Physics For Scientists And Engineers 6E - part 214

19. (a) Make an order-of-magnitude estimate of the resis-

tance  between  the  ends  of  a  rubber  band.  (b)  Make  an
order-of-magnitude  estimate  of  the  resistance  between
the ‘heads’ and ‘tails’ sides of a penny. In each case state
what  quantities  you  take  as  data  and  the  values  you
measure or estimate for them. (c) WARNING! Do not try
this  at  home!  What  is  the  order  of  magnitude  of  the
current that each would carry if it were connected across
a 120-V power supply?

20.

A solid cube of silver (density " 10.5 g/cm

) has a mass of

90.0 g. (a) What is the resistance between opposite faces of
the  cube?  (b)  Assume  each  silver  atom  contributes  one
conduction  electron.  Find  the  average  drift  speed  of
electrons  when  a  potential  difference  of  1.00 # 10

V is

applied to opposite faces. The atomic number of silver is
47, and its molar mass is 107.87 g/mol.

21. A metal wire of resistance R is cut into three equal pieces

that are then connected side by side to form a new wire
the length of which is equal to one-third the original
length. What is the resistance of this new wire?

22.

Aluminum and copper wires of equal length are found to
have the same resistance. What is the ratio of their radii?

23. A current density of 6.00 # 10

A/m

exists in the at-

mosphere at a location where the electric field is 100 V/m.
Calculate the electrical conductivity of the Earth’s atmos-
phere in this region.

24.

The rod in Figure P27.24 is made of two materials. The fig-
ure  is  not  drawn  to  scale.  Each  conductor  has  a  square
cross  section  3.00 mm  on  a  side.  The  first  material  has  a
resistivity  of  4.00 # 10

' (

m and is 25.0 cm long, while

the second material has a resistivity of 6.00 # 10

' (

and is 40.0 cm long. What is the resistance between the
ends of the rod?

same voltage to the same wire. What current does he
register there if the temperature is $ 88.0°C? Assume that
no change occurs in the wire’s shape and size.

29. A certain lightbulb has a tungsten filament with a

resistance  of  19.0 ' when  cold  and  140 ' when  hot.
Assume that the resistivity of tungsten varies linearly with
temperature  even  over  the  large  temperature  range
involved  here,  and  find  the  temperature  of  the  hot
filament. Assume the initial temperature is 20.0°C.

30.

A  carbon  wire  and  a  Nichrome  wire  are  connected  in
series, so that the same current exists in both wires. If the
combination has a resistance of 10.0 k' at 0°C, what is the
resistance  of  each  wire  at  0°C  so  that  the  resistance  of
the combination does not change with temperature? The
total  or  equivalent  resistance  of  resistors  in  series  is
the sum of their individual resistances.

An  aluminum  wire  with  a  diameter  of  0.100 mm  has  a
uniform  electric  field  of  0.200 V/m  imposed  along  its
entire  length.  The  temperature  of  the  wire  is  50.0°C.
Assume one free electron per atom. (a) Use the informa-
tion in Table 27.1 and determine the resistivity. (b) What is
the  current  density  in  the  wire?  (c)  What  is  the  total
current  in  the  wire?  (d)  What  is  the  drift  speed  of  the
conduction electrons? (e) What potential difference must
exist  between  the  ends  of  a  2.00-m  length  of  the  wire  to
produce the stated electric field?

32.

Review problem. An aluminum rod has a resistance of
1.234 ' at 20.0°C. Calculate the resistance of the rod at
120°C by accounting for the changes in both the resistivity
and the dimensions of the rod.

What is the fractional change in the resistance of an iron
filament when its temperature changes from 25.0°C to
50.0°C?

34.

The  resistance  of  a  platinum  wire  is  to  be  calibrated  for
low-temperature  measurements.  A  platinum  wire  with
resistance 1.00 ' at 20.0°C is immersed in liquid nitrogen
at 77 K ($ 196°C). If the temperature response of the plat-
inum wire is linear, what is the expected resistance of the
platinum  wire  at  $ 196°C?  (-

platinum

3.92 # 10

/°C)

35.

The temperature of a sample of tungsten is raised while a
sample of copper is maintained at 20.0°C. At what temper-
ature will the resistivity of the tungsten be four times that
of the copper?

Section 27.6 Electrical Power

36. A toaster is rated at 600 W when connected to a 120-V

source. What current does the toaster carry, and what is its
resistance?

37. A Van de Graaff generator (see Figure 25.29) is operating

so  that  the  potential  difference  between  the  high-voltage
electrode  B  and  the  charging  needles  at  A  is  15.0 kV.
Calculate  the  power  required  to  drive  the  belt  against
electrical  forces  at  an  instant  when  the  effective  current
delivered to the high-voltage electrode is 500 /A.

38. In a hydroelectric installation, a turbine delivers 1 500 hp

to a generator, which in turn transfers 80.0% of the
mechanical energy out by electrical transmission. Under

33.

31.

Problems

853

Figure P27.24

25.0 cm

40.0 cm

Section 27.3 A Model for Electrical Conduction

If the magnitude of the drift velocity of free electrons

in a copper wire is 7.84 # 10

m/s, what is the electric

field in the conductor?

26. If the current carried by a conductor is doubled, what

happens to the (a) charge carrier density? (b) current
density? (c) electron drift velocity? (d) average time
interval between collisions?

27.

Use data from Example 27.1 to calculate the collision
mean free path of electrons in copper. Assume the average
thermal speed of conduction electrons is 8.60 # 10

m/s.

Section 27.4 Resistance and Temperature

28. While taking photographs in Death Valley on a day when

the  temperature  is  58.0°C,  Bill  Hiker  finds  that  a  certain
voltage  applied  to  a  copper  wire  produces  a  current  of
1.000 A.  Bill  then  travels  to  Antarctica  and  applies  the

25.

these conditions, what current does the generator deliver
at a terminal potential difference of 2 000 V?

What is the required resistance of an immersion heater

that increases the temperature of 1.50 kg of water from
10.0°C to 50.0°C in 10.0 min while operating at 110 V?

40. One rechargeable battery of mass 15.0 g delivers to a CD

player  an  average  current  of  18.0 mA  at  1.60 V  for  2.40 h
before  the  battery  needs  to  be  recharged.  The  recharger
maintains a potential difference of 2.30 V across the battery
and  delivers  a  charging  current  of  13.5 mA  for  4.20 h.
(a) What is the efficiency of the battery as an energy storage
device?  (b)  How  much  internal  energy  is  produced  in  the
battery during one charge–discharge cycle? (b) If the battery
is surrounded by ideal thermal insulation and has an overall
effective  specific  heat  of  975 J/kg°C,  by  how  much  will  its
temperature increase during the cycle?

Suppose that a voltage surge produces 140 V for a
moment. By what percentage does the power output of a
120-V, 100-W lightbulb increase? Assume that its resistance
does not change.

42.

A 500-W heating coil designed to operate from 110 V is
made of Nichrome wire 0.500 mm in diameter. (a) Assum-
ing that the resistivity of the Nichrome remains constant at
its 20.0°C value, find the length of wire used. (b) What If?
Now consider the variation of resistivity with temperature.
What power will the coil of part (a) actually deliver when it
is heated to 1 200°C?

43. A coil of Nichrome wire is 25.0 m long. The wire has a

diameter  of  0.400 mm  and  is  at  20.0°C.  If  it  carries  a
current  of  0.500 A,  what  are  (a)  the  magnitude  of  the
electric field in the wire, and (b) the power delivered to it?
(c) What If? If the temperature is increased to 340°C and
the  voltage  across  the  wire  remains  constant,  what  is  the
power delivered?

44. Batteries are rated in terms of ampere-hours (A ( h). For

example,  a  battery  that  can  produce  a  current  of  2.00 A
for  3.00 h  is  rated  at  6.00 A ( h.  (a)  What  is  the  total
energy,  in  kilowatt-hours,  stored  in  a  12.0-V  battery  rated
at 55.0 A ( h? (b) At $0.060 0 per kilowatt-hour, what is the
value of the electricity produced by this battery?

45. A 10.0-V battery is connected to a 120-' resistor. Ignoring

the internal resistance of the battery, calculate the power
delivered to the resistor.

46. Residential building codes typically require the use of

12-gauge  copper  wire  (diameter  0.205 3 cm)  for  wiring
receptacles.  Such  circuits  carry  currents  as  large  as  20 A.
A wire of smaller diameter (with a higher gauge number)
could carry this much current, but the wire could rise to a
high temperature and cause a fire. (a) Calculate the rate
at which internal energy is produced in 1.00 m of 12-gauge
copper  wire  carrying  a  current  of  20.0 A.  (b)  What  If?
Repeat  the  calculation  for  an  aluminum  wire.  Would  a
12-gauge aluminum wire be as safe as a copper wire?

47. An 11.0-W energy-efficient fluorescent lamp is designed to

produce the same illumination as a conventional 40.0-W
incandescent lightbulb. How much money does the user
of the energy-efficient lamp save during 100 hours of use?
Assume a cost of $0.080 0/kWh for energy from the power
company.

41.

39.

48.

We estimate that 270 million plug-in electric clocks are in
the  United  States,  approximately  one  clock  for  each
person.  The  clocks  convert  energy  at  the  average  rate
2.50 W.  To  supply  this  energy,  how  many  metric  tons  of
coal are burned per hour in coal-fired electric generating
plants  that  are,  on  average,  25.0%  efficient?  The  heat  of
combustion for coal is 33.0 MJ/kg.

Compute the cost per day of operating a lamp that draws a
current of 1.70 A from a 110-V line. Assume the cost of
energy from the power company is $0.060 0/kWh.

50.

Review  problem.  The  heating  element  of  a  coffee  maker
operates  at  120 V  and  carries  a  current  of  2.00 A.
Assuming  that  the  water  absorbs  all  of  the  energy  deliv-
ered to the resistor, calculate how long it takes to raise the
temperature of 0.500 kg of water from room temperature
(23.0°C) to the boiling point.

A certain toaster has a heating element made of Nichrome
wire. When the toaster is first connected to a 120-V source
(and  the  wire  is  at  a  temperature  of  20.0°C),  the  initial
current is 1.80 A. However, the current begins to decrease
as  the  heating  element  warms  up.  When  the  toaster
reaches its final operating temperature, the current drops
to 1.53 A. (a) Find the power delivered to the toaster when
it  is  at  its  operating  temperature.  (b)  What  is  the  final
temperature of the heating element?

52. The cost of electricity varies widely through the United

States;  $0.120/kWh  is  one  typical  value.  At  this  unit
price,  calculate  the  cost  of  (a)  leaving  a  40.0-W  porch
light  on  for  two  weeks  while  you  are  on  vacation,
(b) making  a  piece  of  dark  toast  in  3.00 min  with  a
970-W  toaster,  and  (c)  drying  a  load  of  clothes  in
40.0 min in a 5 200-W dryer.

53.

Make  an  order-of-magnitude  estimate  of  the  cost  of  one
person’s  routine  use  of  a  hair  dryer  for  1 yr.  If  you  do
not  use  a  blow  dryer  yourself,  observe  or  interview
someone who does. State the quantities you estimate and
their values.

Additional Problems

54. One lightbulb is marked ‘25 W 120 V,’ and another ‘100 W

120 V’; this means that each bulb has its respective power
delivered  to  it  when  plugged  into  a  constant  120-V
potential difference. (a) Find the resistance of each bulb.
(b) How long does it take for 1.00 C to pass through the
dim bulb? Is the charge different in any way upon its exit
from the bulb versus its entry? (c) How long does it take
for  1.00 J  to  pass  through  the  dim  bulb?  By  what mecha-
nisms does this energy enter and exit the bulb? (d) Find
how  much  it  costs  to  run  the  dim  bulb  continuously  for
30.0  days  if  the  electric  company  sells  its  product  at
$0.070 0 per kWh. What product does the electric company
sell? What is its price for one SI unit of this quantity?

55.

A charge Q is placed on a capacitor of capacitance C. The
capacitor is connected into the circuit shown in Figure
P27.55, with an open switch, a resistor, and an initially un-
charged capacitor of capacitance 3C. The switch is then
closed and the circuit comes to equilibrium. In terms of Q
and C, find (a) the final potential difference between the
plates of each capacitor, (b) the charge on each capacitor,

51.

49.

854

CHAPTE R 27 • Current and Resistance

Problems

855

and (c) the final energy stored in each capacitor. (d) Find
the internal energy appearing in the resistor.

Figure P27.55

56.

A high-voltage transmission line with a diameter of 2.00 cm
and a length of 200 km carries a steady current of 1 000 A.
If the conductor is copper wire with a free charge density
of 8.49 # 10

electrons/m

, how long does it take one

electron to travel the full length of the line?

A more general definition of the temperature coefficient
of resistivity is

where % is the resistivity at temperature T. (a) Assuming
that - is constant, show that

where %

is the resistivity at temperature T

. (b) Using

the series expansion e

' 1 , x for x 22 1, show that

the resistivity is given approximately by the expression
% " %

[1 , -(T $ T

)] for -(T $ T

) 22 1.

58. A high-voltage transmission line carries 1 000 A starting at

700 kV for a distance of 100 mi. If the resistance in the
wire is 0.500 '/mi, what is the power loss due to resistive
losses?

An experiment is conducted to measure the

electrical resistivity of Nichrome in the form of wires with
different  lengths  and  cross-sectional  areas.  For  one  set  of
measurements,  a  student  uses  30-gauge  wire,  which  has  a
cross-sectional  area  of  7.30 # 10

. The student

measures the potential difference across the wire and the
current  in  the  wire  with  a  voltmeter  and  an  ammeter,
respectively.  For  each  of  the  measurements  given  in  the
table  taken  on  wires  of  three  different  lengths,  calculate
the resistance of the wires and the corresponding values of
the  resistivity.  What  is  the  average  value  of  the  resistivity,
and  how  does  this  value  compare  with  the  value  given  in
Table 27.1?

L (m)

V (V)

I (A)

R (%)

(% & m)

0.540

5.22

0.500

1.028

5.82

0.276

1.543

5.94

0.187

60.

An electric utility company supplies a customer’s house
from the main power lines (120 V) with two copper wires,
each of which is 50.0 m long and has a resistance of
0.108 ' per 300 m. (a) Find the voltage at the customer’s

59.

% " %

(T$T

)

- "

1
%

57.

house for a load current of 110 A. For this load current,
find (b) the power the customer is receiving and (c) the
electric power lost in the copper wires.

61. A straight cylindrical wire lying along the x axis has a

length of 0.500 m and a diameter of 0.200 mm. It is made
of a material that obeys Ohm’s law with a resistivity of % "
4.00 # 10

' (

m. Assume that a potential of 4.00 V is

maintained at x " 0, and that V " 0 at x " 0.500 m. Find
(a) the electric field E in the wire, (b) the resistance of the
wire, (c) the electric current in the wire, and (d) the
current density J in the wire. Express vectors in vector
notation. (e) Show that E " %J.

62.

A  straight  cylindrical  wire  lying  along  the  x axis  has  a
length  L and  a  diameter  d.  It  is  made  of  a  material  that
obeys Ohm’s law with a resistivity %. Assume that potential
V is maintained at x " 0, and that the potential is zero at
x " L. In terms of L, d, V, %, and physical constants, derive
expressions  for  (a)  the  electric  field  in  the  wire,  (b) the
resistance of the wire, (c) the electric current in the wire,
and (d) the current density in the wire. Express vectors in
vector notation. (e) Prove that E " %J.

63.

The  potential  difference  across  the  filament  of  a  lamp  is
maintained at a constant level while equilibrium tempera-
ture  is  being  reached.  It  is  observed  that  the  steady-state
current in the lamp is only one tenth of the current drawn
by the lamp when it is first turned on. If the temperature
coefficient  of  resistivity  for  the  lamp  at  20.0°C  is
0.004 50 (°C)

, and if the resistance increases linearly

with increasing temperature, what is the final operating
temperature of the filament?

64.

The current in a resistor decreases by 3.00 A when the
voltage applied across the resistor decreases from 12.0 V to
6.00 V. Find the resistance of the resistor.

An electric car is designed to run off a bank of 12.0-V bat-
teries with total energy storage of 2.00 # 10

J. (a) If the

electric motor draws 8.00 kW, what is the current delivered
to the motor? (b) If the electric motor draws 8.00 kW as
the car moves at a steady speed of 20.0 m/s, how far will
the car travel before it is “out of juice”?

66.

Review problem. When a straight wire is heated, its
resistance is given by R " R

[1 , -(T $ T

)] according to

Equation 27.21, where - is the temperature coefficient of
resistivity. (a) Show that a more precise result, one that
includes the fact that the length and area of the wire
change when heated, is

where  -3 is  the  coefficient  of  linear  expansion  (see
Chapter  19).  (b)  Compare  these  two  results  for  a
2.00-m-long  copper  wire  of  radius  0.100 mm,  first  at
20.0°C and then heated to 100.0°C.

67.

The temperature coefficients of resistivity in Table 27.1 were
determined at a temperature of 20°C. What would they be at
0°C? Note that the temperature coefficient of resistivity at
20°C satisfies % " %

[1 , -(T $ T

)], where %

is the

resistivity of the material at T

20°C. The temperature

R "

[1 , -(T $ T

)][1 , -3(T $ T

)]

[1 , 2-3(T $ T

)]

65.

coefficient of resistivity -3 at 0°C must satisfy the expression
% " %3

[1 , -3T ], where %3

is the resistivity of the material

at 0°C.

68.

An oceanographer is studying how the ion concentration
in sea water depends on depth. She does this by lowering
into  the  water  a  pair  of  concentric  metallic  cylinders
(Fig. P27.68)  at  the  end  of  a  cable  and  taking  data  to
determine  the  resistance  between  these  electrodes  as  a
function  of  depth.  The  water  between  the  two  cylinders
forms a cylindrical shell of inner radius r

, outer radius r

and length L much larger than r

. The scientist applies a

potential  difference  !V between  the  inner  and  outer
surfaces,  producing  an  outward  radial  current  I.  Let  %
represent  the  resistivity  of  the  water.  (a)  Find  the
resistance of the water between the cylinders in terms of L,
%

, r

, and r

. (b) Express the resistivity of the water in

terms of the measured quantities L, r

, r

, !V, and I.

Material with uniform resistivity % is formed into a wedge
as shown in Figure P27.71. Show that the resistance
between face A and face B of this wedge is

R " %

w(y

)

71.

856

CHAPTE R 27 • Current and Resistance

Figure P27.71

Figure P27.72

Face A

Face B

69. In a certain stereo system, each speaker has a resistance of

4.00 '. The system is rated at 60.0 W in each channel, and
each speaker circuit includes a fuse rated 4.00 A. Is this
system adequately protected against overload? Explain
your reasoning.

70.

A close analogy exists between the flow of energy by heat
because  of  a  temperature  difference  (see  Section  20.7)
and  the  flow  of  electric  charge  because  of  a  potential
difference. The energy dQ and the electric charge dq can
both  be  transported  by  free  electrons  in  the  conducting
material.  Consequently,  a  good  electrical  conductor  is
usually a good thermal conductor as well. Consider a thin
conducting  slab  of  thickness  dx, area  A, and  electrical
conductivity  &, with  a  potential  difference  dV between
opposite faces. Show that the current I " dq/dt is given by
the equation on the left below:

Thermal Conduction

Charge Conduction

(Eq. 20.14)

In  the  analogous  thermal  conduction  equation  on  the
right, the rate of energy flow dQ /dt (in SI units of joules
per second) is due to a temperature gradient dT/dx, in a
material  of  thermal  conductivity  k. State  analogous  rules
relating the direction of the electric current to the change
in  potential,  and  relating  the  direction  of  energy  flow  to
the change in temperature.

(

Figure P27.68

72.

A  material  of  resistivity  % is  formed  into  the  shape  of  a
truncated  cone  of  altitude  h as  shown  in  Figure  P27.72.
The bottom end has radius b, and the top end has radius
a. Assume  that  the  current  is  distributed  uniformly  over
any circular cross section of the cone, so that the current
density does not depend on radial position. (The current
density does vary with position along the axis of the cone.)
Show that the resistance between the two ends is described
by the expression

R "

73.

The  dielectric  material  between  the  plates  of  a  parallel-
plate  capacitor  always  has  some  nonzero  conductivity  &.
Let A represent the area of each plate and d the distance
between  them.  Let  4 represent  the  dielectric  constant  of
the  material.  (a)  Show  that  the  resistance  R and  the
capacitance C of the capacitor are related by

(b) Find the resistance between the plates of a 14.0-nF
capacitor with a fused quartz dielectric.

74.

The current–voltage characteristic curve for a semi-

conductor diode as a function of temperature T is given by
the equation

I " I

∆V/k

RC "

Content .. 212 213 214 215 ..