Physics for Scientists and Engineers

Raymond A. Serway, John W. Jewett

Chapter 28

Direct Current Circuits - all with Video Answers

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Chapter Questions

Problem 1

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

Yaqub Khan

Yaqub Khan

Numerade Educator

Problem 2

(a) What is the current in a $5.60-\Omega$ resistor connected to a battery that has a $0.200-\Omega$ internal resistance if the terminal voltage of the battery is $10.0 \mathrm{V} ?$ (b) What is the emf of the battery?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 3

Two $1.50-\mathrm{V}$ batteries - with their positive terminals in the same direction - are inserted in series into the barrel of a flashlight. One battery has an internal resistance of $0.255 \Omega$ the other an internal resistance of $0.153 \Omega .$ When the switch is closed, a current of 600 mA occurs in the lamp.
(a) What is the lamp's resistance? (b) What fraction of the chemical energy transformed appears as internal energy in the batteries?

Sophie S

Numerade Educator

Problem 4

An automobile battery has an emf of $12.6 \mathrm{V}$ and an internal resistance of $0.0800 \Omega .$ The headlights together present equivalent resistance $5.00 \Omega$ (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, taking an additional 35.0 A from the battery?

Shahab Ullah

Numerade Educator

Problem 5

The current in a loop circuit that has a resistance of $R_{1}$ is 2.00 A. The current is reduced to $1.60 \mathrm{A}$ when an additional resistor $R_{2}=3.00 \Omega$ is added in series with $R_{1}$ What is the value of $R_{1} ?$

Shahab Ullah

Numerade Educator

Problem 6

(a) Find the equivalent resistance between points $a$ and $b$ in Figure P28.6. (b) A potential difference of $34.0 \mathrm{V}$ is applied between points $a$ and $b .$ Calculate the current in each resistor.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 7

A lightbulb marked "75 W [at] $120 \mathrm{V}^{\text {m }}$ 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. Draw a circuit diagram and find the actual power delivered to the bulb in this circuit.

Artemisa Mazón

Numerade Educator

Problem 8

Four copper wires of equal length are connected in series. Their cross-sectional areas are $1.00 \mathrm{cm}^{2}, 2.00 \mathrm{cm}^{2}$ $3.00 \mathrm{cm}^{2},$ and $5.00 \mathrm{cm}^{2} .$ A potential difference of $120 \mathrm{V}$ is applied across the combination. Determine the voltage across the $2.00-\mathrm{cm}^{2}$ wire.

Shahab Ullah

Numerade Educator

Problem 9

Consider the circuit shown in Figure P28.9. Find
(a) the current in the $20.0-\Omega$ resistor and (b) the potential difference between points $a$ and $b$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 10

For the purpose of measuring the electric resistance of shoes through the body of the wearer to a metal ground plate, the American National Standards Institute (ANSI) specifies the circuit shown in Figure P28.10. The potential difference $\Delta$Vacross the $1.00-$ M\Omega resistor is measured with a high-resistance voltmeter. (a) Show that the resistance of the footwear is given by
$$R_{\text {shoes }}=1.00 \mathrm{M} \Omega\left(\frac{50.0 \mathrm{V}-\Delta V}{\Delta V}\right)$$
(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.

Artemisa Mazón

Numerade Educator

Problem 11

Three $100-\Omega$ resistors are connected as shown in Figure P28.11. The maximum power that can safely be delivered to any one resistor is $25.0 \mathrm{W}$. (a) What is the maximum voltage that can be applied to the terminals $a$ and b? For the voltage determined in part (a), what is the power delivered to each resistor? What is the total power delivered?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 12

Using only three resistors- $2.00 \Omega, 3.00 \Omega,$ and $4.00 \Omega$ find 17 resistance values that may be obtained by various combinations of one or more resistors. Tabulate the combinations in order of increasing resistance.

Sophie S

Numerade Educator

Problem 13

The current in a circuit is tripled by connecting a $500-\Omega$ resistor in parallel with the resistance of the circuit. Determine the resistance of the circuit in the absence of the $500-\Omega$ resistor.

Shahab Ullah

Numerade Educator

Problem 14

A $6.00-\mathrm{V}$ battery supplies current to the circuit shown in Figure P28.14. When the double-throw switch 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 $1,$ the current in the battery is $1.20 \mathrm{mA}$. When the switch is closed in position 2 the current in the battery is 2.00 mA. Find the resistances $R_{1}, R_{2},$ and $R_{3}$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 15

Calculate the power delivered to each resistor in the circuit shown in Figure P28.15.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 16

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 17

An electric teakettle has a multiposition switch and two heating coils. When only one of the coils 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 requires 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.

Shahab Ullah

Numerade Educator

Problem 18

In Figures 28.4 and $28.6,$ let $R_{1}=11.0 \Omega, R_{2}=22.0 \Omega$ and let the battery have a terminal voltage of $33.0 \mathrm{V} .$ (a) In the parallel circuit shown in Figure $28.6,$ to which resistor is more power delivered? (b) Verify that the sum of the power $\left(I^{2} R\right)$ delivered to each resistor equals the power supplied by the battery $(\mathscr{P}=I \Delta V) .$ (c) In the series circuit, which resistor uses more power? (d) Verify that the sum of the power $\left(I^{2} R\right)$ used by each resistor equals the power supplied by the battery $(\mathscr{P}=I \Delta V) .$ (e) Which circuit configuration uses more power?

Artemisa Mazón

Numerade Educator

Problem 19

Four resistors are connected to a battery as shown in Figure P28.19. The current in the battery is $I$, the battery emf is $\boldsymbol{\varepsilon},$ and the resistor values are $R_{1}=R, R_{2}=2 R$ $R_{3}=4 R, R_{4}=3 R .$ (a) Rank the resistors according to the potential difference across them, from largest to smallest. Note any cases of equal potential differences. (b) Determine the potential difference across each resistor in terms of $\mathcal{E} .$ (c) Rank the resistors according to the current in them, from largest to smallest. Note any cases of equal currents. (d) Determine the current in each resistor in terms of $I$ (e) What If? If $R_{3}$ is increased, what happens to the current in each of the resistors? (f) 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?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 20

Note: The currents are not necessarily in the direction shown for some circuits.
The ammeter shown in Figure $\mathrm{P} 28.20$ reads $2.00 \mathrm{A}$. Find $I_{1}, I_{2},$ and $\mathcal{E}$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 21

Determine the current in each branch of the circuit shown in Figure P28.21.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 22

In Figure $\mathrm{P} 28.21,$ show how to add just enough ammeters to measure every different current. Show how to add just enough voltmeters to measure the potential difference across each resistor and across each battery.

Vishal Gupta

Numerade Educator

Problem 23

The circuit considered in Problem 21 and shown in Figure P28.21 is connected for 2.00 min. (a) Find the energy delivered by each battery. (b) Find the energy delivered to each resistor. (c) Identify the types of energy transformations that occur in the operation of the circuit and the total amount of energy involved in each type of transformation.

Artemisa Mazón

Numerade Educator

Problem 24

Using Kirchhoff's rules, (a) find the current in each resistor in Figure P28.24. (b) Find the potential difference between points $c$ and $f .$ Which point is at the higher potential?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 25

Taking $R=1.00 \mathrm{k} \Omega$ and $\varepsilon=250 \mathrm{V}$ in Figure $\mathrm{P} 28.25$
determine the direction and magnitude of the current in the horizontal wire between $a$ and $e$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 26

In the circuit of Figure P28.26, determine the current in each resistor and the voltage across the $200-\Omega$ resistor.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 27

A dead battery is charged by connecting it to the live battery of another car with jumper cables (Fig. P28.27). Determine the current in the starter and in the dead battery.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 28

For the network shown in Figure $\mathrm{P} 28.28,$ show that the resistance $R_{a b}=(27 / 17) \Omega$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 29

$\rightarrow$ For the circuit shown in Figure $\mathrm{P} 28.29,$ calculate (a) the current in the $2.00-\Omega$ resistor and (b) the potential difference between points $a$ and $b$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 30

Calculate the power delivered to each resistor shown in Figure P28.30.
FIGURE CANT COPY

Brandy Heflin

Numerade Educator

Problem 31

Consider a series $R C$ circuit (see Fig. 28.19 ) for which $R=1.00 \mathrm{M} \Omega, C=5.00 \mu \mathrm{F},$ and $\boldsymbol{\varepsilon}=30.0 \mathrm{V} .$ Find (a) the
time constant of the circuit and (b) the maximum charge on the capacitor after the switch is closed. (c) Find the current in the resistor 10.0 s after the switch is closed.

Shahab Ullah

Numerade Educator

Problem 32

A 2.00 -nF capacitor with an initial charge of $5.10 \mu \mathrm{C}$ is discharged through a $1.30-\mathrm{k} \Omega$ resistor. (a) Calculate the current in the resistor $9.00 \mu \mathrm{s}$ 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?

Sophie S

Numerade Educator

Problem 33

A fully charged capacitor stores energy $U_{0} .$ How much energy remains when its charge has decreased to half its original value?

Shahab Ullah

Numerade Educator

Problem 34

A capacitor in an $R C$ circuit is charged to $60.0 \%$ of its maximum value in 0.900 s. What is the time constant of the circuit?

Shahab Ullah

Numerade Educator

Problem 35

Show that the integral in Equation (1) of Example 28.14 has the value $R C / 2$

Artemisa Mazón

Numerade Educator

Problem 36

In the circuit of Figure $\mathrm{P} 28.36,$ 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.
FIGURE CANT COPY

Supratim Pal

Numerade Educator

Problem 37

The circuit in Figure $\mathrm{P} 28.37$ has been connected for a long time. (a) What is the voltage across the capacitor? (b) If the battery is disconnected, how long does it take the capacitor to discharge to one tenth of its initial voltage?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 38

In places such as a hospital operating room and a factory 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 $150 \mathrm{pF}$, in parallel with a foot capacitance of $80.0 \mathrm{pF}$ produced by the dielectric soles of his or her shoes. The person acquires static electric charge from interactions with furniture, clothing, equipment, packaging materials, and essentially everything else. The static charge is conducted 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 5000 M\Omega. A pair of shoes with special static-dissipative soles can have an equivalent resistance of 1.00 M\Omega. Consider the person's body and shoes as forming an $R C$ circuit with the ground. (a) How long does it take the rubber-soled shoes to reduce a $3000-\mathrm{V}$ static charge to $100 \mathrm{V} ?$ (b) How long does it take the staticdissipative shoes to do the same thing?

Artemisa Mazón

Numerade Educator

Problem 39

A 4.00 -M\Omega resistor and a 3.00 - $\mu$ F capacitor are connected in series with a $12.0-\mathrm{V}$ power supply. (a) What is the time constant for the circuit? (b) Express the current in the circuit and the charge on the capacitor as functions of time.

Shahab Ullah

Numerade Educator

Problem 40

Dielectric materials used in the manufacture of capacitors are characterized by conductivities that are small but not zero. Therefore, a charged capacitor slowly loses its charge by "leaking" across the dielectric. If a capacitor having capacitance $C$ leaks charge such that the potential difference has decreased to half its initial $(t=0)$ value at a time $t,$ what is the equivalent resistance of the dielectric?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 41

Assume that a galvanometer has an internal resistance of $60.0 \Omega$ and requires a current of $0.500 \mathrm{mA}$ to produce fullscale deflection. What resistance must be connected in parallel with the galvanometer if the combination is to serve as an ammeter that has a full-scale deflection for a current of $0.100 \mathrm{A} ?$

Artemisa Mazón

Numerade Educator

Problem 42

A typical galvanometer, which requires a current of $1.50 \mathrm{mA}$ for full-scale deflection and has a resistance of $75.0 \Omega,$ may be used to measure currents of much greater values. To enable an operator to measure large currents without damage to the galvanometer, a relatively small shunt resistor is wired in parallel with the galvanometer, as suggested in Figure $28.27 .$ Most of the current then goes through the shunt resistor. Calculate the value of the shunt resistor that allows the galvanometer to be used to measure a current of $1.00 \mathrm{A}$ at full-scale deflection. (Suggestion: use Kirchhoff's rules.)

Artemisa Mazón

Numerade Educator

Problem 43

The same galvanometer described in the previous problem may be used to measure voltages. In this case a large resistor is wired in series with the galvanometer, as suggested in Figure $28.29 .$ The effect is to limit the current in the galvanometer when large voltages are applied. Most of the potential drop occurs across the resistor placed in series. Calculate the value of the resistor that allows the galvanometer to measure an applied voltage of $25.0 \mathrm{V}$ at full-scale deflection.

Artemisa Mazón

Numerade Educator

Problem 44

Meter loading. Work this problem to five-digit precision. Refer to Figure P28.44. (a) When a $180.00-\Omega$ resistor is connected across a battery of emf $6.0000 \mathrm{V}$ and internal resistance $20.000 \Omega,$ what is the current in the resistor? What is the potential difference across it? (b) Suppose now an ammeter of resistance $0.50000 \Omega$ and a voltmeter of resistance $20000 \Omega$ are added to the circuit as shown in Figure P28.44b. Find the reading of each. (c) What If? Now one terminal of one wire is moved, as shown in Figure P28.44c. Find the new meter readings.

Artemisa Mazón

Numerade Educator

Problem 45

Design a multirange ammeter capable of full-scale deflection for $25.0 \mathrm{mA}, 50.0 \mathrm{mA},$ and $100 \mathrm{mA} .$ Assume the meter movement is a galvanometer that has a resistance of $25.0 \Omega$ and gives a full-scale deflection for $1.00 \mathrm{mA}$

Artemisa Mazón

Numerade Educator

Problem 46

Design a multirange voltmeter capable of full-scale deflection for $20.0 \mathrm{V}, 50.0 \mathrm{V},$ and $100 \mathrm{V} .$ Assume the meter movement is a galvanometer that has a resistance of $60.0 \Omega$ and gives a full-scale deflection for a current of $1.00 \mathrm{mA}$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 47

A particular galvanometer serves as a $2.00-\mathrm{V}$ full-scale voltmeter when a $2500-\Omega$ resistor is connected in series with it. It serves as a $0.500-$ A full-scale ammeter when a $0.220-\Omega$ resistor is connected in parallel with it. Determine the internal resistance of the galvanometer and the current required to produce full-scale deflection.

Sophie S

Numerade Educator

Problem 48

An $8.00-$ ft extension cord has two 18 -gauge copper wires, each having a diameter of $1.024 \mathrm{mm}$. At what rate is energy delivered to the resistance in the cord when it is carrying a current of (a) $1.00 \mathrm{A}$ and (b) $10.0 \mathrm{A} ?$

Artemisa Mazón

Numerade Educator

Problem 49

An electric heater is rated at $1500 \mathrm{W},$ a toaster at $750 \mathrm{W},$ and an electric grill at $1000 \mathrm{W}$. The three appliances are connected to a common $120-\mathrm{V}$ household circuit. (a) How much current does each draw? (b) Is a circuit with a 25.0 -A circuit breaker sufficient in this situation? Explain your answer.

Sophie S

Numerade Educator

Problem 50

Aluminum wiring has sometimes been used instead of copper for economy. According to the National Electrical Code, the maximum allowable current for 12 -gauge copper wire with rubber insulation is $20 \mathrm{A}$. What should be the maximum allowable current in a 12 -gauge aluminum wire if the power per unit length delivered to the resistance in the aluminum wire is the same as that delivered in the copper wire?

Artemisa Mazón

Numerade Educator

Problem 51

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. You may assume that at a typical instant the conductor inside the lamp cord next to your thumb is at potential $\sim 10^{2} \mathrm{V}$ and that the conductor next to your index finger is at ground potential (0 $\mathrm{V}$ ). The resistance of your hand depends strongly on the thickness and the moisture content of the outer layers of your skin. Assume that the resistance of your hand between fingertip and thumb tip is $\sim 10^{4} \Omega .$ 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 that your body is isolated from any other charges or currents. In order-of-magnitude terms describe the potential of your thumb where it contacts the cord, and the potential of your finger where it touches the cord.

Dominador Tan

Numerade Educator

Problem 52

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

Sophie S

Numerade Educator

Problem 53

A battery has an emf of $9.20 \mathrm{V}$ and an internal resistance of $1.20 \Omega$. (a) What resistance across the battery will extract from it a power of $12.8 \mathrm{W} ?$ (b) a power of $21.2 \mathrm{W} ?$

Artemisa Mazón

Numerade Educator

Problem 54

Calculate the potential difference between points $a$ and $b$ in Figure $\mathrm{P} 28.54$ and identify which point is at the higher potential.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 55

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

Shahab Ullah

Numerade Educator

Problem 56

A group of students on spring break manages to reach a deserted island in their wrecked sailboat. They splash ashore with fuel, a European gasoline-powered $240-\mathrm{V}$ generator, a box of North American $100-\mathrm{W}$ 120-V lightbulbs, a 500-W 120-V hot pot, lamp sockets, and some insulated wire. While waiting to be rescued, they decide to use the generator to operate some lightbulbs. (a) Draw a diagram of a circuit they can use, containing the minimum number of lightbulbs with $120 \mathrm{V}$ across each bulb, and no higher voltage. Find the current in the generator and its power output. (b) One student catches a fish and wants to cook it in the hot pot. Draw a diagram of a circuit containing the hot pot and the minimum number of lightbulbs with $120 \mathrm{V}$ across each device, and not more. Find the current in the generator and its power output.

Mayukh Banik

Numerade Educator

Problem 57

A battery has an emf $\mathcal{E}$ and internal resistance $r$. A variable load resistor $R$ is connected across the terminals of the battery. (a) Determine the value of $R$ such that the potential difference across the terminals is a maximum. (b) Determine the value of $R$ so that the current in the circuit is a maximum. (c) Determine the value of $R$ so that the power delivered to the load resistor is a maximum. Choosing the load resistance for maximum power transfer is a case of what is called impedance matching in general. Impedance matching is important in shifting gears on a bicycle, in connecting a loudspeaker to an audio amplifier, in connecting a battery charger to a bank of solar photoelectric cells, and in many other applications.

Artemisa Mazón

Numerade Educator

Problem 58

A 10.0 - $\mu$ 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 $3.00 \mathrm{s}$ after charging begins. Find $R$

Shahab Ullah

Numerade Educator

Problem 59

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

Shahab Ullah

Numerade Educator

Problem 60

When two unknown resistors are connected in series with a battery, the battery delivers total power $\mathscr{P}_{s}$ and carries a total current of $I .$ For the same total current, a total power $\mathscr{P}_{p}$ is delivered when the resistors are connected in parallel. Determine the values of the two resistors.

Shahab Ullah

Numerade Educator

Problem 61

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? (b) At what rate does the internal energy increase in the supply, in the batteries, and in the added series resistance? (c) At what rate does the chemical energy increase in the batteries?

Sophie S

Numerade Educator

Problem 62

Two resistors $R_{1}$ and $R_{2}$ 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.

Sophie S

Numerade Educator

Problem 63

The value of a resistor $R$ is to be determined using the ammeter-voltmeter setup shown in Figure P28.63. The ammeter has a resistance of $0.500 \Omega,$ and the voltmeter has a resistance of $20000 \Omega$. Within what range of actual values of $R$ will the measured values be correct to within $5.00 \%$ if the measurement is made using the circuit shown in (a) Figure $\mathrm{P} 28.63 \mathrm{a}$ and $(\mathrm{b})$ Figure $\mathrm{P} 28.63 \mathrm{b} ?$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 64

A battery is used to charge a capacitor through a resistor, as shown in Figure $28.19 .$ Show that half the energy supplied by the battery appears as internal energy in the resistor and that half is stored in the capacitor.

Artemisa Mazón

Numerade Educator

Problem 65

The values of the components in a simple series $R C$ circuit containing a switch (Fig. 28.19 ) are $C=1.00 \mu \mathrm{F}, R=2.00 \times$ $10^{6} \Omega,$ and $\mathcal{E}=10.0 \mathrm{V} .$ At the instant $10.0 \mathrm{s}$ 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.

Artemisa Mazón

Numerade Educator

Problem 66

The switch in Figure $\mathrm{P} 28.66 \mathrm{a}$ closes when $\Delta V_{c}>2 \Delta V / 3$ and opens when $\Delta V_{c}<\Delta V / 3 .$ The voltmeter reads a voltage as plotted in Figure P28.66b. What is the period $T$ of the waveform in terms of $R_{1}, R_{2},$ and $C ?$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 67

Three $60.0-\mathrm{W}, 120-\mathrm{V}$ lightbulbs are connected across a 120-V power source, as shown in Figure P28.67. Find
(a) the total power delivered to the three bulbs and
(b) the voltage across each. Assume that the resistance of each bulb is constant (even though in reality the resistance might increase markedly with current).
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 68

Switch S has been closed for a long time, and the electric circuit shown in Figure $\mathrm{P} 28.68$ carries a constant current. Take $\quad C_{1}=3.00 \mu \mathrm{F}, \quad C_{2}=6.00 \mu \mathrm{F}, \quad R_{1}=4.00 \mathrm{k} \Omega, \quad$ 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?
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 69

Four resistors are connected in parallel across a $9.20-\mathrm{V}$ battery. They carry currents of $150 \mathrm{mA}, \quad 45.0 \mathrm{mA}$ $14.00 \mathrm{mA},$ and $4.00 \mathrm{mA} .$ (a) If the resistor with the largest resistance is replaced with one having twice the resistance, 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 heat leaks, including the following: $1500 \mathrm{W}$ 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?

Artemisa Mazón

Numerade Educator

Problem 70

$.$ Figure $\mathrm{P} 28.70$ 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 $R_{L}$ 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 itself between the connection points of different subscribers is modeled as the constant resistance $R_{T} .$ Show that the equivalent resistance across the signal source is
$$R_{\mathrm{eq}}=\frac{1}{2}\left[\left(4 R_{T} R_{L}+R_{T}^{2}\right)^{1 / 2}+R_{T}\right]$$
Suggestion: Because the number of subscribers is large, the equivalent resistance would not change noticeably if the first subscriber cancelled his service. Consequently, the equivalent resistance of the section of the circuit to the right of the first load resistor is nearly equal to $R_{\mathrm{eq}}$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 71

$\Rightarrow$ In Figure $\mathrm{P} 28.71,$ suppose the switch has been closed for a time 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 $I_{R_{2}}$ through $R_{2}$ 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.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 72

A regular tetrahedron is a pyramid with a triangular base. Six $10.0-\Omega$ resistors are placed along its six edges, with junctions at its four vertices. A $12.0-\mathrm{V}$ 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.

Sophie S

Numerade Educator

Problem 73

The circuit shown in Figure $\mathrm{P} 28.73$ is set up in the laboratory to measure an unknown capacitance $C$ with the use of a voltmeter of resistance $R=10.0 \mathrm{M} \Omega$ and a battery whose emf is $6.19 \mathrm{V} .$ The data given in the table are the measured voltages across the capacitor as a function of time, where $t=0$ represents the instant at which the switch is opened. (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.
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 74

The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P28.74). The unknown resistance $R_{x}$ is between points $C$ and $E .$ Point $E$ is a true ground but is inaccessible for direct measurement since 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) 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 $R_{2}=6.00 \Omega ?$
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 75

The circuit in Figure $P 28.75$ contains two resistors, $R_{1}=2.00 \mathrm{k} \Omega \quad$ and $\quad R_{2}=3.00 \mathrm{k} \Omega, \quad$ and two capacitors,
$C_{1}=2.00 \mu \mathrm{F}$ and $C_{2}=3.00 \mu \mathrm{F},$ connected to a battery with emf $\boldsymbol{\varepsilon}=120 \mathrm{V} .$ No charge is on either capacitor before switch $\mathrm{S}$ is closed. Determine the charges $q_{1}$ and $q_{2}$ on capacitors $C_{1}$ and $C_{2},$ respectively, after the switch is closed. (Suggestion: First reconstruct the circuit so that it becomes a simple $R C$ circuit containing a single resistor and single capacitor in series, connected to the battery, and then determine the total charge $q$ stored in the equivalent circuit.)
FIGURE CANT COPY

Artemisa Mazón

Numerade Educator

Problem 76

This problem $^{6}$ illustrates how a digital voltmeter affects the voltage across a capacitor in an $R C$ circuit. A digital voltmeter of internal resistance $r$ is used to measure the voltage across a capacitor after the switch in Figure P28.76 is closed. Because the meter has finite resistance, part of the current supplied by the battery passes through the meter. (a) Apply Kirchhoff's rules to this circuit, and use the fact that $i_{C}=d q / d t$ to show that this leads to the differential equation
$$R_{\mathrm{eq}} \frac{d q}{d t}+\frac{q}{C}=\frac{r}{r+R} \boldsymbol{\varepsilon}$$
where $R_{\mathrm{eq}}=r R /(r+R) .$ (b) Show that the solution to this differential equation is
$$q=\frac{r}{r+R} C \mathcal{E}\left(1-e^{-t / R_{e q} C}\right)$$ and that the voltage across the capacitor as a function of time is
$$V_{C}=\frac{r}{r+R} \boldsymbol{\varepsilon}\left(1-e^{-t / R_{e q} C}\right)$$
(c) What If? If the capacitor is fully charged, and the switch is then opened, how does the voltage across the capacitor behave in this case?

Artemisa Mazón

Numerade Educator