Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 28

Direct-Current Circuits - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

Problem 1

A battery has an emf of 15.0 V. The terminal voltage of the battery is 11.6 V when it is delivering 20.0 W of power to an external load resistor R. (a) What is the value of R ? (b) What is the internal resistance of the battery?

Chris Johnson

Chris Johnson

Numerade Educator

Problem 2

Two $1.50-\mathrm{V}$ batteries - with their positive terminals in the same direction-are inserted in series into a flashlight. One battery has an internal resistance of $0.255 \Omega,$ and the other has an internal resistance of 0.153 . When the switch is closed, the bulb carries a current of 600 $\mathrm{mA}$ .

Chris Johnson

Numerade Educator

Problem 3

An automobile battery has an emf of 12.6 $\mathrm{V}$ and an internal resistance of 0.0800$\Omega$ . The headlights together have an equivalent resistance of 5.00$\Omega$ (assumed constant). What is the potential difference across the headlight. bulbs (a) when they are the only load on the battery and
(b) when the starter motor is operated, requiring an additional 35.0 A from the battery?

Chris Johnson

Numerade Educator

Problem 4

As in Example 28.2, consider a power supply with fixed emf e and internal resistance r causing current in a load resistance R. In this problem, R is fixed and r is a variable. The efficiency is defined as the energy delivered to the load divided by the energy delivered by the emf. (a) When the internal resistance is adjusted for maximum power transfer, what is the efficiency? (b) What should be the internal resistance for maximum possible efficiency? (c) When the electric company sells energy to a customer, does it have a goal of high efficiency or of maximum power transfer? Explain. (d) When a student connects a loudspeaker to an amplifier, does she most want high efficiency or high power transfer? Explain.

Chris Johnson

Numerade Educator

Problem 5

What is the equivalent resistance of the combination of identical resistors between points $a$ and $b$ in Figure $\mathrm{P} 28.5 ?$

Chris Johnson

Numerade Educator

Problem 6

A lightbulb marked " 75 $\mathrm{W}$ [at] 120 $\mathrm{V}^{n}$ is screwed into a socket at one end of a long extension cord, in which each of the two conductors has resistance 0.800$\Omega$ . The other
end of the extension cord is plugged into a $120-\mathrm{V}$ outlet. (a) Explain why the actual power delivered to the lightbulb cannot be 75 $\mathrm{W}$ in this situation. (b) Draw a circuit diagram. (c) Find the actual power delivered to the light bulb in this circuit.

Chris Johnson

Numerade Educator

Problem 7

Three $100-\Omega$ resistors are connected as shown in Figure P28.7. The maximum power that can safely be delivered to any one resistor is 25.0 $\mathrm{W}$ . (a) What is the maximum potential difference that can be applied to the terminals $a$ and $b ?$ (b) For the voltage determined in part (a), what is the power delivered to each resistor? (c) What is the total power delivered to the combination of resistors?

Chris Johnson

Numerade Educator

Problem 8

Consider the two circuits shown in Figure P28.8 in which the batteries are identical. The resistance of each light-bulb is R. Neglect the internal resistances of the batteries. (a) Find expressions for the currents in each light-bulb. (b) How does the brightness of B compare with that of C? Explain. (c) How does the brightness of A compare with that of B and of C? Explain.

Chris Johnson

Numerade Educator

Problem 9

Consider the circuit shown in Figure $\mathrm{P} 28.9 .$ Find (a) the current in the $20.0-\Omega$ resistor and $(\mathrm{b})$ the potential difference between points $a$ and $b .$

Chris Johnson

Numerade Educator

Problem 10

(a) You need a $45-\Omega$ resistor, but the stockroom has only $20-\Omega$ and $50-\Omega$ resistors. How can the desired resistance be achieved under these circumstances? (b) What can you do if you need a $35-\Omega$ resistor?

Chris Johnson

Numerade Educator

Problem 11

A battery with $\varepsilon=6.00 \mathrm{V}$ and no internal resistance supplies current to the circuit shown in Figure $\mathrm{P} 28.11 .$ When the double-throw switch $\mathrm{S}$ is open as shown in the figure, the current in the battery is 1.00 $\mathrm{mA}$ . When the switch is closed in position $a,$ the current in the battery is 1.20 $\mathrm{mA}$ . When the switch is closed in position $b$ , the current in the battery is 2.00 $\mathrm{mA}$ . Find the resistances (a) $R_{1},(\mathrm{b}) R_{2},$ and (c) $R_{3}$ .

PR

Patrick Ramsey

Numerade Educator

Problem 12

If A battery with emf $\varepsilon$ and no internal resistance supplies current to the circuit shown in Figure $\mathrm{P} 28.11$ . When the double-throw switch $\mathrm{S}$ is open as shown in the figure,
position $a,$ the current in the battery is $I_{a}$ . When the switch is closed in position $b$ , the current in the battery is $I_{b}$ . Find the resistances (a) $R_{1},$ (b) $R_{2},$ and $(c) R_{3}$ .

Sheh Lit Chang

University of Washington

Problem 13

Consider the combination of resistors shown in Figure $\mathrm{P} 28.13$ . (a) Find the equivalent resistance between points $a$ and $b .$ (b) If a voltage of 35.0 $\mathrm{V}$ is applied between points $a$ and $b$ , find the current in each resistor.

Sheh Lit Chang

University of Washington

Problem 14

(a) When the switch Sin the circuit of Figure $\mathrm{P} 28.14$ is closed, will the equivalent resistance between points $a$ and $b$ increase or decrease? State your reasoning. (b) Assume the equivalent resistance drops by 50.0$\%$ when the switch is closed. Determine the value of $R .$

Sheh Lit Chang

University of Washington

Problem 15

Two resistors connected in series have an equivalent resistance of $690 \Omega .$ When they are connected in parallel, their equivalent resistance is $150 \Omega .$ Find the resistance of each resistor.

Chris Johnson

Numerade Educator

Problem 16

Four resistors are connected to a battery as shown in Figure $\mathrm{P} 28.16 .$ (a) Determine the potential difference across each resistor in terms of $\mathcal{E} .$ (b) Determine the current in each resistor in terms of $I .$ (c) What If? If $R_{3}$ is increased, explain what happens to the current in each
of the resistors. (d) In the limit that $R_{3} \rightarrow \infty,$ what are the new values of the current in each resistor in terms of $I,$ the original current in the battery?

Sheh Lit Chang

University of Washington

Problem 17

Calculate the power delivered to each resistor in the circuit shown in Figure P28.17.

Sheh Lit Chang

University of Washington

Problem 18

For the purpose of measuring the electric resistance of shoes through the body of the wearer standing on a metal ground plate, the American National Standards Institute (ANSI) specifies the circuit shown in Figure $\mathrm{P} 28.18$ . The potential difference $\Delta V$ across the $1.00-\mathrm{M} \Omega$ resistor is measured with an ideal voltmeter. (a) Show that the resistance of the footwear is $$
R_{\text { shoes }}=\frac{50.0 \mathrm{V}-\Delta V}{\Delta V} $$ (b) In a medical test, a current through the human body should not exceed 150$\mu \mathrm{A}$ . Can the current delivered by the ANSI-specified circuit exceed 150$\mu \mathrm{A}$ ? To decide, consider a person standing barefoot on the ground plate.

Sheh Lit Chang

University of Washington

Problem 20

Why is the following situation impossible? A technician is testing a circuit that contains a resistance $R$ . He realizes that a better design for the circuit would include a resistance $\frac{7}{3} R$ rather than R. He has three additional resistors, each with resistance R. By combining these additional resistors in a
certain combination that is then placed in series with the original resistor, he achieves the desired resistance.

Sheh Lit Chang

University of Washington

Problem 21

The circuit shown in Figure P28.21 is connected for 2.00 min. (a) Determine the current in each branch of
the circuit. (b) Find the energy delivered by each battery. (c) Find the energy delivered to each resistor. (d) Identify the type of energy storage transformation that occurs in the operation of the circuit. (e) Find the total amount of energy transformed into internal energy in the resistors.

Sheh Lit Chang

University of Washington

Problem 22

For the circuit shown in Figure $\mathrm{P} 28.22$ , calculate (a) the current in the $2.00-\Omega$ resistor and $(\mathrm{b})$ the potential difference between points $a$ and $b .$

Sophie S

Numerade Educator

Problem 23

The ammeter shown in Figure $\mathrm{P} 28.23$ reads 2.00 A. Find (a) $I_{1},$ (b) $I_{2},$ and $(\mathrm{c}) \varepsilon$

Sheh Lit Chang

University of Washington

Problem 24

Jumper cables are connected from a fresh battery in one car to charge a dead battery in another car. Figure P28.24 shows the circuit diagram for this situation. While the cables are connected, the ignition switch of the car with the dead battery is closed and the starter is activated to start the engine. Determine the current in (a) the starter and (b) the dead battery. (c) Is the dead battery being charged while the starter is operating?

Sheh Lit Chang

University of Washington

Problem 25

What are the expected readings of (a) the ideal ammeter and (b) the ideal voltmeter in Figure $\mathrm{P} 28.25 ?$

Sheh Lit Chang

University of Washington

Problem 26

The following equations describe an electric circuit:
$$\begin{aligned}-I_{1}(220 \Omega)+5.80 \mathrm{V}-I_{2}(370 \Omega) &=0 \\+I_{2}(370 \Omega)+I_{3}(150 \Omega)-3.10 \mathrm{V} &=0 \\ I_{1}+I_{3}-I_{2}=0 \\ \text { (a) Draw a diagram of the circuit. (b) Calculate the } \\ \text { unknowns and identify the physical meaning of each } \\ \text { unknown. } \end{aligned} $$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

Taking $R=1.00 \mathrm{k} \Omega$ and $\varepsilon=250 \mathrm{V}$ in Figure $\mathrm{P} 28.27$ , determine the direction and magnitude of the current in the horizontal wire between $a$ and $e .$

Sheh Lit Chang

University of Washington

Problem 28

In the circuit of Figure $\mathrm{P} 28.28$ , determine (a) the current in each resistor and $(\mathrm{b})$ the potential difference across the $200-\Omega$ resistor.

Andrija Isakov

Numerade Educator

Problem 29

In Figure $\mathrm{P} 28.29,$ find $(\mathrm{a})$ the current in each resistor and
(b) the power delivered to each resistor.

Sheh Lit Chang

University of Washington

Problem 30

In the circuit of Figure $\operatorname{P} 28.30,$ the current $I_{1}=$ 3.00 $\mathrm{A}$ and the values of $\varepsilon$ for the ideal battery and $R$ are unknown. What are the currents (a) $I_{2}$ and $(\mathrm{b}) I_{3} ?(\mathrm{c})$ Can you find the values of $\varepsilon$ and $R$ ? If so, find their values. If not, explain.

Sheh Lit Chang

University of Washington

Problem 31

(a) Can the circuit shown in Figure $\mathrm{P} 28.31$ be reduced to a single resistor connected to the battery? Explain. Calculate the currents (b) $I_{1},(\mathrm{c}) I_{2},$ and (d) $I_{3} .$

Sheh Lit Chang

University of Washington

Problem 32

For the circuit shown in Figure $\mathrm{P} 28.32$ , we wish to find the currents $I_{1}, I_{2},$ and $I_{3}$ . Use Kirchhoff's rules to obtain equations for $(a)$ the upper loop, (b) the lower loop, and (c) the junction on the left side. In each case, suppress units for clarity and simplify, combining the terms. (d) Solve the junction equation for $I_{3}$ . (e) Using the equation found in part (d), eliminate $I_{3}$ from the equation found in part (b). (f). (f). (f) Solve the equations found in parts (a) and (e) simultaneously for the two unknowns $I_{1}$ and $I_{2}$ . (g) Substitute the answers found in part (f) into the junction equation found in part (d), solving for $I_{3} .$ (h) What is the significance of the negative answer for $I_{2} ?$

Sheh Lit Chang

University of Washington

Problem 33

An uncharged capacitor and a resistor are connected in series to a source of emf. If $\varepsilon=9.00 \mathrm{V}, C=20.0 \mu \mathrm{F}$ , and $R=100 \Omega,$ find $(\mathrm{a})$ the time constant of the circuit, (b) the maximum charge on the capacitor, and $(\mathrm{c})$ the charge on the capacitor at a time equal to one time constant after the battery is connected.

Sheh Lit Chang

University of Washington

Problem 34

Consider a series $R C$ circuit as in Figure $\mathrm{P} 28.34$ for which $R=1.00 \mathrm{M} \Omega, C=5.00 \mu \mathrm{F},$ and $\varepsilon=30.0 \mathrm{V}$ . Find $(\mathrm{a})$ the time constant of the circuit and $(\mathrm{b})$ the maximum charge on the capacitor after the switch is thrown closed. (c) Find the current in the resistor 10.0 s after the switch is closed.

Sheh Lit Chang

University of Washington

Problem 35

A. 00 -nF capacitor with an initial charge of 5.10$\mu \mathrm{C}$ is dis- charged through a $1.30-\mathrm{k} \Omega$ resistor. (a) Calculate the current in the resistor 9.00$\mu$ s after the resistor is connected across the terminals of the capacitor. (b) What charge remains on the capacitor after 8.00$\mu \mathrm{s}$ ? (c) What is the maximum current in the resistor?

Sheh Lit Chang

University of Washington

Problem 36

A $10.0-\mu \mathrm{F}$ capacitor is charged by a $10.0-\mathrm{V}$ battery through a resistance $R .$ The capacitor reaches a potential difference of 4.00 $\mathrm{V}$ in a time interval of 3.00 $\mathrm{s}$ after charging begins. Find $R .$

Sophie S

Numerade Educator

Problem 37

The circuit in Figure P28.37 has been connected for a long time. (a) What is the potential difference across the capacitor? (b) If the battery is disconnected from the circuit, over what time interval does the capacitor discharge to one-tenth its initial voltage?

Sheh Lit Chang

University of Washington

Problem 38

Show that the integral $\int_{0}^{\infty} e^{-2 t / R C} d t$ in Example 28.11 has the value $\frac{1}{2} R C .$

Sophie S

Numerade Educator

Problem 39

In the circuit of Figure $P 28.39$ , the switch $S$ has been open for a long time. It is then suddenly closed. Take $\varepsilon=10.0 \mathrm{V}$ , $R_{1}=50.0 \mathrm{k} \Omega, R_{2}=100 \mathrm{k} \Omega,$ and $C=10.0 \mu \mathrm{F}$ . Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at $t=0$ . Determine the current in the switch as a function of time.

Sheh Lit Chang

University of Washington

Problem 40

In the circuit of Figure $\mathrm{P} 28.39$ , the switch $\mathrm{S}$ has been open for a long time. It is then suddenly closed. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at $t=0 .$ Determine the current in the switch as a function of time.

Artemisa Mazón

Numerade Educator

Problem 41

A charged capacitor is connected to a resistor and switch as in Figure $\mathrm{P} 28.41 .$ The circuit has a time constant of 1.50 $\mathrm{s}$ .Soon after the switch is closed, the charge on the capaci-
tor is 75.0$\%$ of its initial charge. (a) Find the time interval required for the capacitor to reach this charge. (b) If $R=$ $250 \mathrm{k} \Omega,$ what is the value of $C ?$

Sheh Lit Chang

University of Washington

Problem 42

An electric heater is rated at $1.50 \times 10^{3} \mathrm{W}, \mathrm{a}$ toaster at 750 $\mathrm{W}$ , and an electric grill at $1.00 \times 10^{3} \mathrm{W}$ . The three appliances are connected to a common $120-\mathrm{V}$ household circuit. (a) How much current does each draw? (b) If the circuit is protected with a $25.0-\mathrm{A}$ circuit breaker, will the circuit breaker be tripped in this situation? Explain your answer.

Sheh Lit Chang

University of Washington

Problem 43

Turn on your desk lamp. Pick up the cord, with your thumb and index finger spanning the width of the cord. (a) Compute an order-of-magnitude estimate for the current in your hand. Assume the conductor inside the lamp cord next to your thumb is at potential ~ 102 V at a typical instant and the conductor next to your index finger is at ground potential (0 V). The resistance of your hand depends strongly on the thickness and the moisture content of the outer layers of your skin. Assume the resistance of your hand between fingertip and thumb tip is ~ 104 V. You may model the cord as having rubber insulation. State the other quantities you measure or estimate and their values. Explain your reasoning. (b) Suppose your body is isolated from any other charges or currents. In order-of-magnitude terms, estimate the potential difference between your thumb where it contacts the cord and your finger where it touches the cord.

Eduard Sanchez

Numerade Educator

Problem 44

Find the equivalent resistance between points $a$ and $b$ in
Figure $\mathrm{P} 28.44 .$

Sheh Lit Chang

University of Washington

Problem 45

Assume you have a battery of emf e and three identical lightbulbs, each having constant resistance R. What is the total power delivered by the battery if the lightbulbs are connected (a) in series and (b) in parallel? (c) For which connection will the lightbulbs shine the brightest?

Sheh Lit Chang

University of Washington

Problem 46

Four resistors are connected in parallel across a 9.20-V battery. They carry currents of 150 mA, 45.0 mA, 14.0 mA, and 4.00 mA. If the resistor with the largest resistance is replaced with one having twice the resistance, (a) what is the ratio of the new current in the battery to the original current? (b) What If? If instead the resistor with the smallest resistance is replaced with one having twice the resistance, what is the ratio of the new total current to the original current? (c) On a February night, energy leaves a house by several energy leaks, including $1.50 \times 10^{3} \mathrm{W}$ by conduction through the ceiling, 450 $\mathrm{W}$ by infiltration (air- flow) around the windows, 140 $\mathrm{W}$ by conduction through the basement wall above the foundation sill, and 40.0 $\mathrm{W}$ by conduction through the plywood door to the attic. To produce the biggest saving in heating bills, which one of these energy transfers should be reduced first? Explain how you decide. Clifford Swartz suggested the idea for this problem.

Sheh Lit Chang

University of Washington

Problem 47

Four $1.50-\mathrm{V}$ AA batteries in series are used to power a small
radio. If the batteries can move a charge of $240 \mathrm{C},$ how long
will they last if the radio has a resistance of 200$\Omega ?$

Narayan Hari

Numerade Educator

Problem 48

The resistance between terminals $a$ and $b$ in Figure $P 28.48$
is 75.0 . $\Omega .$ If the resistors labeled $R$ have the same value,
determine $R .$

Shoukat Ali

Other Schools

Problem 49

The circuit in Figure $\mathrm{P} 28.49$ has been connected for several seconds. Find the current (a) in the $4.00-\mathrm{V}$ battery, (b) in the $3.00-\Omega$ resistor, $(\mathrm{c})$ in the $8.00-\mathrm{V}$ battery, and $(\mathrm{d})$ in the $3.00-\mathrm{V}$ battery. (e) Find the charge on the capacitor.

Sheh Lit Chang

University of Washington

Problem 50

The circuit in Figure P28.50a consists of three resistors and one battery with no internal resistance. (a) Find the current in the 5.00-V resistor. (b) Find the power delivered to the 5.00-V resistor. (c) In each of the circuits in Figures P28.50b, P28.50c, and P28.50d, an additional 15.0-V battery has been inserted into the circuit. Which diagram or diagrams represent a circuit that requires the use of Kirchhoff’s rules to find the currents? Explain why. (d) In which of these three new circuits is the smallest amount of power delivered to the 10.0-V resistor? (You need not

Sheh Lit Chang

University of Washington

Problem 51

For the circuit shown in Figure P28.51, the ideal volt meter reads 6.00 V and the ideal ammeter reads 3.00 mA. Find (a) the value of R, (b) the emf of the battery, and (c) the voltage across the 3.00-kV resistor.

Sheh Lit Chang

University of Washington

Problem 52

Why is the following situation impossible? A battery has an emf of $\varepsilon=9.20 \mathrm{V}$ and an internal resistance of $r=1.20 \Omega .$ A resistance $R$ is connected across the battery and extracts
from it a power of $P=21.2 \mathrm{W}$ .

Sheh Lit Chang

University of Washington

Problem 53

(a) Calculate the potential difference between points $a$ and
$b$ in Figure $\mathrm{P} 28.53$ and $(\mathrm{b})$ identify which point is at the
higher potential.

Ajay Singhal

Numerade Educator

Problem 54

Find (a) the equivalent resistance of the circuit in Figure P28.54, (b) the potential difference across each resistor, (c) each current indicated in Figure P28.54, and (d) the power delivered to each resistor.

Sheh Lit Chang

University of Washington

Problem 55

A rechargeable battery has an emf of 13.2 $\mathrm{V}$ and an internal resistance of $0.850 \Omega .$ It is charged by a $14.7-\mathrm{V}$ power supply for a time interval of 1.80 $\mathrm{h}$ . After charg-ing, the battery returns to its original state as it delivers a constant current to a load resistor over 7.30 $\mathrm{h}$ . Find the efficiency of the battery as an energy storage device. (The efficiency here is defined as the energy delivered to the load during discharge divided by the energy delivered by the $14.7-\mathrm{V}$ power supply during the charging process.)

Sheh Lit Chang

University of Washington

Problem 56

A power supply has an open-circuit voltage of 40.0 $\mathrm{V}$ and an internal resistance of 2.00$\Omega$ . It is used to charge two storage batteries connected in series, each having an emf
of 6.00 $\mathrm{V}$ and internal resistance of 0.300$\Omega$ . If the charging current is to be $4.00 \mathrm{A},$ (a) what additional resistance should be added in series? At what rate does the internal
energy increase in (b) the supply, (c) in the batteries, and (d) in the added series resistance? (e) At what rate does the chemical energy increase in the batteries?

Sheh Lit Chang

University of Washington

Problem 57

When two unknown resistors are connected in series with a battery, the battery delivers 225 $\mathrm{W}$ and carries a total current of 5.00 A. For the same total current, 50.0 $\mathrm{W}$ is delivered when the resistors are connected in parallel. Deter- mine the value of each resistor.

Sheh Lit Chang

University of Washington

Problem 58

When two unknown resistors are connected in series with a battery, the battery delivers 225 $\mathrm{W}$ and carries a total current of 5.00 A. For the same total current, 50.0 $\mathrm{W}$ is delivered when the resistors are connected in parallel. Deter- mine the value of each resistor.

Sheh Lit Chang

University of Washington

Problem 59

The pair of capacitors in Figure $\mathrm{P} 28.59$ are fully charged by a $12.0-\mathrm{V}$ battery. The battery is disconnected, and the switch is then closed. After 1.00 $\mathrm{ms}$ has elapsed, (a) how much charge remains on the $3.00-\mu \mathrm{F}$ capacitor? (b) How much charge remains on the $2.00-\mu \mathrm{F}$ capacitor? (c) What is the current in the resistor at this time?

Sheh Lit Chang

University of Washington

Problem 61

Two resistors R1 and R2 are in parallel with each other. Together they carry total current I. (a) Determine the current in each resistor. (b) Prove that this division of the total current I between the two resistors results in less power delivered to the combination than any other division. It is a general principle that current in a direct current circuit distributes itself so that the total power delivered to the circuit is a
minimum.

Sheh Lit Chang

University of Washington

Problem 62

(a) Determine the equilibrium charge on the capaci- tor in the circuit of Figure P28.62 as a function of R.
(b) Evaluate the charge when R 5 10.0 V. (c) Can the charge on the capacitor be zero? If so, for what value of R? (d) What is the maximum possible magnitude of thecharge on the capacitor? For what value of R is it achieved?
(e) Is it experimentally meaningful to take R 5 `? Explain your answer. If so, what charge magnitude does it imply?

Sheh Lit Chang

University of Washington

Problem 63

The values of the components in a simple series $R C$ circuit containing a switch (Fig. P28.34) are $C=1.00 \mu \mathrm{F}, R=$ $2.00 \times 10^{6} \Omega,$ and $\varepsilon=10.0 \mathrm{V}$ . At the instant 10.0 s after the switch is closed, calculate (a) the charge on the capacitor, (b) the current in the resistor, (c) the rate at which energy is being stored in the capacitor, and (d) the rate at which
energy is being delivered by the battery.

Keshav Singh

Numerade Educator

Problem 64

1. A battery is used to charge a capacitor through a resistor as shown in Figure $\mathrm{P} 28.34$ . Show that half the energy supplied by the battery appears as internal energy in the resistor and half is stored in the capacitor.

Sheh Lit Chang

University of Washington

Problem 65

A young man owns a canister vacuum cleaner marked $^{2} 535 \mathrm{W}$ [at] 120 $\mathrm{V}^{n}$ and a Volkswagen Beetle, which he wishes to clean. He parks the car in his apartment parking lot and uses an inexpensive extension cord 15.0 m long to plug in the vacuum cleaner. You may assume the cleaner has constant resistance. (a) If the resistance of each of the two conductors in the extension cord is 0.900 V, what is the actual power delivered to the cleaner? (b) If instead the power is to be at least 525 W, what must be the diameter of each of two identical copper conductors in the cord he
buys? (c) Repeat part (b) assuming the power is to be at\ least 532 W.

Sheh Lit Chang

University of Washington

Problem 66

Three identical 60.0-W, 120-V lightbulbs are connected across a 120-V power source as shown in Figure P28.66. Assuming the resistance of each lightbulb is constant (even though in reality the resistance might increase markedly with current), find (a) the total power supplied by the power source and (b) the potential difference across each lightbulb.

Sheh Lit Chang

University of Washington

Problem 67

Switch S shown in Figure $P 28.67$ has been closed for a long time, and the electric circuit carries a constant current. Take $C_{1}=3.00 \mu \mathrm{F}, C_{2}=6.00 \mu \mathrm{F}, R_{1}=4.00 \mathrm{k} \Omega,$ and $R_{2}=$ $7.00 \mathrm{k} \Omega .$ The power delivered to $R_{2}$ is 2.40 $\mathrm{W}$ . (a) Find the charge on $C_{1} .$ (b) Now the switch is opened. After many milliseconds, by how much has the charge on $C_{2}$ changed?

Artemisa Mazón

Numerade Educator

Problem 68

An ideal voltmeter connected across a certain fresh 9-V battery reads 9.30 V, and an ideal ammeter briefly connected across the same battery reads 3.70 A. We say the battery has an open-circuit voltage of 9.30 V and a short-circuit current of 3.70 A. Model the battery as a source of emf e in series with an internal resistance r as in Active Figure 28.1a. Determine both (a) e and (b) r. An experimenter connects two of these identical batteries together as shown in Figure P28.68. Find (c) the open-circuit voltage
and (d) the short-circuit current of the pair of connected batteries. (e) The experimenter connects a 12.0-V resistor between the exposed terminals of the connected batteries. Find the current in the resistor. (f) Find the power delivered to the resistor. (g) The experimenter connects a

Sheh Lit Chang

University of Washington

Problem 69

A regular tetrahedron is a pyramid with a triangular base and triangular sides as shown in Figure P28.69. Imagine the six straight lines in Figure P28.69 are each 10.0-V resistors, with junctions at the four vertices. A 12.0-V battery is connected to any two of the vertices. Find (a) the equivalent resistance of the tetrahedron between these vertices and (b) the current in the battery.

Sheh Lit Chang

University of Washington

Problem 70

Figure P28.70 shows a circuit model for the transmission of an electrical signal such as cable TV to a large number of subscribers. Each subscriber connects a load resistance RL between the transmission line and the ground. The ground is assumed to be at zero potential and able to carry any current between any ground connections with negligible resistance. The resistance of the transmission line between the connection points of different subscribers is modeled as the constant resistance RT . Show that the
equivalent resistance across the signal source is
$R_{\mathrm{eq}}=\frac{1}{2}\left[\left(4 R_{T} R_{L}+R_{T}^{2}\right)^{1 / 2}+R_{T}\right]$

Sheh Lit Chang

University of Washington

Problem 71

In Figure $\mathrm{P} 28.71,$ suppose the switch has been closed for a time interval sufficiently long for the capacitor to become fully charged. Find (a) the steady-state current in each resistor and (b) the charge $Q$ on the capacitor. (c) The switch is now opened at $t=0 .$ Write an equation for the current in $R_{2}$ as a function of time and (d) find the time interval required for the charge on the capacitor to fall to one-fifth its initial value.

Sheh Lit Chang

University of Washington

Problem 72

The circuit shown in Figure $\mathrm{P} 28.72$ is set up in the laboratory to measure an unknown capacitance $\mathrm{Cin}$ series with a resistance $R=10.0 \mathrm{M} \Omega$ powered by a battery whose emf is 6.19 $\mathrm{V}$ . The data given in the table are the measured volt-
ages across the capacitor as a function of time, where $t=$ 0 represents the instant at which the switch is thrown to position $b$ . (a) Construct a graph of $\ln (\mathcal{E} / \Delta V)$ versus $t$ and
perform a linear least-squares fit to the data. (b) From the slope of your graph, obtain a value for the time constant of the circuit and a value for the capacitance.

Sheh Lit Chang

University of Washington

Problem 73

The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P28.73). The unknown resistance $R_{x}$ is between points $C$ and $E .$ Point $E$ is a true ground, but it is inaccessible for direct measurement because this stratum is several meters below the Earth's surface. Two identical rods are driven into the ground at $A$ and $B$ , introducing an unknown resistance $R_{y}$ . The procedure is as follows. Measure resistance $R_{1}$ between points $A$ and $B,$ then connect $A$ and $B$ with a heavy conducting wire and measure resistance $R_{2}$ between points $A$ and $C .$ (a) Derive an equation for $R_{x}$ in terms of the observable resistances, $R_{1}$ and $R_{2}$ . (b) $\mathrm{A}$ satisfactory ground resistance would be $R_{x}<2.00 \Omega$ . Is the grounding of the station adequate if measurements give $R_{1}=13.0 \Omega$ and $R_{2}=6.00 \Omega$ ? Explain.

Sheh Lit Chang

University of Washington

Problem 74

In places such as hospital operating rooms or factories for electronic circuit boards, electric sparks must be avoided. A person standing on a grounded floor and touching nothing else can typically have a body capacitance of 150 pF, in parallel with a foot capacitance of 80.0 pF produced by the dielectric soles of his or her shoes. The person acquires static electric charge from interactions with his or her surroundings. The static charge flows to ground through the equivalent resistance of the two shoe soles in parallel with each other. A pair of rubber-soled street shoes can present an equivalent resistance of 5.00 3 103 MV. A pair of shoes with special static-dissipative soles can have an equivalent resistance of 1.00 MV. Consider the person’s body and shoes as forming an RC circuit with the ground. (a) How long does it take the rubber-soled shoes to reduce a person's potential from $3.00 \times 10^{3} \mathrm{V}$ to 100 $\mathrm{V} ?$ (b) How long does it take the static-dissipative shoes to do the same thing?

Sheh Lit Chang

University of Washington

Problem 75

An electric teakettle has a multi position switch and two heating coils. When only one coil is switched on, the well-insulated kettle brings a full pot of water to a boil over the time interval $\Delta t$ . When only the other coil is switched on, it takes a time interval of 2$\Delta t$ to boil the same amount of water. Find the time interval required to boil the same amount of water if both coils are switched on (a) in a parallel connection and (b) in a series connection.

Sheh Lit Chang

University of Washington

Problem 76

A voltage $\Delta V$ is applied to a series configuration of $n$ resistors, each of resistance $R$ . The circuit components are reconnected in a parallel configuration, and voltage $\Delta V$ is again applied. Show that the power delivered to the series configuration is 1$/ n^{2}$ times the power delivered to the parallel configuration.

Sheh Lit Chang

University of Washington

Problem 77

The resistor $R$ in Figure $\mathrm{P} 28.77$ receives 20.0 $\mathrm{W}$ of power. Determine the value of $R .$

Sheh Lit Chang

University of Washington

Problem 78

The switch in Figure $\mathrm{P} 28.78 \mathrm{a}$ closes when $\Delta V_{c}>\frac{2}{3} \Delta V$ and opens when $\Delta V_{c}<\frac{1}{3} \Delta V .$ The ideal voltmeter reads a potential difference as plotted in Figure $\mathrm{P} 28.78 \mathrm{b}$ . What is the period $T$ of the waveform in terms of $R_{1}, R_{2},$ and $C ?$

Sheh Lit Chang

University of Washington