For Scientists And Engineers 6E - part 249

Problems

993

10.

A coil of 15 turns and radius 10.0 cm surrounds a long
solenoid of radius 2.00 cm and 1.00 + 10

turns/meter

(Fig. P31.10). The current in the solenoid changes as
I " (5.00 A) sin(120t). Find the induced emf in the
15-turn coil as a function of time.

11. Find the current through section PQ of length a " 65.0 cm

in Figure P31.11. The circuit is located in a magnetic field
whose magnitude varies with time according to the expres-
sion B " (1.00 + 10

T/s)t. Assume the resistance per

length of the wire is 0.100 (/m.

12. A 30-turn circular coil of radius 4.00 cm and resistance

1.00 ( is placed in a magnetic field directed perpen-
dicular to the plane of the coil. The magnitude of the
magnetic field varies in time according to the expression
B " 0.010 0t 2 0.040 0t

, where t is in seconds and

B is in tesla. Calculate the induced emf in the coil at
t " 5.00 s.

A long solenoid has n " 400 turns per meter and carries a
current given by I " (30.0 A)(1 % e

1.60t

). Inside the sole-

noid and coaxial with it is a coil that has a radius of
6.00 cm and consists of a total of N " 250 turns of fine
wire (Fig. P31.13). What emf is induced in the coil by the
changing current?

13.

14. An instrument based on induced emf has been used to

measure projectile speeds up to 6 km/s. A small magnet is
imbedded  in  the  projectile,  as  shown  in  Figure  P31.14.
The  projectile  passes  through  two  coils  separated  by  a
distance  d.  As  the  projectile  passes  through  each  coil  a
pulse  of  emf  is  induced  in  the  coil.  The  time  interval
between pulses can be measured accurately with an oscillo-
scope, and thus the speed can be determined. (a) Sketch a
graph of 'V versus t for the arrangement shown. Consider
a  current  that  flows  counterclockwise  as  viewed  from  the
starting point of the projectile as positive. On your graph,
indicate which pulse is from coil 1 and which is from coil
2.  (b)  If  the  pulse  separation  is  2.40 ms  and  d " 1.50 m,
what is the projectile speed?

A coil formed by wrapping 50 turns of wire in the shape of
a square is positioned in a magnetic field so that the
normal to the plane of the coil makes an angle of 30.0°
with the direction of the field. When the magnetic field is
increased uniformly from 200 /T to 600 /T in 0.400 s, an
emf of magnitude 80.0 mV is induced in the coil. What is
the total length of the wire?

16.

When a wire carries an AC current with a known frequency,
you can use a Rogowski coil to determine the amplitude I

max

of the current without disconnecting the wire to shunt the

15.

Figure P31.9 Problems 9 and 71.

15-turn coil

Figure P31.10

×
×
×
×

Figure P31.11

N turns

n turns/m

× × × × × × × × × × × × × × × × × × × × × ×

Figure P31.13

S N

Figure P31.14

I(t )

Figure P31.16

994

CHAPTE R 31 • Faraday’s Law

current  in  a  meter.  The  Rogowski  coil,  shown  in  Figure
P31.16,  simply  clips  around  the  wire.  It  consists  of  a
toroidal conductor wrapped around a circular return cord.
The toroid has n turns per unit length and a cross-sectional
area  A. The  current  to  be  measured  is  given  by  I(t) "
I

max

sin *t. (a) Show that the amplitude of the emf

induced in the Rogowski coil is

max

nA* I

max

(b) Explain why the wire carrying the unknown current
need not be at the center of the Rogowski coil, and why the
coil will not respond to nearby currents that it does not
enclose.

17.

A  toroid  having  a  rectangular  cross  section  (a " 2.00 cm
by  b " 3.00 cm)  and  inner  radius  R " 4.00 cm  consists
of 500 turns  of  wire  that  carries  a  sinusoidal  current
I " I

max

sin *t, with I

max

50.0 A and a frequency

f " */2, " 60.0 Hz. A coil that consists of 20 turns of
wire links with the toroid, as in Figure P31.17. Determine
the emf induced in the coil as a function of time.

18.

A  piece  of  insulated  wire  is  shaped  into  a  figure  8,  as  in
Figure  P31.18.  The  radius  of  the  upper  circle  is  5.00 cm
and that of the lower circle is 9.00 cm. The wire has a uni-
form  resistance  per  unit  length  of  3.00  (/m.  A  uniform
magnetic field is applied perpendicular to the plane of the
two circles, in the direction shown. The magnetic  field is
increasing at a constant rate of 2.00 T/s. Find the magni-
tude and direction of the induced current in the wire.

Section 31.2 Motional emf
Section 31.3 Lenz’s Law

19.

An automobile has a vertical radio antenna 1.20 m long.
The automobile travels at 65.0 km/h on a horizontal road

where  the  Earth’s  magnetic  field  is  50.0 /T  directed
toward the north and downward at an angle of 65.0° below
the horizontal. (a) Specify the direction that the automo-
bile  should  move  in  order  to  generate  the  maximum
motional emf in the antenna, with the top of the antenna
positive  relative  to  the  bottom.  (b)  Calculate  the  magni-
tude of this induced emf.

20. Consider the arrangement shown in Figure P31.20. Assume

that R " 6.00 (, ! " 1.20 m, and a uniform 2.50-T mag-
netic field is directed into the page. At what speed should
the bar be moved to produce a current of 0.500 A in the
resistor?

Figure  P31.20  shows  a  top  view  of  a  bar  that  can  slide
without friction. The resistor is 6.00 ( and a 2.50-T mag-
netic  field  is  directed  perpendicularly  downward,  into
the  paper.  Let  ! " 1.20 m.  (a)  Calculate  the  applied
force required to move the bar to the right at a constant
speed  of  2.00 m/s.  (b)  At  what  rate  is  energy  delivered
to the resistor?

22.

A conducting rod of length ! moves on two horizontal,
frictionless rails, as shown in Figure P31.20. If a constant
force of 1.00 N moves the bar at 2.00 m/s through a mag-
netic field

B that is directed into the page, (a) what is the

current through the 8.00-( resistor R ? (b) What is the rate
at which energy is delivered to the resistor? (c) What is the
mechanical power delivered by the force

app

23.

Very large magnetic fields can be produced using a proce-
dure called flux compression. A metallic cylindrical tube of
radius R is placed coaxially in a long solenoid of somewhat
larger  radius.  The  space  between  the  tube  and  the  sole-
noid  is  filled  with  a  highly  explosive  material.  When  the
explosive  is  set  off,  it  collapses  the  tube  to  a  cylinder  of
radius r 0 R. If the collapse happens very rapidly, induced
current  in  the  tube  maintains  the  magnetic  flux  nearly
constant inside the tube. If the initial magnetic field in the
solenoid  is  2.50 T,  and  R/r " 12.0,  what  maximum  value
of magnetic field can be achieved?

24.

The  homopolar  generator,  also  called  the  Faraday  disk,  is  a
low-voltage, high-current electric generator. It consists of
a  rotating  conducting  disk  with  one  stationary  brush  (a
sliding  electrical  contact)  at  its  axle  and  another  at  a
point  on  its  circumference,  as  shown  in  Figure  P31.24.
A magnetic field is applied perpendicular to the plane of
the disk. Assume the field is 0.900 T, the angular speed is
3 200 rev/min, and the radius of the disk is 0.400 m. Find
the emf generated between the brushes. When supercon-
ducting coils are used to produce a large magnetic field, a
homopolar generator can have a power output of several

21.

′ = 20

= 500

Figure P31.17

Figure P31.18

Problem 71 in Chapter 29 can be assigned with this section.

app

Figure P31.20 Problems 20, 21, and 22.

Problems

995

megawatts. Such a generator is useful, for example, in
purifying metals by electrolysis. If a voltage is applied to
the output terminals of the generator, it runs in reverse as
a homopolar motor capable of providing great torque, useful
in ship propulsion.

25.

Review problem. A flexible metallic wire with linear den-
sity 3.00 + 10

kg/m is stretched between two fixed

clamps  64.0 cm  apart  and  held  under  tension  267 N.  A
magnet is placed near the wire as shown in Figure P31.25.
Assume  that  the  magnet  produces  a  uniform  field  of
4.50 mT  over  a  2.00-cm  length  at  the  center  of  the  wire,
and a negligible field elsewhere. The wire is set vibrating
at its fundamental (lowest) frequency. The section of the
wire in the magnetic field moves with a uniform amplitude
of 1.50 cm. Find (a) the frequency and (b) the amplitude
of  the  electromotive  force  induced  between  the  ends  of
the wire.

26.

The  square  loop  in  Figure  P31.26  is  made  of  wires  with
total  series  resistance  10.0 (.  It  is  placed  in  a  uniform
0.100-T magnetic field directed perpendicularly into the
plane  of  the  paper.  The  loop,  which  is  hinged  at  each
corner,  is  pulled  as  shown  until  the  separation  between
points  A and  B is  3.00 m.  If  this  process  takes  0.100 s,
what is the average current generated in the loop? What
is the direction of the current?

A helicopter (Figure P31.27) has blades of length 3.00 m,
extending out from a central hub and rotating at
2.00 rev/s. If the vertical component of the Earth’s mag-
netic field is 50.0 /T, what is the emf induced between the
blade tip and the center hub?

27.

28. Use Lenz’s law to answer the following questions concern-

ing  the  direction  of  induced  currents.  (a)  What  is  the
direction  of  the  induced  current  in  resistor  R in  Figure
P31.28a  when  the  bar  magnet  is  moved  to  the  left?
(b) What  is  the  direction  of  the  current  induced  in  the
resistor R immediately after the switch S in Figure P31.28b

Figure P31.24

Figure P31.25

3.00 m

Figure P31.26

Figure P31.27

Ross Harrison Koty/Getty Images

S N

(a)

(b)

(c)

(d)

–

Figure P31.28

996

CHAPTE R 31 • Faraday’s Law

is closed? (c) What is the direction of the induced current
in R when the current I in Figure P31.28c decreases
rapidly to zero? (d) A copper bar is moved to the right
while its axis is maintained in a direction perpendicular to
a magnetic field, as shown in Figure P31.28d. If the top of
the bar becomes positive relative to the bottom, what is the
direction of the magnetic field?

A rectangular coil with resistance R has N turns, each of
length ! and width w as shown in Figure P31.29. The coil
moves into a uniform magnetic field

B with constant velocity

v. What are the magnitude and direction of the total
magnetic force on the coil (a) as it enters the magnetic field,
(b) as it moves within the field, and (c) as it leaves the field?

29.

31.

Two parallel rails with negligible resistance are 10.0 cm apart
and  are  connected  by  a  5.00-( resistor.  The  circuit  also
contains  two  metal  rods  having  resistances  of  10.0 ( and
15.0 ( sliding  along  the  rails  (Fig.  P31.31).  The  rods  are
pulled away from the resistor at constant speeds of 4.00 m/s
and  2.00 m/s,  respectively.  A  uniform  magnetic  field  of
magnitude  0.010 0 T  is  applied  perpendicular  to  the  plane
of the rails. Determine the current in the 5.00 -( resistor.

Section 31.4 Induced emf and Electric Fields

32.

For the situation shown in Figure P31.32, the magnetic
field changes with time according to the expression
B " (2.00t

4.00t

0.800)T, and r

2R " 5.00 cm.

(a) Calculate the magnitude and direction of the force
exerted on an electron located at point P

when t " 2.00 s.

(b) At what time is this force equal to zero?

A magnetic field directed into the page changes with time
according to B " (0.030 0t

1.40)T, where t is in

seconds.  The  field  has  a  circular  cross  section  of  radius
R " 2.50 cm  (Fig.  P31.32).  What  are  the  magnitude  and
direction  of  the  electric  field  at  point  P

when t " 3.00 s

and r

0.020 0 m ?

34.

A  long  solenoid  with  1 000  turns  per  meter  and
radius 2.00 cm  carries  an  oscillating  current  given  by
I " (5.00 A) sin(100,t). What is the electric field induced
at  a  radius  r " 1.00 cm  from  the  axis  of  the  solenoid?
What is the direction of this electric field when the current
is increasing counterclockwise in the coil?

Section 31.5 Generators and Motors

33.

A coil of area 0.100 m

is rotating at 60.0 rev/s with

the axis of rotation perpendicular to a 0.200-T magnetic
field. (a) If the coil has 1 000 turns, what is the maximum
emf generated in it? (b) What is the orientation of the coil
with respect to the magnetic field when the maximum
induced voltage occurs?

36.

In a 250-turn automobile alternator, the magnetic flux in
each turn is !

(2.50 + 10

Wb) cos(*t), where * is

the  angular  speed  of  the  alternator.  The  alternator  is
geared  to  rotate  three  times  for  each  engine  revolution.
When  the  engine  is  running  at  an  angular  speed  of
1 000 rev/min,  determine  (a)  the  induced  emf  in  the
alternator as a function of time and (b) the maximum emf
in the alternator.

37.

A long solenoid, with its axis along the  x axis, consists of
200 turns per meter of wire that carries a steady current of
15.0 A. A coil is formed by wrapping 30 turns of thin wire
around a circular frame that has a radius of 8.00 cm. The
coil  is  placed  inside  the  solenoid  and  mounted  on  an
axis that  is  a  diameter  of  the  coil  and  coincides  with  the
y axis.  The  coil  is  then  rotated  with  an  angular  speed
of 4.00, rad/s.  (The  plane  of  the  coil  is  in  the  yz plane

35.

Figure P31.29

Motion toward

the loop

Figure P31.30

×
×
×
×
×
×

10.0

Ω

15.0

Ω

5.00

Ω

2.00 m/s

4.00 m/s

Figure P31.31

× × × × × × × ×
× × ×

× × × ×

× × × × ×

× ×

× × × × × ×

× × × × ×

Figure P31.32 Problems 32 and 33.

30.

In Figure P31.30, the bar magnet is moved toward the
loop. Is V

positive, negative, or zero? Explain.

Problems 28 and 62 in Chapter 29 can be assigned with
this section.

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