Physics For Scientists And Engineers 6E - part 230

Questions

917

force exerted on a current-carrying conductor of arbitrary shape in a uniform mag-
netic field

(29.7)

where

L( is a vector directed from one end of the conductor to the opposite end. Be-

cause integration of Equation 29.4 for a closed loop yields a zero result, the net mag-
netic force on any closed loop carrying a current in a uniform magnetic field is zero.

The

magnetic dipole moment " of a loop carrying a current I is

(29.10)

where the area vector

A is perpendicular to the plane of the loop and ! A ! is equal to

the area of the loop. The SI unit of " is A # m

The torque # on a current loop placed in a uniform magnetic field

B is

(29.11)

The potential energy of the system of a magnetic dipole in a magnetic field is

(29.12)

If a charged particle moves in a uniform magnetic field so that its initial velocity is

perpendicular to the field, the particle moves in a circle, the plane of which is perpen-
dicular to the magnetic field. The radius of the circular path is

(29.13)

where m is the mass of the particle and q is its charge. The angular speed of the
charged particle is

(29.14)

. "

r "

U " $" %

# " " !

" "

L( ! B

1. At a given instant, a proton moves in the positive x direction

through a magnetic field in the negative z direction. What
is the direction of the magnetic force? Does the proton
continue to move in the positive x direction? Explain.
Two charged particles are projected into a magnetic field
perpendicular to their velocities. If the charges are deflected
in opposite directions, what can you say about them?

3. If a charged particle moves in a straight line through some

region of space, can you say that the magnetic field in that
region is zero?

4. Suppose an electron is chasing a proton up this page when

they suddenly enter a magnetic field perpendicular to the
page. What happens to the particles?

5. How can the motion of a moving charged particle be used

to distinguish between a magnetic field and an electric
field? Give a specific example to justify your argument.

6. List several similarities and differences between electric

and magnetic forces.

7. Justify the following statement: “It is impossible for a

constant (in other words, a time-independent) magnetic
field to alter the speed of a charged particle.”

8. In view of your answer to Question 7, what is the role of a

magnetic field in a cyclotron?

9. The electron beam in Figure Q29.9 is projected to the

right. The beam deflects downward in the presence of a
magnetic field produced by a pair of current-carrying coils.
(a) What is the direction of the magnetic field? (b) What
would happen to the beam if the magnetic field were
reversed in direction?

Q U E S T I O N S

Figure Q29.9

Courtesy of Central Scientific Company

918

C H A P T E R 2 9 • Magnetic Fields

10. A current-carrying conductor experiences no magnetic

force when placed in a certain manner in a uniform mag-
netic field. Explain.
Is it possible to orient a current loop in a uniform
magnetic field such that the loop does not tend to rotate?
Explain.

12. Explain why it is not possible to determine the

charge and  the  mass  of  a  charged  particle  separately
by measuring  accelerations  produced  by  electric  and
magnetic forces on the particle.
How  can  a  current  loop  be  used  to  determine  the
presence  of  a  magnetic  field  in  a  given  region  of
space?

14. Charged particles from outer space, called cosmic rays,

strike the Earth more frequently near the poles than near
the equator. Why?

15. What is the net force on a compass needle in a uniform

magnetic field?

16. What type of magnetic field is required to exert a resultant

force on a magnetic dipole? What is the direction of the
resultant force?

17. A proton moving horizontally enters a uniform magnetic

field  perpendicular  to  the  proton’s  velocity,  as  shown  in
Figure  Q29.17.  Describe  the  subsequent  motion  of  the
proton.  How  would  an  electron  behave  under  the  same
circumstances?

13.

11.

18. In the cyclotron, why do particles having different speeds

take the same amount of time to complete a one-half circle
trip around one dee?

19. The bubble chamber is a device used for observing tracks of

particles that pass through the chamber, which is immersed
in a magnetic field. If some of the tracks are spirals and
others are straight lines, what can you say about the
particles?

20. Can a constant magnetic field set into motion an electron

initially at rest? Explain your answer.

21. You are designing a magnetic probe that uses the Hall

effect  to  measure  magnetic  fields.  Assume  that  you  are
restricted  to  using  a  given  material  and  that  you  have
already  made  the  probe  as  thin  as  possible.  What,  if
anything,  can  be  done  to  increase  the  Hall  voltage
produced for a given magnitude of magnetic field?

Figure Q29.17

Section 29.1 Magnetic Fields and Forces

Determine the initial direction of the deflection of

charged particles as they enter the magnetic fields as
shown in Figure P29.1.

(a) downward, (b) northward, (c) westward, or (d) south-
eastward?

3. An electron moving along the positive x axis perpendicu-

lar to a magnetic field experiences a magnetic deflection
in the negative y direction. What is the direction of the
magnetic field?

4. A proton travels with a speed of 3.00 % 10

m/s at an angle

of 37.0° with the direction of a magnetic field of 0.300 T
in the ' y direction. What are (a) the magnitude of the
magnetic force on the proton and (b) its acceleration?
A proton moves perpendicular to a uniform magnetic field

B at 1.00 % 10

m/s and experiences an acceleration of

2.00 % 10

m/s

in the ' x direction when its velocity is in

the ' z direction. Determine the magnitude and direction
of the field.

6. An electron is accelerated through 2 400 V from rest and

then  enters  a  uniform  1.70-T  magnetic  field.  What  are
(a) the  maximum  and  (b)  the  minimum  values  of  the
magnetic force this charge can experience?
A  proton  moving  at  4.00 % 10

m/s through a magnetic

field of 1.70 T experiences a magnetic force of magnitude
8.20 % 10

N. What is the angle between the proton’s

velocity and the field?

= straightforward, intermediate, challenging

= full solution available in the Student Solutions Manual and Study Guide

= coached solution with hints available at http://www.pse6.com

= computer useful in solving problem

= paired numerical and symbolic problems

P R O B L E M S

(a)

–

(c)

(b)

(d)

right

at 45

Figure P29.1

2. Consider an electron near the Earth’s equator. In which di-

rection does it tend to deflect if its velocity is directed

Problems

919

Figure P29.14

Figure P29.15 Problems 15 and 16.

Figure P29.17

8. At the equator, near the surface of the Earth, the magnetic

field is approximately 50.0 +T northward, and the electric
field is about 100 N/C downward in fair weather. Find the
gravitational, electric, and magnetic forces on an electron
in this environment, assuming the electron has an instan-
taneous velocity of 6.00 % 10

m/s directed to the east.

A proton moves with a velocity of

v " (2iˆ $ 4jˆ ' kˆ) m/s in

a region in which the magnetic field is

B " (iˆ ' 2jˆ $ 3kˆ) T.

What is the magnitude of the magnetic force this charge
experiences?

10.

An electron has a velocity of 1.20 % 10

m/s (in the posi-

tive x direction), and an acceleration of 2.00 % 10

m/s

(in the positive z direction) in a uniform electric and mag-
netic field. If the electric field has a magnitude of
20.0 N/C (in the positive z direction), what can you deter-
mine about the magnetic field in the region? What can
you not determine?

Section 29.2 Magnetic Force Acting on a Current-

Carrying Conductor

A wire having a mass per unit length of 0.500 g/cm

carries a 2.00-A current horizontally to the south. What are
the direction and magnitude of the minimum magnetic
field needed to lift this wire vertically upward?

12. A wire carries a steady current of 2.40 A. A straight section

of the wire is 0.750 m long and lies along the x axis within
a uniform magnetic field,

B " 1.60kˆ T. If the current is

in the ' x direction, what is the magnetic force on the
section of wire?

13. A wire 2.80 m in length carries a current of 5.00 A in a

region where a uniform magnetic field has a magnitude of
0.390 T. Calculate the magnitude of the magnetic force on
the wire assuming the angle between the magnetic field
and the current is (a) 60.0°, (b) 90.0°, (c) 120°.

14.

A  conductor  suspended  by  two  flexible  wires  as  shown  in
Figure P29.14 has a mass per unit length of 0.040 0 kg/m.
What  current  must  exist  in  the  conductor  in  order  for
the tension  in  the  supporting  wires  to  be  zero  when  the
magnetic field is 3.60 T into the page? What is the required
direction for the current?

11.

15.

Review  Problem.  A  rod  of  mass  0.720 kg  and  radius
6.00 cm  rests  on  two  parallel  rails  (Fig.  P29.15)  that  are
d " 12.0 cm apart and L " 45.0 cm long. The rod carries a
current  of  I " 48.0 A  (in  the  direction  shown)  and  rolls
along the rails without slipping. A uniform magnetic field

16.

Review  Problem.  A  rod  of  mass  m  and  radius  R  rests  on
two parallel rails (Fig. P29.15) that are a distance d apart
and  have  a  length  L.  The  rod  carries  a  current  I (in  the
direction shown) and rolls along the rails without slipping.
A  uniform  magnetic  field  B  is  directed  perpendicular  to
the rod and the rails. If it starts from rest, what is the speed
of the rod as it leaves the rails?

A nonuniform magnetic field exerts a net force on a mag-

netic dipole. A strong magnet is placed under a horizontal
conducting ring of radius r that carries current I as
shown in Figure P29.17. If the magnetic field

B makes an

angle ! with the vertical at the ring’s location, what are
the magnitude and direction of the resultant force on
the ring?

17.

18.

In Figure P29.18, the cube is 40.0 cm on each edge. Four
straight  segments  of  wire—ab,  bc,  cd,  and  da—form  a
closed loop that carries a current I " 5.00 A, in the direc-
tion  shown.  A  uniform  magnetic  field  of  magnitude
B " 0.020 0 T is in the positive y direction. Determine the
magnitude  and  direction  of  the  magnetic  force  on  each
segment.

of magnitude 0.240 T is directed perpendicular to the rod
and the rails. If it starts from rest, what is the speed of the
rod as it leaves the rails?

920

C H A P T E R 2 9 • Magnetic Fields

Figure P29.18

0.300 m

30.0

I = 1.20 A

0.400 m

Figure P29.23

19.

Assume that in Atlanta, Georgia, the Earth’s magnetic field
is 52.0 +T northward at 60.0° below the horizontal. A tube
in a neon sign carries current 35.0 mA, between two diago-
nally  opposite  corners  of  a  shop  window,  which  lies  in  a
north–south  vertical  plane.  The  current  enters  the  tube  at
the bottom south corner of the window. It exits at the oppo-
site  corner,  which  is  1.40 m  farther  north  and  0.850 m
higher up. Between these two points, the glowing tube spells
out DONUTS. Use the theorem proved as Case 1 in the text
to determine the total vector magnetic force on the tube.

Section 29.3 Torque on a Current Loop in a Uniform

Magnetic Field

20. A current of 17.0 mA is maintained in a single circular loop

of 2.00 m circumference. A magnetic field of 0.800 T is
directed parallel to the plane of the loop. (a) Calculate the
magnetic moment of the loop. (b) What is the magnitude
of the torque exerted by the magnetic field on the loop?

21. A small bar magnet is suspended in a uniform 0.250-T

magnetic field. The maximum torque experienced by the
bar magnet is 4.60 % 10

N # m. Calculate the magnetic

moment of the bar magnet.

22.

A long piece of wire with a mass of 0.100 kg and a total
length of 4.00 m is used to make a square coil with a side of
0.100 m. The coil is hinged along a horizontal side, carries
a 3.40-A current, and is placed in a vertical magnetic field
with a magnitude of 0.010 0 T. (a) Determine the angle
that the plane of the coil makes with the vertical when the
coil is in equilibrium. (b) Find the torque acting on the coil
due to the magnetic force at equilibrium.

A rectangular coil consists of N " 100 closely wrapped

turns and has dimensions a " 0.400 m and b " 0.300 m.
The coil is hinged along the y axis, and its plane makes an
angle ! " 30.0° with the x axis (Fig. P29.23). What is the

23.

magnitude of the torque exerted on the coil by a uniform
magnetic field B " 0.800 T directed along the x axis when
the current is I " 1.20 A in the direction shown? What is
the expected direction of rotation of the coil?

24. A 40.0-cm length of wire carries a current of 20.0 A. It is

bent into a loop and placed with its normal perpendicular
to a magnetic field with a magnitude of 0.520 T. What is
the torque on the loop if it is bent into (a) an equilateral
triangle? What If? What is the torque if the loop is (b) a
square or (c) a circle? (d) Which torque is greatest?

25.

A current loop with magnetic dipole moment " is placed
in a uniform magnetic field

B, with its moment making

angle ! with the field. With the arbitrary choice of U " 0
for ! " 90°, prove that the potential energy of the
dipole–field system is U " $ " !

B. You may imitate the

discussion in Chapter 26 of the potential energy of an
electric dipole in an electric field.

26. The needle of a magnetic compass has magnetic moment

9.70 mA # m

. At its location, the Earth’s magnetic field is

55.0 +T  north  at  48.0° below  the  horizontal.  (a)  Identify
the orientations of the compass needle that represent min-
imum potential energy and maximum potential energy of
the  needle–field  system.  (b)  How  much  work  must  be
done  on  the  needle  to  move  it  from  the  former  to  the
latter orientation?

27.

A wire is formed into a circle having a diameter of 10.0 cm
and  placed  in  a  uniform  magnetic  field  of  3.00 mT.  The
wire  carries  a  current  of  5.00 A.  Find  (a)  the  maximum
torque on the wire and (b) the range of potential energies
of  the  wire–field  system  for  different  orientations  of  the
circle.

28.

The rotor in a certain electric motor is a flat rectangular
coil  with  80  turns  of  wire  and  dimensions  2.50 cm  by
4.00 cm. The rotor rotates in a uniform magnetic field of
0.800 T. When the plane of the rotor is perpendicular to
the direction of the magnetic field, it carries a current of
10.0 mA.  In  this  orientation,  the  magnetic  moment
of the rotor is directed opposite the magnetic field. The
rotor  then  turns  through  one-half  revolution.  This
process is repeated to cause the rotor to turn steadily at
3 600 rev/min. (a) Find the maximum torque acting on
the rotor. (b) Find the peak power output of the motor.
(c)  Determine  the  amount  of  work  performed  by  the
magnetic  field  on  the  rotor  in  every  full  revolution.
(d) What is the average power of the motor?

Section 29.4 Motion of a Charged Particle in a Uniform

Magnetic Field

29. The magnetic field of the Earth at a certain location

is directed vertically downward and has a magnitude of
50.0 +T. A proton is moving horizontally toward the west
in this field with a speed of 6.20 % 10

m/s. (a) What

are the direction and magnitude of the magnetic force
the field exerts on this charge? (b) What is the radius of
the circular arc followed by this proton?

30.

A singly charged positive ion has a mass of 3.20 % 10

kg.

After  being  accelerated  from  rest  through  a  potential
difference  of  833 V,  the  ion  enters  a  magnetic  field  of
0.920 T  along  a  direction  perpendicular  to  the  direction

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