• Resistance has units of ohms ( Ω ), related to volts and amperes by 1 Ω = 1 V/A .
• There is a voltage or IR drop across a resistor, caused by the current flowing through it, given by V = IR .
20.3 Resistance and Resistivity
724 CHAPTER 20 | ELECTRIC CURRENT, RESISTANCE, AND OHM'S LAW
• The resistance R of a cylinder of length L and cross-sectional area A is R = ρL
A , where ρ is the resistivity of the material.
• Values of ρ in Table 20.1 show that materials fall into three groups— conductors, semiconductors, and insulators.
• Temperature affects resistivity; for relatively small temperature changes Δ T , resistivity is ρ = ρ 0(1 + αΔ T) , where ρ 0 is the original resistivity and α is the temperature coefficient of resistivity.
• Table 20.2 gives values for α , the temperature coefficient of resistivity.
• The resistance R of an object also varies with temperature: R = R 0(1 + αΔ T) , where R 0 is the original resistance, and R is the resistance after the temperature change.
20.4 Electric Power and Energy
• Electric power P is the rate (in watts) that energy is supplied by a source or dissipated by a device.
• Three expressions for electrical power are
P = IV,
P = V 2
R ,
and
P = I 2 R.
• The energy used by a device with a power P over a time t is E = Pt .
20.5 Alternating Current versus Direct Current
• Direct current (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
• The voltage source of an alternating current (AC) system puts out V = V 0 sin 2 π ft , where V is the voltage at time t , V 0 is the peak voltage, and f is the frequency in hertz.
• In a simple circuit, I = V/R and AC current is I = I 0 sin 2 π ft , where I is the current at time t , and I 0 = V 0 /R is the peak current.
• The average AC power is P ave = 12 I 0 V 0 .
• Average (rms) current I rms and average (rms) voltage V rms are I rms = I 02 and V rms = V 02 , where rms stands for root mean square.
• Thus, P ave = I rms V rms .
• Ohm’s law for AC is I rms = V rms
R .
2
• Expressions for the average power of an AC circuit are P
2
ave = I rms V rms , P ave = V rms
R , and P ave = I rms R , analogous to the expressions
for DC circuits.
20.6 Electric Hazards and the Human Body
• The two types of electric hazards are thermal (excessive power) and shock (current through a person).
• Shock severity is determined by current, path, duration, and AC frequency.
• Table 20.3 lists shock hazards as a function of current.
• Figure 20.25 graphs the threshold current for two hazards as a function of frequency.
20.7 Nerve Conduction–Electrocardiograms
• Electric potentials in neurons and other cells are created by ionic concentration differences across semipermeable membranes.
• Stimuli change the permeability and create action potentials that propagate along neurons.
• Myelin sheaths speed this process and reduce the needed energy input.
• This process in the heart can be measured with an electrocardiogram (ECG).
Conceptual Questions
1. Can a wire carry a current and still be neutral—that is, have a total charge of zero? Explain.
2. Car batteries are rated in ampere-hours ( A ⋅ h ). To what physical quantity do ampere-hours correspond (voltage, charge, . . .), and what
relationship do ampere-hours have to energy content?
3. If two different wires having identical cross-sectional areas carry the same current, will the drift velocity be higher or lower in the better conductor?
Explain in terms of the equation v d = I
nqA , by considering how the density of charge carriers n relates to whether or not a material is a good
conductor.
4. Why are two conducting paths from a voltage source to an electrical device needed to operate the device?
5. In cars, one battery terminal is connected to the metal body. How does this allow a single wire to supply current to electrical devices rather than
two wires?
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6. Why isn’t a bird sitting on a high-voltage power line electrocuted? Contrast this with the situation in which a large bird hits two wires simultaneously
with its wings.
20.2 Ohm’s Law: Resistance and Simple Circuits
7. The IR drop across a resistor means that there is a change in potential or voltage across the resistor. Is there any change in current as it passes
through a resistor? Explain.
8. How is the IR drop in a resistor similar to the pressure drop in a fluid flowing through a pipe?
20.3 Resistance and Resistivity
9. In which of the three semiconducting materials listed in Table 20.1 do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)
10. Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar—is its resistance the
same along its length as across its width? (See Figure 20.37.)
Figure 20.37 Does current taking two different paths through the same object encounter different resistance?
11. If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?
12. Explain why R = R 0(1 + αΔ T) for the temperature variation of the resistance R of an object is not as accurate as ρ = ρ 0(1 + αΔ T) , which gives the temperature variation of resistivity ρ .
20.4 Electric Power and Energy
13. Why do incandescent lightbulbs grow dim late in their lives, particularly just before their filaments break?
14. The power dissipated in a resistor is given by P = V 2 / R , which means power decreases if resistance increases. Yet this power is also given by
P = I 2 R , which means power increases if resistance increases. Explain why there is no contradiction here.
20.5 Alternating Current versus Direct Current
15. Give an example of a use of AC power other than in the household. Similarly, give an example of a use of DC power other than that supplied by
batteries.
16. Why do voltage, current, and power go through zero 120 times per second for 60-Hz AC electricity?
17. You are riding in a train, gazing into the distance through its window. As close objects streak by, you notice that the nearby fluorescent lights make
dashed streaks. Explain.
20.6 Electric Hazards and the Human Body
18. Using an ohmmeter, a student measures the resistance between various points on his body. He finds that the resistance between two points on
the same finger is about the same as the resistance between two points on opposite hands—both are several hundred thousand ohms. Furthermore,
the resistance decreases when more skin is brought into contact with the probes of the ohmmeter. Finally, there is a dramatic drop in resistance (to a
few thousand ohms) when the skin is wet. Explain these observations and their implications regarding skin and internal resistance of the human
body.
19. What are the two major hazards of electricity?
20. Why isn’t a short circuit a shock hazard?
21. What determines the severity of a shock? Can you say that a certain voltage is hazardous without further information?
22. An electrified needle is used to burn off warts, with the circuit being completed by having the patient sit on a large butt plate. Why is this plate
large?
23. Some surgery is performed with high-voltage electricity passing from a metal scalpel through the tissue being cut. Considering the nature of
electric fields at the surface of conductors, why would you expect most of the current to flow from the sharp edge of the scalpel? Do you think high- or
low-frequency AC is used?
24. Some devices often used in bathrooms, such as hairdryers, often have safety messages saying “Do not use when the bathtub or basin is full of
water.” Why is this so?
25. We are often advised to not flick electric switches with wet hands, dry your hand first. We are also advised to never throw water on an electric fire.
Why is this so?
26. Before working on a power transmission line, linemen will touch the line with the back of the hand as a final check that the voltage is zero. Why
the back of the hand?
27. Why is the resistance of wet skin so much smaller than dry, and why do blood and other bodily fluids have low resistances?
726 CHAPTER 20 | ELECTRIC CURRENT, RESISTANCE, AND OHM'S LAW
28. Could a person on intravenous infusion (an IV) be microshock sensitive?
29. In view of the small currents that cause shock hazards and the larger currents that circuit breakers and fuses interrupt, how do they play a role in
preventing shock hazards?
20.7 Nerve Conduction–Electrocardiograms
30. Note that in Figure 20.28, both the concentration gradient and the Coulomb force tend to move Na+ ions into the cell. What prevents this?
31. Define depolarization, repolarization, and the action potential.
32. Explain the properties of myelinated nerves in terms of the insulating properties of myelin.
CHAPTER 20 | ELECTRIC CURRENT, RESISTANCE, AND OHM'S LAW 727
Problems & Exercises
13. A large cyclotron directs a beam of He++ nuclei onto a target with a
beam current of 0.250 mA. (a) How many He++ nuclei per second is
this? (b) How long does it take for 1.00 C to strike the target? (c) How
1. What is the current in milliamperes produced by the solar cells of a
long before 1.00 mol of He++ nuclei strike the target?
pocket calculator through which 4.00 C of charge passes in 4.00 h?
14. Repeat the above example on Example 20.3, but for a wire made of
2. A total of 600 C of charge passes through a flashlight in 0.500 h. What
silver and given there is one free electron per silver atom.
is the average current?
15. Using the results of the above example on Example 20.3, find the
3. What is the current when a typical static charge of 0.250 µ C moves
drift velocity in a copper wire of twice the diameter and carrying 20.0 A.
from your finger to a metal doorknob in 1.00 µ s ?
16. A 14-gauge copper wire has a diameter of 1.628 mm. What
magnitude current flows when the drift velocity is 1.00 mm/s? (See above
4. Find the current when 2.00 nC jumps between your comb and hair
example on Example 20.3 for useful information.)
over a 0.500 - µ s time interval.
17. SPEAR, a storage ring about 72.0 m in diameter at the Stanford
5. A large lightning bolt had a 20,000-A current and moved 30.0 C of
Linear Accelerator (closed in 2009), has a 20.0-A circulating beam of
charge. What was its duration?
electrons that are moving at nearly the speed of light. (See Figure
20.39.) How many electrons are in the beam?
6. The 200-A current through a spark plug moves 0.300 mC of charge.
How long does the spark last?
7. (a) A defibrillator sends a 6.00-A current through the chest of a patient
by applying a 10,000-V potential as in the figure below. What is the
resistance of the path? (b) The defibrillator paddles make contact with the
patient through a conducting gel that greatly reduces the path resistance.
Discuss the difficulties that would ensue if a larger voltage were used to
produce the same current through the patient, but with the path having
perhaps 50 times the resistance. (Hint: The current must be about the
same, so a higher voltage would imply greater power. Use this equation
for power: P = I 2 R .)
Figure 20.39 Electrons circulating in the storage ring called SPEAR constitute a
20.0-A current. Because they travel close to the speed of light, each electron
completes many orbits in each second.
20.2 Ohm’s Law: Resistance and Simple Circuits
18. What current flows through the bulb of a 3.00-V flashlight when its hot
resistance is 3.60 Ω ?
19. Calculate the effective resistance of a pocket calculator that has a
1.35-V battery and through which 0.200 mA flows.
20. What is the effective resistance of a car’s starter motor when 150 A
flows through it as the car battery applies 11.0 V to the motor?
21. How many volts are supplied to operate an indicator light on a DVD
player that has a resistance of 140 Ω , given that 25.0 mA passes
through it?
22. (a) Find the voltage drop in an extension cord having a 0.0600- Ω
resistance and through which 5.00 A is flowing. (b) A cheaper cord
utilizes thinner wire and has a resistance of 0.300 Ω . What is the
Figure 20.38 The capacitor in a defibrillation unit drives a current through the heart of
voltage drop in it when 5.00 A flows? (c) Why is the voltage to whatever
a patient.
appliance is being used reduced by this amount? What is the effect on
8. During open-heart surgery, a defibrillator can be used to bring a patient
the appliance?
out of cardiac arrest. The resistance of the path is 500 Ω and a
23. A power transmission line is hung from metal towers with glass
10.0-mA current is needed. What voltage should be applied?
insulators having a resistance of 1.00 × 109 Ω . What current flows
9. (a) A defibrillator passes 12.0 A of current through the torso of a
through the insulator if the voltage is 200 kV? (Some high-voltage lines
person for 0.0100 s. How much charge moves? (b) How many electrons
are DC.)
pass through the wires connected to the patient? (See figure two
problems earlier.)
20.3 Resistance and Resistivity
10. A clock battery wears out after moving 10,000 C of charge through
the clock at a rate of 0.500 mA. (a) How long did the clock run? (b) How
24. What is the resistance of a 20.0-m-long piece of 12-gauge copper
many electrons per second flowed?
wire having a 2.053-mm diameter?
11. The batteries of a submerged non-nuclear submarine supply 1000 A
25. The diameter of 0-gauge copper wire is 8.252 mm. Find the
at full speed ahead. How long does it take to move Avogadro’s number (
resistance of a 1.00-km length of such wire used for power transmission.
6.02 × 1023 ) of electrons at this rate?
26. If the 0.100-mm diameter tungsten filament in a light bulb is to have a
resistance of 0.200 Ω at 20.0ºC , how long should it be?
12. Electron guns are used in X-ray tubes. The electrons are accelerated
through a relatively large voltage and directed onto a metal target,
27. Find the ratio of the diameter of aluminum to copper wire, if they have
producing X-rays. (a) How many electrons per second strike the target if
the same resistance per unit length (as they might in household wiring).
the current is 0.500 mA? (b) What charge strikes the target in 0.750 s?
728 CHAPTER 20 | ELECTRIC CURRENT, RESISTANCE, AND OHM'S LAW
28. What current flows through a 2.54-cm-diameter rod of pure silicon
that is 20.0 cm long, when 1.00 × 103 V is applied to it? (Such a rod
may be used to make nuclear-particle detectors, for example.)
29. (a) To what temperature must you raise a copper wire, originally at
20.0ºC , to double its resistance, neglecting any changes in
dimensions? (b) Does this happen in household wiring under ordinary
circumstances?
30. A resistor made of Nichrome wire is used in an application where its
resistance cannot change more than 1.00% from its value at 20.0ºC .
Over what temperature range can it be used?
31. Of what material is a resistor made if its resistance is 40.0% greater
at 100ºC than at 20.0ºC ?
32. An electronic device designed to operate at any temperature in the
Figure 20.40 The strip of solar cells just above the keys of this calculator convert light
range from –10.0ºC to 55.0ºC contains pure carbon resistors. By what to electricity to supply its energy needs. (credit: Evan-Amos, Wikimedia Commons)
factor does their resistance increase over this range?
43. How many watts does a flashlight that has 6.00×102 C pass
33. (a) Of what material is a wire made, if it is 25.0 m long with a 0.100
through it in 0.500 h use if its voltage is 3.00 V?
mm diameter and has a resistance of 77.7 Ω at 20.0ºC ? (b) What is
44. Find the power dissipated in each of these extension cords: (a) an
its resistance at 150ºC ?
extension cord having a 0.0600 - Ω resistance and through which
34. Assuming a constant temperature coefficient of resistivity, what is the
5.00 A is flowing; (b) a cheaper cord utilizing thinner wire and with a
maximum percent decrease in the resistance of a constantan wire
resistance of 0.300 Ω .
starting at 20.0ºC ?
45. Verify that the units of a volt-ampere are watts, as implied by the
35. A wire is drawn through a die, stretching it to four times its original
equation P = IV .
length. By what factor does its resistance increase?
46. Show that the units 1 V2 / Ω = 1W , as implied by the equation
36. A copper wire has a resistance of 0.500 Ω at 20.0ºC , and an
iron wire has a resistance of 0.525 Ω at the same temperature. At
P = V 2 / R .
what temperature are their resistances equal?
47. Show that the units 1 A2 ⋅ Ω = 1 W , as implied by the equation
37. (a) Digital medical thermometers determine temperature by
measuring the resistance of a semiconductor device called a thermistor
P = I 2 R .
(which has α = – 0.0600 / ºC ) when it is at the same temperature as
48. Verify the energy unit equivalence that 1 kW ⋅ h = 3.60 × 106 J .
the patient. What is a patient’s temperature if the thermistor’s resistance
at that temperature is 82.0% of its value at 37.0ºC (normal body
49. Electrons in an X-ray tube are accelerated through 1.00×102 kV
temperature)? (b) The negative value for α may not be maintained for
and directed toward a target to produce X-rays. Calculate the power of
very low temperatures. Discuss why and whether this is the case here.
the electron beam in this tube if it has a current of 15.0 mA.
(Hint: Resistance can’t become negative.)
50. An electric water heater consumes 5.00 kW for 2.00 h per day. What
38. Integrated Concepts
is the cost of running it for one year if electricity costs
(a) Redo Exercise 20.25 taking into account the thermal expansion of
12.0 cents/kW ⋅ h ? See Figure 20.41.
the tungsten filament. You may assume a thermal expansion coefficient
of 12 × 10−6 / ºC . (b) By what percentage does your answer differ from
that in the example?
39. Unreasonable Results
(a) To what temperature must you raise a resistor made of constantan to
double its resistance, assuming a constant temperature coefficient of
resistivity? (b) To cut it in half? (c) What is unreasonable about these
results? (d) Which assumptions are unreasonable, or which premises are
inconsistent?
20.4 Electric Power and Energy
Figure 20.41 On-demand electric hot water heater. Heat is supplied to water only
when needed. (credit: aviddavid, Flickr)
40. What is the power o