E
LECTRICC
URRENTS18
Responses to Questions
1. In the circuit (not in the battery), electrons flow from high potential energy (at the negative terminal) to low potential energy (at the positive terminal). Inside the battery, the chemical reaction does work on the electrons to take them from low potential energy to high potential energy (to the negative terminal). A chemical description could say that the chemical reaction taking place at the negative electrode leaves electrons behind on the terminal, and the positive ions created at the negative electrode pull electrons off the positive electrode.
2. Battery energy is what is being “used up.” As charges leave the battery terminal, they have a relatively high potential energy. Then as the charges move through the flashlight bulb, they lose potential energy. The battery uses a chemical reaction to replace the potential energy of the charges, by lowering the battery’s chemical potential energy. When a battery is “used up,” it is unable to give potential energy to charges.
3. Ampere-hours measures charge. The ampere is a charge per unit time, and the hour is a time, so the product is charge. One ampere-hour of charge is 3600 coulombs of charge.
4. Resistance is given by the relationship If the ratio of resistivity to area is the same for the copper wire and the aluminum wire, then the resistances will be the same. Thus if
or then the resistances will be the same.
Also, resistance changes with temperature. By having the two wires at different temperatures, it might be possible to have their resistance be the same.
5. The terminal of the battery (usually the negative one) is connected to the metal chassis, frame, and engine block of the car. This means that all voltages used for electrical devices in the car are measured with respect to the car’s frame. Also, since the frame is a large mass of metal, it can supply charges for current without significantly changing its electric potential.
7. When a lightbulb burns out, its filament burns to the point of breakage. Once the filament (part of the conducting path for the electricity flowing through the bulb) is broken, current can no longer flow through the bulb. It no longer gives off any light.
8. We assume that the voltage is the same in both cases. If the resistance increases, then the power delivered to the heater will decrease according to If the power decreases, then the heating process will slow down.
9. Resistance is given by the relationship Thus, to minimize the resistance, you should have a small length and a large cross-sectional area. Likewise, to maximize the resistance, you should have a large length and a small cross-sectional area.
(a) For the least resistance, connect the wires to the faces that have dimensions of 2a by 3a, which maximizes the area and minimizes the length (a).
(b) For the greatest resistance, connect the wires to the faces that have dimensions of a by 2a, which minimizes the area and maximizes the length (3a).
10. When a lightbulb is first turned on, it will be cool and the filament will have a lower resistance than when it is hot. This lower resistance means that there will be more current through the bulb while it is cool. This momentary high current will make the filament quite hot. If the temperature is too high, then the filament will vaporize, and the current will no longer be able to flow in the bulb. After the light has been on for some time, the filament is at a constant high temperature, with a higher resistance and a lower current. Since the temperature is constant, there is less thermal stress on the filament than when the light is first turned on.
11. Assuming that both lightbulbs have the same voltage, since the higher-power bulb will draw the most current. Likewise, assuming that both lightbulbs have the same voltage, since the higher-power bulb will have the lower resistance. So the 100-W bulb will draw the most current, and the 75-W bulb will have the higher resistance.
12. Transmission lines have resistance and therefore will change some electrical energy to thermal energy (heat) as the electrical energy is transmitted. We assume that the resistance of the transmission lines is constant. Then the “lost” power is given by where I is the current carried by the
transmission lines. The transmitted power is given by where V is the voltage across the transmission lines. For a given value of the higher the voltage is, the lower the current has to be. As the current is decreased, is also decreased, so there is a lower amount of power lost.
13. The 15-A fuse is blowing because the circuit is carrying more than 15 A of current. The circuit is probably designed to only carry 15 A, so there might be a “short” or some other malfunction causing the current to exceed 15 A. Replacing the 15-A fuse with a 25-A fuse will allow more current to flow and thus make the wires carrying the current get hotter. A fire or damage to certain kinds of electrical equipment might result. The blown fuse is a warning that something is wrong with the circuit.
much lower frequencies, such as 5 Hz, the eye and brain are able to process the on-off cycle of the lights, and they will appear to flicker.
15. There are several factors which can be considered. As the voltage reverses with each cycle of AC, the potential energy of the electrons is raised again. Thus with each “pass” through the light, the electrons lose their potential energy and then get it back again. Secondly, the heating of the filament (which causes the light) does not depend on the direction of the current, but only on the fact that a current exists, so the light occurs as the electrons move in both directions. Also, at 60 Hz, the current peaks 120 times per second. The small amount of time while the magnitude of the current is small is not long enough for the hot metal filament to cool down, so it stays lit the entire cycle. Finally, the human eye sees anything more rapid than about 20 Hz as continuous, even if it is not. So even if the light was to go dim during part of the cycle, our eyes would not detect it.
16. When the toaster is first turned on, the Nichrome wire is at room temperature. The wire starts to heat up almost immediately. Since the resistance increases with temperature, the resistance will be increasing as the wire heats. Assuming the voltage supplied is constant, the current will be decreasing as the resistance increases.
17. Current is NOT used up in a resistor. The current that flows into the resistor is the same as the current that flows out of the resistor. If that were not the case, then there would be either an increase or decrease in the charge of the resistor, but the resistor actually stays neutral, indicating equal charge flow both in and out. What does get “used up” is potential energy. The charges that leave a resistor have lower potential energy than the charges that enter a resistor. The amount of energy decrease per unit time is given by
18. If you turn on an electric appliance when you are outside with bare feet, and the appliance shorts out through you, the current has a direct path to ground through your feet, and you will receive a severe shock. If you are inside wearing socks and shoes with thick soles, and the appliance shorts out, the current will not have an easy path to ground through you and will most likely find an alternate path. You might receive a mild shock, but not a severe one.
19. In the two wires described, the drift velocities of the electrons will be about the same, but the current density, and therefore the current, in the wire with twice as many free electrons per atom will be twice as large as in the other wire.
20. (a) If the length of the wire doubles, then its resistance also doubles, so the current in the wire will be reduced by a factor of 2. Drift velocity is proportional to current, so the drift velocity will be halved.
(b) If the wire’s radius is doubled, then the drift velocity remains the same. (Although, since there are more charge carriers, the current will quadruple.)
(c) If the potential difference doubles while the resistance remains constant, then the drift velocity and current will also double.
Responses to MisConceptual Questions
1. (c) It may be thought that the orientation of the battery is important for the bulb to work properly. But the lightbulb and it will glow equally bright regardless of the direction in which current flows through it.
point in the loop. Therefore, the amount of current that flows through the lightbulb is the same as the amount that flows through the battery.
3. (a) Ohm’s law is an empirical law showing that for some materials the current through the material is proportional to the voltage across the material. For other materials, such as diodes, fluorescent lightbulbs, and superconductors, Ohm’s law is not valid. Since it is not valid for all objects, it cannot be a fundamental law of physics.
4. (e) A common misconception is that the electrons are “used up” by the lightbulb. A good analogy would be water flowing across a water wheel in a flour mill. The water flows onto the wheel at the top (high potential) and causes the wheel to rotate as the water descends along the wheel. The amount of water that leaves the bottom of the wheel is the same as the amount that entered at the top, but it does so at a lower point (low potential). The change in potential energy goes into work in the wheel. In a lightbulb, electrons at higher potential energy enter the lightbulb and give off that energy as they pass through the bulb. As with the water on the wheel, the number of electrons exiting and entering the bulb is the same.
5. (e) A misconception commonly found when dealing with electric circuits is that electrons are “used up” by the lightbulb. The current is a measure of the rate that electrons pass a given point. If the current were different at two points in the circuit, then electrons would be building up (or being depleted) between those two points. The buildup of electrons would cause the circuit to be time dependent and not a steady state system. The flow of electrons (current) must be the same at all points in a loop.
6. (b) Ohm’s law requires that the ratio of voltage to current be constant. Since it is not constant in this case, the material does not obey Ohm’s law.
7. (d) A common misconception is that the charge or current is “used up” in a resistor. The resistor removes energy from the system, such that the electrons exiting the resistor have less potential energy than the electrons entering, but the number of electrons (charge carriers) entering and exiting the resistor is the same. The rate of electrons entering and exiting is equal to the constant current in the circuit.
8. (c) Since the unit of kilowatt-hour contains the word “watt,” it is often incorrectly thought to be a unit of power. However, the kilowatt-hour is the product of the unit of power (kilowatt) and a unit of time (hour), resulting in a unit of energy.
9. (b) Each device added to the circuit is added in parallel. The voltage across the circuit does not change as the devices are added. Each new device, however, creates a path for additional current to flow, causing the current in the circuit breaker to increase. When the current becomes too high, the circuit breaker will open.
11. (b) A common misconception is that the electrons must travel from the switch to the lightbulb for the lightbulb to turn on. This is incorrect because there are electrons throughout the circuit, not just at the switch. When the switch is turned on, the electric potential across the circuit creates an electric field in the wire that causes all of the conduction electrons in the wire to move.
Solutions to Problems
1. Use the definition of current, Eq. 18–1.
2. Use the definition of current, Eq. 18–1.
3. Use the definition of current, Eq. 18–1.
4. Use Eq. 18–2 for resistance.
5. Use Eq. 18–2 for the voltage.
6. The ampere-hour is a unit of charge.
7. (a) Use Eq. 18–2 to find the current. (b) Use the definition of current, Eq. 18–1.
9. (a) Use Eq. 18–2 for resistance.
(b) Use the definition of current, Eq. 18–1.
10. Find the current from the voltage and resistance and then find the number of electrons from the current.
11. (a) If the voltage drops by 15%, and the resistance stays the same, then by Eq. 18–2, the current will also drop by 15%.
(b) If the resistance drops by 15% (the same as being multiplied by 0.85), and the voltage stays the same, then by Eq. 18–2, the current must be divided by 0.85.
12. Use Eq. 18–3 to find the diameter, with the area
13. Use Eq. 18–3 to calculate the resistance, with the area
15. Use Eq. 18–3 to express the resistances, with the area so
The diameter of the tungsten should be 4.0 mm.
16. Since the resistance is directly proportional to the length, the length of the long piece must be 4.0 times the length of the short piece.
Make the cut at 20% of the length of the wire.
17. Calculate the voltage drop by combining Ohm’s law (Eq. 18–2) with the expression for resistance, Eq. 18–3.
18. The wires have the same resistance and the same resistivity.
19. In each case calculate the resistance by using Eq. 18–3 for resistance.
(a) (b) (c)
20. The length of the new wire is half the length of the original wire, and the cross-sectional area of the new wire is twice that of the original wire. Use Eq. 18–3.
21. Use Eq. 18–4 multiplied by so that it expresses resistance instead of resistivity.
So raise the temperature by to a final temperature of .
22. Use Eq. 18–4 for the resistivity.
23. Use Eq. 18–4 multiplied by so that it expresses resistances instead of resistivity.
24. The original resistance is and the high-temperature resistance is where the two voltages are the same. The two resistances are related by Eq. 18–4, multiplied by so that it expresses resistance instead of resistivity.
25. The total resistance is to be 3200 ohms at all temperatures. Write each resistance in terms of Eq. 18–4 (with multiplied by to express resistance instead of resistivity.
26. (a) Calculate each resistance separately using Eq. 18–3 and then add the resistances together to find the total resistance.
(b) The current through the wire is the voltage divided by the total resistance.
(c) For each segment of wire, Ohm’s law is true. Both wires have the current found in (b) above. Notice that the total voltage is 95 mV.
27. Use Eq. 18–5 to find the power from the voltage and the current.
28. Use Eq. 18–6b to find the resistance from the voltage and the power.
29. Use Eq. 18–6b to find the voltage from the power and the resistance.
30. Use Eq. 18–6b to find the resistance and Eq. 18–5 to find the current.
31. The battery’s potential energy is equal to the charge that it can deliver times its voltage.
32. The power needed is the voltage times the current, Eq. 18–5.
33. (a) Since says that the resistance is inversely proportional to the power for a constant voltage, we predict that the 950-W setting has the higher resistance.
(b) (c)
34. (a) Use Eq. 18–2 to find the resistance.
(b) An amount of charge loses a potential energy of as it passes through the resistor. The amount of charge is found from Eq. 18–1.
35. (a) Use Eq. 18–5 to find the current. (b) Use Eq. 18–6b to find the resistance.
36. The power (and thus the brightness) of the bulb is proportional to the square of the voltage, according
to Eq. 18–6a, Since the resistance is assumed to be constant, if the voltage is cut in half from
37. To find the kWh of energy, multiply the kilowatts of power consumption by the number of hours in operation.
To find the cost of the energy used in a month, multiply by 4 days per week of usage, 4 weeks per month, and the cost per kWh.
38. To find the cost of the energy, multiply the kilowatts of power consumption by the number of hours in operation by the cost per kWh.
39. The rating is the amount of charge that the battery can deliver. The potential energy of the charge is the charge times the voltage.
40. (a) Calculate the resistance from Eq. 18–2 and the power from Eq. 18–5.
(b) If four D-cells are used, then the voltage will double, to 6.0 V. Assuming that the resistance of the bulb stays the same (by ignoring heating effects in the filament), the power that the bulb
would need to dissipate is given by Eq. 18–6b, A doubling of the voltage means the
power is increased by a factor of 4. This should not be tried because the bulb is probably not rated for such a high wattage. The filament in the bulb would probably burn out, and the glass bulb might even explode if the filament burns violently.
41. Each bulb will draw an amount of current found from Eq. 18–5.
The number of bulbs to draw 15 A is the total current divided by the current per bulb.
43. Use Eqs. 18–3 and 18–6b to calculate the length of the wire.
If the voltage increases by a factor of 6 without the resistance changing, then the power will increase by a factor of 36. The blanket would theoretically be able to deliver 540 W of power, which might make the material catch on fire or burn the occupant.
44. Find the current used to deliver the power in each case and then find the power dissipated in the resistance at the given current.
45. (a) By conservation of energy and the efficiency claim, 85% of the electrical power emitted by the heater must be the rate at which energy is absorbed by the water. The power emitted by the heater is given by Eq. 18–5.
(b) Use Ohm’s law to find the resistance of the heater.
47. Use Ohm’s law, Eq. 18–2, and the relationship between peak and rms values, Eq. 18–8a.
48. Find the peak current from Ohm’s law and then find the rms current from the relationship between peak and rms values.
49. (a) When everything electrical is turned off, no current will be flowing into the house, even though a voltage is being supplied. Since for a given voltage, the more resistance, the lower the current, a zero current corresponds to an infinite resistance.
(b) Assuming that the voltage is 120 V, use Eq. 18–6ba to calculate the resistance.
50. The power and current can be used to find the peak voltage and then the rms voltage can be found from the peak voltage.
51. Use the average power and rms voltage to calculate the peak voltage and peak current.
(a) (b)
52. (a) We use Eqs. 18–9c and 18–9a.
53. (a) We assume that the 2.2 hp is the average power, so the maximum power is twice that, or 4.4 hp, as seen in Fig. 18–22.
(b) Use the average power and the rms voltage to find the peak current.
54. (a) The average power used can be found from the resistance and the rms voltage by Eq. 18–9c. (b) The maximum power is twice the average power, and the minimum power is 0.
55. We follow exactly the derivation in Example 18–14, which results in an expression for the drift velocity.
56. (a) Use Ohm’s law to find the resistance. (b) Find the resistivity from Eq. 18–3.
57. We are given a net charge, a concentration, and a speed (like the drift speed) for both types of ions. From that we can use Eq. 18–10 to determine the current per unit area. Both currents are in the same direction in terms of conventional current—positive charge moving north has the same effect as negative charge moving south—so they can be added.
58. The magnitude of the electric field is the voltage change per unit meter.
59. The speed is the change in position per unit time.
Two measurements are needed because there may be a time delay from the stimulation of the nerve to the generation of the action potential.
60. The power is the work done per unit time. The work done to move a charge through a potential difference is the charge times the potential difference. The charge density must be multiplied by the surface area of the cell (the surface area of an open tube, length times circumference) to find the actual charge moved.
61. The energy supplied by the battery is the energy consumed by the lights.
62. (a) If the wire obeys Ohm’s law, then or showing a linear relationship between
(b) From the graph and the calculated linear fit, we see that the wire obeys Ohm’s law.
(c) Use Eq. 18–3 to find the resistivity.
From Table 18–1, the material is Nichrome.
63. Use Eq. 18–5 to calculate the current. We assume that 120 V has 3 significant figures.
64. From Eq. 18–2, if then
65. Use Eq. 18–6b to express the resistance in terms of the power and Eq. 18–3 to express the resistance in terms of the wire geometry.
66. (a) Calculate the total kWh used per day and then multiply by the number of days and the cost per kWh.
67. To deliver 15 MW of power at 120 V, a current of is required.
Calculate the power dissipated in the resistors using the current and the resistance. There are two wires, so we consider 2.0 m of wire.
68. (a) Use Eq. 18–6b to relate the power to the voltage for a constant resistance.
(b) The lower power output means that the resistor is generating less heat, so the resistor’s temperature would be lower. The lower temperature results in a lower value of the resistance, which would increase the power output at the lower voltages. Thus the decrease would be smaller than the value given in the first part of the problem.
69. Assume that we have a meter of wire, carrying 35 A of current, and dissipating 1.5 W of heat. The
power dissipated is and the resistance is
70. (a) The resistance at the operating temperature can be calculated directly from Eq. 18–6b. (b) The resistance at room temperature is found by converting Eq. 18–4 into an equation for
71. (a) The angular frequency is (b) The maximum current is 1.40 A. (c) For a resistor,
72. (a) The power delivered to the interior is 65% of the power drawn from the source.
(b) The current drawn is current from the source. The source power is used to calculate the current. 73. The volume of wire is unchanged by the stretching. The volume is equal to the length of the wire times
its cross-sectional area. Since the length was increased by a factor of 1.5, the area was decreased by a factor of 1.5. Use Eq. 18–3.
74. The long, thick conductor is labeled as conductor number 1, and the short, thin conductor is labeled as number 2. The power transformed by a resistor is given by Eq. 18–6, and both have the same voltage applied. Use Eq. 18–3 for resistance.
75. (a) From Eq. 18–5, if power P is delivered to the transmission line at voltage V, then there must be a current As this current is carried by the transmission line, there will be power losses of
due to the resistance of the wire. This power loss can be expressed as
Equivalently, there is a voltage drop across the transmission lines of Thus the voltage available to the users is and the power available to the users is
(b) Since V should be as large as possible to minimize 76. (a) Use Eq. 18–6b.
(b) Only 75% of the heat from the oven is used to heat the water.
(c) (1/5 of a cent)
77. (a) The horsepower required is the power dissipated by the frictional force, since we are neglecting the energy used for acceleration.
(b) The charge available by each battery is so the total charge available is 24 times that. The potential energy of that charge is the charge times the voltage. That energy must be delivered (batteries discharged) in a certain amount of time to produce the 3000 W necessary. The speed of the car times the discharge time is the range of the car between recharges.
79. The resistance can be calculated from the power and voltage, and then the diameter of the wire can be calculated from the resistance.
80. Use Eq. 18–6b.
(a)
(b) (2 significant figures)
81. Use Eq. 18–6b for the power in each case, assuming the resistance is constant.
82. The resistance of the filament when the flashlight is on is That can be used
with a combination of Eqs. 18–3 and 18–4 to find the temperature.
83. The solution assumes that the 40-W values are precise to 2 significant figures. (a) The current can be found from Eq. 18–5.
(b) The resistance can be found from Eq. 18–6b. (c) The charge is the current times the time.
(d) The energy is the power times the time, and the power is the same for both bulbs.
84. (a) The power is given by Eq. 18–5.
(b) The power dissipated by the resistor is calculated by Eq. 18–6b, and the resistance is calculated by Eq. 18–3.
(c) Recalculate using the alternate wire. (d) The savings is due to the power difference.
85. (a) The energy stored is electrical potential energy, found by multiplying the available charge by the potential.
(b) The work done by the potential energy applies a force to overcome the retarding force and thus does the same amount of work as the retarding force.
86. The volume of wire remains constant as the wire is stretched from the original length of to the final length of Thus the cross-sectional area changes from
to Use Eq. 18–3 for the resistance and Eq. 18–6b for the power dissipated.
The power is reduced by a factor of 4.
87. The wasted power is due to losses in the wire. The current in the wire can be found from Eq. 18–5.
88. (a) The D-cell provides 25 mA at 1.5 V for 820 h, at a cost of $1.70.
(b) The AA-cell provides 25 mA at 1.5 V for 120 h, at a cost of $1.25. (c) Compare the battery costs with a normal commercial electric rate.
D-cell:
AA-cell:
89. Eq. 18–3 can be used. The area to be used is the cross-sectional area of the pipe.
90. Model the protons as moving in a continuous beam of cross-sectional area A. Then by Eq. 18–10, The variable n is the number of protons per unit volume, so where N is the number of protons in the beam and is the circumference of the ring. The “drift” velocity in this case is the speed of light.
91. When the tank is empty, the entire length of the wire is in a nonsuperconducting state and has a nonzero resistivity, which we call Then the resistance of the wire when the tank is empty is given
by When a length x of the wire is superconducting, that portion of the wire has 0 resistance. Then the resistance of the wire is only due to the length and
Solutions to Search and Learn Problems
1. Ohm’s law is an empirical law that applies only to certain materials. Newton’s laws are general laws that apply to all objects.
2. (a) The efficiency of the bulb is proportional to the power radiated from the bulb. The power radiated is given by Eq. 14–6. The subscript “H” is used for the halogen bulb and “T” for the traditional incandescent bulb.
The halogen bulb is about 33% more efficient than the traditional incandescent bulb.
(b) The radiated power is proportional to the power consumed by the lightbulb. Since the halogen bulb is 33% more efficient than the incandescent, the power input can be decreased by a factor of 1.33 to provide the same output intensity.
3. The resistor code in Fig. 18–12 shows that the resistance is Solve for the resistivity by inserting this resistance into Eq. 18–3 along with the length and diameter of the resistor.
Using Table 18–1, the only material that has a resistivity in this range is germanium.
4. For the cylindrical wire, its (constant) volume is given by so Combine this
relationship with Eq. 18–3. We assume that and use that fact in the third line where we drop
the term
This is true for any initial conditions, so
6. (a) We use Eq. 18–3.
(b) Use Eq. 18–3 with the resistivity of copper to find the resistance. (c) Use Eq. 18–3 with the resistivity of aluminum.
(d) Use Eq. 18–6a to determine the power dissipated in each wire.
(e) Set the resistances in Eq. 18–3 equal and solve for the ratio of the wire diameters.
(f) Use the density of copper to find the mass of the copper wire in part (b):
Set the mass of aluminum equal to the mass of the copper and then calculate the diameter of the aluminum wire.
Using the length, diameter, and resistivity of aluminum, calculate the resistance of the aluminum wire.
In part (b) above, the copper has a resistance of In Section 18–4 it states that an
7. The electrons are assumed to be moving with simple harmonic motion. During one cycle, an object in simple harmonic motion will move a distance equal to twice its amplitude—it will move from its minimum position to its maximum position. From Eq. 11–7, we know that where is the angular frequency of oscillation and A is the amplitude. From Eq. 18–10, we see that
Finally, the maximum current can be related to the power by Eqs. 18–8 and 18–9. The charge carrier density, n, is calculated in Example 18–14.
The electron will move this distance in both directions from its equilibrium point, so its maximum displacement is from one extreme to the other, which is twice the amplitude
8. (a) We model the axon as a cylindrically shaped parallel plate capacitor, with opposite charges on each side. The area of the capacitor plates is the area of the walls of the cylindrical surface area.
(b) In one action, the voltage changes from –70 mV to +30 mV, or a difference of 100 mV. From the capacitance and change in voltage we calculate the change in charge during the action.
Each sodium ion is singly ionized and carries a charge of Use this to calculate the number of ions that have moved.
Table 18–2 gives the concentration of sodium ions inside the axon as Multiplying this concentration by the volume of the axon and Avogadro’s number gives the number of sodium ions in the axon. We divide the change in ions by the total number to determine the relative change during the action.
