Physics For Scientists And Engineers 6E - part 241

Problems

961

33.

A  single-turn  square  loop  of  wire,  2.00 cm  on  each  edge,
carries a clockwise current of 0.200 A. The loop is inside a
solenoid,  with  the  plane  of  the  loop  perpendicular  to
the magnetic  field  of  the  solenoid.  The  solenoid  has
30 turns/cm  and  carries  a  clockwise  current  of  15.0 A.
Find  the  force  on  each  side  of  the  loop  and  the  torque
acting on the loop.

Section 30.5 Magnetic Flux

34. Consider the hemispherical closed surface in Figure

P30.34. The hemisphere is in a uniform magnetic field
that makes an angle ! with the vertical. Calculate the
magnetic flux through (a) the flat surface S

and (b) the

hemispherical surface S

Figure P30.35

Figure P30.36

Figure P30.34

(b)

1.25 cm

(a)

A cube of edge length ! " 2.50 cm is positioned as shown
in Figure P30.35. A uniform magnetic field given
by

B " (5ˆi * 4ˆj * 3ˆk)T exists throughout the region.

(a) Calculate the flux through the shaded face. (b) What is
the total flux through the six faces?

35.

36.

A  solenoid  2.50 cm  in  diameter  and  30.0 cm  long  has
300 turns  and  carries  12.0 A.  (a)  Calculate  the  flux
through  the  surface  of  a  disk  of  radius  5.00 cm  that  is
positioned  perpendicular  to  and  centered  on  the  axis  of

the solenoid, as shown in Figure P30.36a. (b) Figure
P30.36b shows an enlarged end view of the same solenoid.
Calculate the flux through the blue area, which is defined
by an annulus that has an inner radius of 0.400 cm and
outer radius of 0.800 cm.

Section 30.7 Displacement Current and the General

Form of Ampère’s Law

A 0.100-A current is charging a capacitor that has square

plates 5.00 cm on each side. The plate separation is
4.00 mm. Find (a) the time rate of change of electric flux
between the plates and (b) the displacement current
between the plates.

38.

A 0.200-A current is charging a capacitor that has circular
plates 10.0 cm in radius. If the plate separation is
4.00 mm, (a) what is the time rate of increase of electric
field between the plates? (b) What is the magnetic field
between the plates 5.00 cm from the center?

Section 30.8 Magnetism in Matter

39.

In Bohr’s 1913 model of the hydrogen atom, the electron is
in a circular orbit of radius 5.29 & 10

m and its speed is

2.19 & 10

m/s. (a) What is the magnitude of the magnetic

moment due to the electron’s motion? (b) If the electron
moves in a horizontal circle, counterclockwise as seen from
above, what is the direction of this magnetic moment vector?

40.

A magnetic field of 1.30 T is to be set up in an iron-core
toroid. The toroid has a mean radius of 10.0 cm, and
magnetic permeability of 5 000 #

. What current is

required if the winding has 470 turns of wire? The
thickness of the iron ring is small compared to 10 cm, so
the field in the material is nearly uniform.

A toroid with a mean radius of 20.0 cm and 630 turns (see

Fig. 30.30) is filled with powdered steel whose magnetic

41.

37.

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CHAPTE R 3 0 • Sources of the Magnetic Field

susceptibility 8 is 100. The current in the windings is
3.00 A. Find B (assumed uniform) inside the toroid.

42.

A  particular  paramagnetic  substance  achieves  10.0%  of
its saturation  magnetization  when  placed  in  a  magnetic
field  of  5.00 T  at  a  temperature  of  4.00 K.  The  density  of
magnetic atoms in the sample is 8.00 & 10

atoms/m

, and

the magnetic moment per atom is 5.00 Bohr magnetons.
Calculate the Curie constant for this substance.

43.

Calculate the magnetic field strength H of a magnetized
substance in which the magnetization is 0.880 & 10

A/m

and the magnetic field has magnitude 4.40 T.

44.

At saturation, when nearly all of the atoms have their mag-
netic  moments  aligned,  the  magnetic  field  in  a  sample
of iron  can  be  2.00 T. If  each  electron  contributes  a
magnetic  moment  of  9.27 & 10

A ( m

(one Bohr

magneton), how many electrons per atom contribute to
the saturated field of iron? Iron contains approximately
8.50 & 10

atoms/m

45.

(a)  Show  that  Curie’s  law  can  be  stated  in  the  following
way:  The  magnetic  susceptibility  of  a  paramagnetic
substance is inversely proportional to the absolute temper-
ature,  according  to  8 " C#

/T, where C is Curie’s

constant. (b) Evaluate Curie’s constant for chromium.

Section 30.9 The Magnetic Field of the Earth

46.

A circular coil of 5 turns and a diameter of 30.0 cm is ori-
ented in a vertical plane with its axis perpendicular to the
horizontal  component  of  the  Earth’s  magnetic  field.  A
horizontal compass placed at the center of the coil is made
to  deflect  45.0° from  magnetic  north  by  a  current  of
0.600 A in the coil. (a) What is the horizontal component
of the Earth’s magnetic field? (b) The current in the coil is
switched  off.  A  “dip  needle”  is  a  magnetic  compass
mounted  so  that  it  can  rotate  in  a  vertical  north–south
plane.  At  this  location  a  dip  needle  makes  an  angle  of
13.0° from the vertical. What is the total magnitude of the
Earth’s magnetic field at this location?

The magnetic moment of the Earth is approximately
8.00 & 10

A ( m

. (a) If this were caused by the complete

magnetization of a huge iron deposit, how many unpaired
electrons would this correspond to? (b) At two unpaired
electrons per iron atom, how many kilograms of iron would
this correspond to? (Iron has a density of 7 900 kg/m

, and

approximately 8.50 & 10

iron atoms/m

Additional Problems

48. The magnitude of the Earth’s magnetic field at either pole

is approximately 7.00 & 10

T. Suppose that the field

fades  away,  before  its  next  reversal.  Scouts,  sailors,  and
conservative politicians around the world join together in
a  program  to  replace  the  field.  One  plan  is  to  use  a
current  loop  around  the  equator,  without  relying  on
magnetization  of  any  materials  inside  the  Earth.
Determine the current that would generate such a field if
this plan were carried out. (Take the radius of the Earth as
R

6.37 & 10

m.)

A very long, thin strip of metal of width w carries a current
I along its length as shown in Figure P30.49. Find the

49.

47.

magnetic field at the point P in the diagram. The point P
is in the plane of the strip at distance b away from it.

Figure P30.49

Figure P30.52

50.

Suppose  you  install  a  compass  on  the  center  of  the
dashboard  of  a  car.  Compute  an  order-of-magnitude
estimate  for  the  magnetic  field  at  this  location  produced
by  the  current  when  you  switch  on  the  headlights.  How
does it compare with the Earth’s magnetic field? You may
suppose the dashboard is made mostly of plastic.

51.

For  a  research  project,  a  student  needs  a  solenoid  that
produces  an  interior  magnetic  field  of  0.030 0 T.  She
decides to use a current of 1.00 A and a wire 0.500 mm in
diameter. She winds the solenoid in layers on an insulating
form  1.00 cm  in  diameter  and  10.0 cm  long.  Determine
the number of layers of wire needed and the total length
of the wire.

52.

A thin copper bar of length ! " 10.0 cm is supported hori-
zontally by two (nonmagnetic) contacts. The bar carries
current I

100 A in the ' x direction, as shown in

Figure P30.52. At a distance h " 0.500 cm below one end
of the bar, a long straight wire carries a current I

200 A

in the z direction. Determine the magnetic force exerted
on the bar.

A nonconducting ring of radius 10.0 cm is uniformly

charged with a total positive charge 10.0 #C. The ring
rotates at a constant angular speed 20.0 rad/s about an

53.

Problems

963

Figure P30.55 Problems 55 and 56.

Figure P30.57 (a) General view of one turn of each saddle coil.

(b) End view of the coils carrying current into the paper on the

left and out of the paper on the right.

× × ×

× ×

× × ×

(b)

(a)

axis through its center, perpendicular to the plane of the
ring. What is the magnitude of the magnetic field on the
axis of the ring 5.00 cm from its center?
A nonconducting ring of radius R is uniformly charged
with a total positive charge q. The ring rotates at a constant
angular speed 4 about an axis through its center, perpen-
dicular to the plane of the ring. What is the magnitude of
the magnetic field on the axis of the ring a distance R/2
from its center?

55.

Two  circular  coils  of  radius  R,  each  with  N turns,  are
perpendicular  to  a  common  axis.  The  coil  centers  are  a
distance R apart. Each coil carries a steady current I in the
same direction, as shown in Figure P30.55. (a) Show that
the  magnetic  field  on  the  axis  at  a  distance  x from  the
center of one coil is

(b) Show that dB/dx and d

B/dx

are both zero at the

point midway between the coils. This means the magnetic
field in the region midway between the coils is uniform.
Coils in this configuration are called Helmholtz coils.

B "

)

3/2

(2R

2Rx)

3/2

(

54.

58.

A very large parallel-plate capacitor carries charge with
uniform charge per unit area * / on the upper plate and
'

on the lower plate. The plates are horizontal and both

move  horizontally  with  speed  v to  the  right.  (a)  What
is the  magnetic  field  between  the  plates?  (b)  What  is
the magnetic  field  close  to  the  plates  but  outside  of  the
capacitor? (c) What is the magnitude and direction of the
magnetic  force  per  unit  area  on  the  upper  plate?  (d)  At
what  extrapolated  speed  v will  the  magnetic  force  on  a
plate balance the electric force on the plate? Calculate this
speed numerically.

59.

Two  circular  loops  are  parallel,  coaxial,  and  almost  in
contact, 1.00 mm apart (Fig. P30.59). Each loop is 10.0 cm
in  radius.  The  top  loop  carries  a  clockwise  current  of
140 A. The bottom loop carries a counterclockwise current
of 140 A. (a) Calculate the magnetic force exerted by the
bottom  loop  on  the  top  loop.  (b)  The  upper  loop  has  a
mass  of  0.021 0 kg.  Calculate  its  acceleration,  assuming
that  the  only  forces  acting  on  it  are  the  force  in  part
(a) and  the  gravitational  force.  Suggestion: Think  about
how one loop looks to a bug perched on the other loop.

60.

What  objects  experience  a  force  in  an  electric  field?
Chapter 23 gives the answer: any electric charge, stationary
or  moving,  other  than  the  charge  that  created  the
field. What  creates  an  electric  field?  Any  electric  charge,
stationary or moving, as you studied in Chapter 23. What
objects experience a force in a magnetic field? An electric
current  or  a  moving  electric  charge,  other  than  the
current  or  charge  that  created  the  field,  as  discussed  in
Chapter  29.  What  creates  a  magnetic  field?  An  electric
current,  as  you  studied  in  Section  30.1,  or  a  moving
electric  charge,  as  shown  in  this  problem.  (a)  To  display
how a moving charge creates a magnetic field, consider a
charge q moving with velocity

v. Define the vector r " r ˆr

56.

Two  identical,  flat,  circular  coils  of  wire  each  have
100 turns and a radius of 0.500 m. The coils are arranged
as a set of Helmholtz coils (see Fig. P30.55), parallel and
with  separation  0.500 m.  Each  coil  carries  a  current  of
10.0 A. Determine the magnitude of the magnetic field at
a  point  on  the  common  axis  of  the  coils  and  halfway
between them.

57.

We  have  seen  that  a  long  solenoid  produces  a  uniform
magnetic  field  directed  along  the  axis  of  a  cylindrical
region.  However,  to  produce  a  uniform  magnetic  field
directed parallel to a diameter of a cylindrical region, one
can  use  the  saddle  coils illustrated  in  Figure  P30.57.  The
loops  are  wrapped  over  a  somewhat  flattened  tube.
Assume the straight sections of wire are very long. The end
view of the tube shows how the windings are applied. The
overall  current  distribution  is  the  superposition  of  two
overlapping  circular  cylinders  of  uniformly  distributed
current,  one  toward  you  and  one  away  from  you.  The
current  density  J is  the  same  for  each  cylinder.  The
position  of  the  axis  of  one  cylinder  is  described  by  a
position vector

a relative to the other cylinder. Prove that

the magnetic field inside the hollow tube is #

Ja/2

downward. Suggestion: The use of vector methods
simplifies the calculation.

140 A

Figure P30.59

964

CHAPTE R 3 0 • Sources of the Magnetic Field

Figure P30.61

Figure P30.63

Figure P30.64

Figure P30.65

to lead from the charge to some location. Show that the
magnetic field at that location is

(b) Find the magnitude of the magnetic field 1.00 mm to
the side of a proton moving at 2.00 & 10

m/s. (c) Find

the magnetic force on a second proton at this point,
moving with the same speed in the opposite direction.
(d) Find the electric force on the second proton.

61.

Rail  guns have  been  suggested  for  launching  projectiles
into space without chemical rockets, and for ground-to-air
antimissile  weapons  of  war.  A  tabletop  model  rail  gun
(Fig. P30.61) consists of two long parallel horizontal rails
3.50 cm apart, bridged by a bar BD of mass 3.00 g. The bar
is originally at rest at the midpoint of the rails and is free
to slide without friction. When the switch is closed, electric
current is quickly established in the circuit ABCDEA. The
rails and bar have low electric resistance, and the current
is  limited  to  a  constant  24.0 A  by  the  power  supply.
(a) Find  the  magnitude  of  the  magnetic  field  1.75 cm
from  a  single  very  long  straight  wire  carrying  current
24.0 A.  (b)  Find  the  magnitude  and  direction  of  the
magnetic field at point C in the diagram, the midpoint of
the bar, immediately after the switch is closed. Suggestion:
Consider  what  conclusions  you  can  draw  from  the
Biot–Savart law. (c) At other points along the bar BD, the
field  is  in  the  same  direction  as  at  point  C,  but  larger  in
magnitude.  Assume  that  the  average  effective  magnetic
field along BD is five times larger than the field at C. With
this assumption, find the magnitude and direction of the
force on the bar. (d) Find the acceleration of the bar when
it  is  in  motion.  (e)  Does  the  bar  move  with  constant
acceleration?  (f)  Find  the  velocity  of  the  bar  after  it  has
traveled 130 cm to the end of the rails.

B "

v & ˆr

62.

Fifty turns of insulated wire 0.100 cm in diameter

are tightly wound to form a flat spiral. The spiral  fills a
disk surrounding a circle of radius 5.00 cm and extend-
ing  to  a  radius  10.00 cm  at  the  outer  edge.  Assume  the
wire  carries  current  I at  the  center  of  its  cross  section.
Approximate  each  turn  of  wire  as  a  circle.  Then  a  loop
of  current  exists  at  radius  5.05 cm,  another  at  5.15 cm,
and  so  on.  Numerically  calculate  the  magnetic  field  at
the center of the coil.

63.

Two long, parallel conductors carry currents in the same
direction as shown in Figure P30.63. Conductor A carries a
current of 150 A and is held firmly in position. Conductor
B carries a current I

and is allowed to slide freely up and

down (parallel to A) between a set of nonconducting
guides. If the mass per unit length of conductor B

is 0.100 g/cm, what value of current I

will result in

equilibrium when the distance between the two conduc-
tors is 2.50 cm?

64.

Charge is sprayed onto a large nonconducting belt above
the left-hand roller in Figure P30.64. The belt carries the
charge with a uniform surface charge density / as it moves
with a speed v between the rollers as shown. The charge is
removed by a wiper at the right-hand roller. Consider a
point just above the surface of the moving belt. (a) Find
an expression for the magnitude of the magnetic field

at this point. (b) If the belt is positively charged, what is
the direction of

B? (Note that the belt may be considered

as an infinite sheet.)

65.

An infinitely long straight wire carrying a current I

partially surrounded by a loop as shown in Figure P30.65.

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