Physics For Scientists And Engineers 6E - part 273

Problems

1089

head.  Assume  that  the  antenna  emits  energy  with
cylindrical  wave  fronts.  (The  actual  radiation  from
antennas  follows a more  complicated  pattern.)  (b)  The
ANSI/IEEE  C95.1-1991  maximum  exposure  standard  is
0.57 mW/cm

for persons living near cellular telephone

base stations, who would be continuously exposed to the
radiation.  Compare  the  answer  to  part  (a)  with  this
standard.
A linearly polarized microwave of wavelength 1.50 cm is
directed  along  the  positive  x axis.  The  electric  field
vector  has  a  maximum  value  of  175 V/m  and  vibrates
in the  xy plane.  (a)  Assume  that  the  magnetic  field
component  of  the  wave  can  be  written  in  the  form
B " B

max

sin(kx ) %t) and give values for B

max

, k, and %.

Also, determine in which plane the magnetic field vector
vibrates. (b) Calculate the average value of the Poynting
vector  for  this  wave.  (c)  What  radiation  pressure
would this  wave  exert  if  it  were  directed  at  normal
incidence  onto  a  perfectly  reflecting  sheet?  (d)  What
acceleration  would  be  imparted  to  a  500-g  sheet
(perfectly  reflecting  and  at  normal  incidence)  with
dimensions of 1.00 m $ 0.750 m?

56.

The Earth reflects approximately 38.0% of the incident
sunlight from its clouds and surface. (a) Given that the
intensity of solar radiation is 1 340 W/m

, what is the radi-

ation  pressure  on  the  Earth,  in  pascals,  at  the  location
where  the  Sun  is  straight  overhead?  (b)  Compare  this  to
normal atmospheric pressure at the Earth’s surface, which
is 101 kPa.
An  astronaut,  stranded  in  space  10.0 m  from  his
spacecraft and at rest relative to it, has a mass (including
equipment)  of  110 kg.  Because  he  has  a  100-W  light
source  that  forms  a  directed  beam,  he  considers  using
the  beam  as  a  photon  rocket  to  propel  himself  continu-
ously  toward  the  spacecraft.  (a)  Calculate  how  long  it
takes  him  to  reach  the  spacecraft  by  this  method.
(b) What  If?  Suppose,  instead,  that  he  decides  to  throw
the  light  source  away  in  a  direction  opposite  the
spacecraft. If the mass of the light source is 3.00 kg and,
after  being  thrown,  it  moves  at  12.0 m/s  relative  to  the
recoiling  astronaut,  how  long  does  it  take  for  the
astronaut to reach the spacecraft?

58.

Review  problem. A  1.00-m-diameter  mirror  focuses  the
Sun’s  rays  onto  an  absorbing  plate  2.00 cm  in  radius,
which  holds  a  can  containing  1.00 L  of  water  at  20.0°C.
(a) If the solar intensity is 1.00 kW/m

, what is the inten-

sity on the absorbing plate? (b) What are the maximum
magnitudes of the fields

E and B? (c) If 40.0% of the

energy is absorbed, how long does it take to bring the
water to its boiling point?

59.

Lasers have been used to suspend spherical glass beads in
the Earth’s gravitational field. (a) A black bead has a mass
of 1.00 &g and a density of 0.200 g/cm

. Determine the

radiation intensity needed to support the bead. (b) If the
beam has a radius of 0.200 cm, what is the power required
for this laser?

60.

Lasers have been used to suspend spherical glass beads in
the Earth’s gravitational field. (a) A black bead has a mass

57.

55.

m and a density 3. Determine the radiation intensity
needed to support the bead. (b) If the beam has a radius r,
what is the power required for this laser?

61.

A  microwave  source  produces  pulses  of  20.0-GHz  radia-
tion,  with  each  pulse  lasting  1.00 ns.  A  parabolic  reflec-
tor  with  a  face  area  of  radius  6.00 cm  is  used  to  focus
the microwaves  into  a  parallel  beam  of  radiation,  as
shown in Figure P34.61. The average power during each
pulse  is  25.0 kW.  (a)  What  is  the  wavelength  of  these
microwaves?  (b)  What  is  the  total  energy  contained
in each  pulse?  (c)  Compute  the  average  energy  density
inside  each  pulse.  (d)  Determine  the  amplitude  of
the electric  and  magnetic  fields  in  these  microwaves.
(e) Assuming  this  pulsed  beam  strikes  an  absorbing
surface,  compute  the  force  exerted  on  the  surface
during the 1.00-ns duration of each pulse.

12.0 cm

Figure P34.61

62.

The electromagnetic power radiated by a nonrelativistic
moving point charge q having an acceleration a is

where #

is the permittivity of free space and c is the speed

of  light  in  vacuum.  (a)  Show  that  the  right  side  of  this
equation has units of watts. (b) An electron is placed in a
constant  electric  field  of  magnitude  100 N/C.  Determine
the  acceleration  of  the  electron  and  the  electromagnetic
power radiated by this electron. (c) What If? If a proton is
placed in a cyclotron with a radius of 0.500 m and a mag-
netic  field  of  magnitude  0.350 T,  what  electromagnetic
power does this proton radiate?

63.

A  thin  tungsten  filament  of  length  1.00 m  radiates
60.0 W of power in the form of electromagnetic waves. A
perfectly absorbing surface in the form of a hollow cylin-
der  of  radius  5.00 cm  and  length  1.00 m  is  placed  con-
centrically  with  the  filament.  Calculate  the  radiation
pressure acting on the cylinder. (Assume that the radia-
tion  is  emitted  in  the  radial  direction,  and  ignore  end
effects.)

64.

The  torsion  balance  shown  in  Figure  34.8  is  used  in
an experiment  to  measure  radiation  pressure.  The  sus-
pension  fiber  exerts  an  elastic  restoring  torque.  Its
torque  constant  is  1.00 $ 10

)

N ! m/degree, and the

length of the horizontal rod is 6.00 cm. The beam from
a 3.00-mW helium–neon laser is incident on the black
disk, and the mirror disk is completely shielded.

" "

6,#

1090

C H A P T E R 3 4 • Electromagnetic Waves

Calculate the angle between the equilibrium positions of
the horizontal bar when the beam is switched from “off”
to “on.”

65.

A  “laser  cannon”  of  a  spacecraft  has  a  beam  of  cross-
sectional area A. The maximum electric field in the beam
is  E.  The  beam  is  aimed  at  an  asteroid  that  is  initially
moving  in  the  direction  of  the  spacecraft.  What  is  the
acceleration of the asteroid relative to the spacecraft if the
laser beam strikes the asteroid perpendicular to its surface,
and the surface is nonreflecting? The mass of the asteroid
is m. Ignore the acceleration of the spacecraft.

66.

A  plane  electromagnetic  wave  varies  sinusoidally  at
90.0 MHz as it travels along the ' x direction. The peak
value of the electric field is 2.00 mV/m, and it is directed
along  the  1 y direction.  (a)  Find  the  wavelength,  the
period,  and  the  maximum  value  of  the  magnetic  field.
(b) Write expressions in SI units for the space and time
variations of the electric field and of the magnetic field.
Include numerical values and include subscripts to indi-
cate  coordinate  directions.  (c)  Find  the  average  power
per  unit  area  that  this  wave  carries  through  space.
(d) Find the average energy density in the radiation (in
joules  per  cubic  meter).  (e)  What  radiation  pressure
would this wave exert upon a perfectly reflecting surface
at normal incidence?

67.

Eliza  is  a  black  cat  with  four  black  kittens:  Penelope,
Rosalita, Sasha, and Timothy. Eliza’s mass is 5.50 kg, and
each  kitten  has  mass  0.800 kg.  One  cool  night  all  five
sleep  snuggled  together  on  a  mat,  with  their  bodies
forming one hemisphere. (a) Assuming that the purring
heap  has  uniform  density  990 kg/m

, find the radius of

the hemisphere. (b) Find the area of its curved surface.
(c) Assume the surface temperature is uniformly 31.0°C
and the emissivity is 0.970. Find the intensity of radiation
emitted  by  the  cats  at  their  curved  surface,  and  (d)  the
radiated  power  from  this  surface.  (e)  You  may  think  of
the emitted electromagnetic wave as having a single pre-
dominant  frequency  (of  31.2 THz).  Find  the  amplitude
of  the  electric  field  just  outside  the  surface  of  the  cozy
pile, and (f) the amplitude of the magnetic field. (g) Are
the  sleeping  cats  charged?  Are  they  current-carrying?
Are they magnetic? Are they a radiation source? Do they
glow  in  the  dark?  Give  an  explanation  for  your  answers
so that they do not seem contradictory. (h) What If? The
next  night  the  kittens  all  sleep  alone,  curling  up  into
separate  hemispheres  like  their  mother.  Find  the  total
radiated  power  of  the  family.  (For  simplicity,  we  ignore
throughout  the  cats’  absorption  of  radiation  from  the
environment.)

68.

Review  problem. (a)  An  elderly  couple  has  a  solar
water heater  installed  on  the  roof  of  their  house  (Fig.
P34.68).  The  heater  consists  of  a  flat  closed  box  with

Note: Section 20.7 introduced electromagnetic radiation
as a mode of energy transfer. The following three
problems use ideas introduced both there and in the
current chapter.

extraordinarily  good  thermal  insulation.  Its  interior  is
painted  black,  and  its  front  face  is  made  of  insulating
glass.  Assume  that  its  emissivity  for  visible  light  is  0.900
and its emissivity for infrared light is 0.700. Assume that
light from the noon Sun is incident perpendicular to the
glass  with  an  intensity  of  1 000 W/m

, and that no

water enters  or  leaves  the  box.  Find  the  steady-state
temperature  of  the  interior  of  the  box.  (b)  What  If?
The couple  builds  an  identical  box  with  no  water
tubes. It  lies  flat  on  the  ground  in  front  of  the  house.
They  use  it  as  a  cold  frame,  where  they  plant  seeds  in
early spring. Assuming the same noon Sun is at an eleva-
tion  angle  of  50.0°,  find  the  steady-state  temperature  of
the  interior  of  this  box  when  its  ventilation  slots  are
tightly closed.

69.

Review problem. The study of Creation suggests a Creator
with an inordinate fondness for beetles and for small red
stars. A small red star radiates electromagnetic waves with
power 6.00 $ 10

W, which is only 0.159% of the lumi-

nosity of the Sun. Consider a spherical planet in a circular
orbit around this star. Assume the emissivity of the planet
is  equal  for  infrared  and  for  visible  light.  Assume  the
planet  has  a  uniform  surface  temperature.  Identify  the
projected area over which the planet absorbs starlight and
the radiating area of the planet. If beetles thrive at a tem-
perature  of  310 K,  what  should  be  the  radius  of  the
planet’s orbit?

Answers to Quick Quizzes

34.1 (c). Figure 34.3b shows that the

B and E vectors

reach their maximum and minimum values at the same
time.

34.2 (c). The

B field must be in the ' z direction in order

that the Poynting vector be directed along the ) y direction.

34.3 (d). The first three choices are instantaneous values and

vary in time. The wave intensity is an average over a full
cycle.

34.4 (b). To maximize the pressure on the sails, they should be

perfectly reflective, so that the pressure is given by Equa-
tion 34.27.

34.5(b), (c). The radiation pressure (a) does not change

because pressure is force per unit area. In (b), the
smaller disk absorbs less radiation, resulting in a smaller

Figure P34.68

Bill Banaszewski/V

isuals Unlimited

Answers to Quick Quizzes

1091

force. For the same reason, the momentum in (c) is
reduced.

34.6 (a), (b) " (c), (d). The closest point along the x axis in

Figure 34.11 (choice a) will represent the highest inten-
sity. Choices (b) and (c) correspond to points equidistant
in different directions. Choice (d) is along the axis of the
antenna and the intensity is zero.

34.7 (a). The best orientation is parallel to the transmitting

antenna because that is the orientation of the electric

field. The electric field moves electrons in the receiving
antenna, thus inducing a current that is detected and
amplified.

34.8 (c). Either Equation 34.13 or Figure 34.12 can be used to

find the order of magnitude of the wavelengths.

34.9 (a). Either Equation 34.13 or Figure 34.12 can be used to

find the order of magnitude of the wavelength.

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