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

+ 1 more educators

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?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

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 . When the switch is closed, a current of 600 $\mathrm{mA}$ 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 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 $\mathcal{E}$ 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.

Sophie S

Numerade Educator

Problem 5

(a) Find the equivalent resistance between points $a$ and $b$
in Figure $\mathrm{P} 28.5 .$ (b) A potential difference of 34.0 $\mathrm{V}$ is
applied between points $a$ and $b .$ Calculate the current in each resistor.

Shoukat Ali

Other Schools

Problem 6

A lightbulb marked "75 W [at] 120 V" 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-V outlet.
(a) Explain why the actual power delivered to the lightbulb cannot be 75 $\mathrm{W}$ in this situation. (b) What can you reasonably model as constant about the lightbulb? Draw a circuit diagram and find the actual power delivered to the lightbulb in this circuit.

Sophie S

Numerade Educator

Problem 7

A Consider the circuit shown in Figure $\mathrm{P} 28.7$ . Find (a) the current in the $20.0-\Omega$ resistor and (b) the potential difference between points $a$ and $b .$

Chris Johnson

Numerade Educator

Problem 8

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 $\mathrm{P} 28.8$ . The potential
difference $\Delta V$ across the $1.00-\mathrm{M} \Omega$ resistor is measured
with a high-resistance voltmeter. (a) Show that the resistance of the footwear is
$$
R_{\text { shoes }}=1.00 \mathrm{M\Omega}\left(\frac{50.0 \mathrm{V}-\Delta V}{\Delta V}\right)
$$
(b) In a medical test, a current through the human body
should not exceed 150$\mu A$ . Can the current delivered by
the ANSI-specified circuit exceed 150$\mu A$ ? To decide, consider a person standing barefoot on the ground plate.

Sophie S

Numerade Educator

Problem 9

Three $100-\Omega$ resistors are connected as shown in Figure
$\mathrm{P} 28.9 .$ 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? What is the total power delivered?

Sophie S

Numerade Educator

Problem 10

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 11

A 6.00 -V battery 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 $R_{1}, R_{2},$ and $R_{3} .$

Sophie S

Numerade Educator

Problem 12

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

Nishant Kumar

Numerade Educator

Problem 13

When the switch $\mathrm{S}$ in the circuit of Figure $\mathrm{P} 28.13$ is
closed, will the equivalent resistance between points $a$ and
$b$ increase or decrease? State your reasoning. Assume the
equivalent resistance changes by a factor of $2 .$ Determine
the value of $R .$

Sophie S

Numerade Educator

Problem 14

Four resistors are connected to a battery as shown in
Figure $\mathrm{P} 28.14$ . The current in the battery is $I$ ; the battery
emf is $\mathcal{E} ;$ and the resistor values are $R_{1}=R, R_{2}=2 R, R_{3}=$
$4 R,$ and $R_{4}=3 R$ (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 $R_{3}$ increased, explain 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?

Sophie S

Numerade Educator

Problem 15

Calculate the power delivered to each resistor in the circuit shown in Figure $\mathrm{P} 28.15$ .

BN

Braden Nyberg

Numerade Educator

Problem 16

The ammeter shown in Figure $\mathrm{P} 28.16$ reads 2.00 A. Find
$I_{1}, I_{2},$ and $\mathcal{E} .$

Sophie S

Numerade Educator

Problem 17

A Determine the current in each branch of the circuit
shown in Figure $\mathrm{P} 28.17$ .

Sophie S

Numerade Educator

Problem 18

In Figure $\mathrm{P} 28.17$ , show how to add only enough ammeters to measure every different current. Show how to add
only enough voltmeters to measure the potential difference across each resistor and across each battery.

Sophie S

Numerade Educator

Problem 19

The circuit considered in Problem 17 and shown in
Figure $\mathrm{P} 28.17$ is connected for $2.00 \mathrm{min} .$ (a) Find the
energy delivered by each battery. (b) Find the energy
delivered to each resistor. (c) Identify the net energy
transformation that occurs in the operation of the circuit.
Find the total amount of energy transformed.

Sophie S

Numerade Educator

Problem 20

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

Sophie S

Numerade Educator

Problem 21

Consider the circuit shown in Figure $\mathrm{P} 28.21 .$ What are
the expected readings of the ideal ammeter and ideal
voltmeter?

Sophie S

Numerade Educator

Problem 22

Taking $R=1.00 \mathrm{k} \Omega$ and $\mathcal{E}=250 \mathrm{V}$ in Figure $\mathrm{P} 28.22$
determine the direction and magnitude of the current in
the horizontal wire between $a$ and $e .$

Sophie S

Numerade Educator

Problem 23

In the circuit of Figure $\mathrm{P} 28.23$ , determine the current in
each resistor and the potential difference across the $200-\Omega$
resistor

Sophie S

Numerade Educator

Problem 24

A dead battery is charged by connecting it to the live battery of another car with jumper cables (Fig. P28.24). Determine the current in the starter and in the dead battery.

Sophie S

Numerade Educator

Problem 25

For the circuit shown in Figure $\mathrm{P} 28.25$ , 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 26

For the network shown in Figure $\mathrm{P} 28.26$ , show that the resistance $R_{a b}=\frac{27}{17} \Omega .$

Sophie S

Numerade Educator

Problem 27

A Consider a series $R C$ circuit (see Active Fig. 28.16 ) for
which $R=1.00 \mathrm{M} \Omega, C=5.00 \mu \mathrm{F},$ and $\varepsilon=30.0 \mathrm{V}$ . Find
(a) the time constant of the circuit and $(\mathrm{b})$ the maximum
charge on the capacitor after the switch is thrown to $a$ ,
connecting the capacitor to the battery. (c) Find the current in the resistor 10.0 $\mathrm{s}$ after the switch is thrown to $a$ .

Sophie S

Numerade Educator

Problem 28

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 29

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$\mu$ 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 30

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

Sophie S

Numerade Educator

Problem 31

The circuit in Figure $\mathrm{P} 28.31$ 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 of its initial voltage?

Vishal Gupta

Numerade Educator

Problem 32

In the circuit of Figure $\mathrm{P} 28.32$ , 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.

Sophie S

Numerade Educator

Problem 33

Assume a galvanometer has an internal resistance of 60.0$\Omega$ and requires a current of 0.500 $\mathrm{mA}$ to produce
full-scale 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 $\mathrm{A}$ ?

Sophie S

Numerade Educator

Problem 34

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 $0.500-\mathrm{A}$ 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 35

A particular galvanometer requires a current of 1.50 $\mathrm{mA}$
for full-scale deflection and has a resistance of 75.0 $\mathrm{A}$ . It
can be used to measure voltages by wiring a large resistor
in series with the galvanometer as suggested in Active Figure 28.22 . The effect is to limit the current in the galvanometer when large voltages are applied. Calculate the
value of the resistor that allows the galvanometer to measure an applied voltage of 25.0 $\mathrm{V}$ at full-scale deflection.

Sophie S

Numerade Educator

Problem 36

O Meter loading. Work this problem to five-digit precision.
Refer to Figure $\mathrm{P} 28.36$ . (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?

Sophie S

Numerade Educator

Problem 37

A 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 38

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 $\sim 10^{2} \mathrm{V}$ 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
$\sim 10^{4} \Omega .$ 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
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 39

The circuit in Figure $\mathrm{P} 28.39$ 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 (d) in the $3.00-\mathrm{V}$ battery. Find $(\mathrm{e})$ the charge on the
capacitor.

Sheh Lit Chang

University of Washington

Problem 40

The circuit in Figure $\mathrm{P} 28.40 \mathrm{a}$ consists of three resistors and one battery with no internal resistance. (a) Find the current in the $5.00-\Omega$ resistor. (b) Find the power delivered to the $5.00-\Omega$ resistor. (c) In each of the circuits in Figures $\mathrm{P} 28.40 \mathrm{b}, \quad \mathrm{P} 28.40 \mathrm{c},$ and $\mathrm{P} 28.40 \mathrm{d},$ an additional
$15.0-\mathrm{V}$ 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.
In which of these three circuits is the smallest amount of
power delivered to the $10.0-\Omega$ resistor? You need not calculate the power in each circuit if you explain your answer.

Sophie S

Numerade Educator

Problem 41

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 42

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 $\mathrm{W}$ ? Explain your answers.

Sophie S

Numerade Educator

Problem 43

Calculate the potential difference between points $a$ and $b$
in Figure $\mathrm{P} 28.43$ and identify which point is at the higher
potential.

Sophie S

Numerade Educator

Problem 44

Assume you have a battery of emf $\boldsymbol{\varepsilon}$ 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? (b) In parallel? (c) For which
connection will the lightbulbs shine the brightest?

Sophie S

Numerade Educator

Problem 45

A rechargeable battery has a constant 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 charging,
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.)

Sophie S

Numerade Educator

Problem 46

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 47

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.
Determine the value of each resistor.

Shoukat Ali

Other Schools

Problem 48

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 value of each resistor.

Sophie S

Numerade Educator

Problem 49

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 50

(a) Determine the equilibrium charge on the capacitor
in the circuit of Figure $\mathrm{P} 28.50$ as a function of $R$ . (b) Eval-
uate the charge when $R=10.0 \Omega .$ (c) Can the charge on
the capacitor be zero? If so, for what value of $R$ ? (d) What
is the maximum possible magnitude of the charge on the
capacitor? For what value of $R$ is it achieved? (e) Is it
experimentally meaningful to take $R=\infty$ ? Explain your
answer. If so, what charge magnitude does it imply? Suggestion: You may do part (b) before part (a), as practice
for it.

Sophie S

Numerade Educator

Problem 51

The value of a resistor $R$ is to be determined using the ammeter-voltmeter setup shown in Figure $\mathrm{P} 28.51$ . The ammeter has a resistance of $0.500 \Omega,$ and the voltmeter has a resistance of 20.0 $\mathrm{k} \Omega$ . 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 $\mathrm{P} 28.51 \mathrm{a}$ and $(\mathrm{b})$ Figure $\mathrm{P} 28.51 \mathrm{b} ?$

Sophie S

Numerade Educator

Problem 52

A battery is used to charge a capacitor through a resistor as shown in Active Figure 28.16 $\mathrm{b}$ . Show that half the energy supplied by the battery appears as internal energy in the resistor and half is stored in the capacitor.

Sophie S

Numerade Educator

Problem 53

The values of the components in a simple series $R C$ circuit containing a switch (Active Fig. 28.16 $\mathrm{b} )$ 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 thrown to $a,$ 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.

Sophie S

Numerade Educator

Problem 54

A young man owns a canister vacuum cleaner marked 535 $\mathrm{W}$ at 120 $\mathrm{V}$ 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 $\mathrm{m}$ 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 $0.900 \Omega,$ what is the actual power delivered to the cleaner? (b) If instead the power is to be at least 525 $\mathrm{W}$ , 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. Suggestion: A symbolic solution can simplify the calculations.

Sophie S

Numerade Educator

Problem 55

Three $60.0-\mathrm{W}, 120-\mathrm{V}$ lightbulbs are connected across a $120-\mathrm{V}$ power source as shown in Figure $\mathrm{P} 28.55 .$ Find (a) the total power delivered to the three lightbulbs and (b) the potential difference across each. Assume the resistance of each lightbulb is constant (even though in reality the resistance might increase markedly with current).

Sophie S

Numerade Educator

Problem 56

Switch S shown in Figure $\mathrm{P} 28.56$ 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?

Sophie S

Numerade Educator

Problem 57

An ideal voltmeter connected across a certain fresh battery reads 9.30 $\mathrm{V}$ , and an ideal ammeter briefly connected across the same battery reads 3.70 $\mathrm{A}$ . We say that the battery has an open-circuit voltage of 9.30 $\mathrm{V}$ and a shortcircuit current of 3.70 A. (a) Model the battery as a source of emf $\mathcal{E}$ in series with an internal resistance $r .$ Determine both $\mathcal{E}$ and $r .$ (b) An irresponsible experimenter connects 20 of these identical batteries together as suggested in Figure $\mathrm{P} 28.57$ . Do not try this experiment yourself! Find the open-circuit voltage and the short-circuit current of the set of connected batteries.
(c) Assume the resistance between the palms of the experimenter's two hands is 120$\Omega$ . Find the current in his body that would result if his palms touched the two exposed terminals of the set of connected batteries. (d) Find the power that would be delivered to his body in this situation. (e) Thinking it is safe to do so, the experimenter threads a copper wire inside his shirt between his hands, like a mitten string. To reduce the current in his body to 5.00 $\mathrm{mA}$ when he presses the ends of the wire against the battery poles, what should the resistance of the copper wire be? ( $f$ ) Find the power delivered to his body in this situation. (g) Find the power delivered to the copper wire. (h) Explain why the sum of the two powers in parts $(f)$ and $(g)$ is much less than the power calculated in part (d). Is it meaningful to ask where the rest of the power is going?

Sophie S

Numerade Educator

Problem 58

Four resistors are connected in parallel across a $9.20-\mathrm{V}$ battery. They carry currents of $150 \mathrm{mA}, 45.0 \mathrm{mA}, 14.0 \mathrm{mA},$ 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 energy leaks, including the following: 1500 $\mathrm{W}$ by conduction through the ceiling, 450 $\mathrm{W}$ by infiltration (airflow) 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.

Sophie S

Numerade Educator

Problem 59

Figure $\mathrm{P} 28.59$ 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 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 canceled the service. Consequently, the equivalent resistance of the section of the circuit to the right of the first load resistor is nearly equal to $R_{\mathrm{eq}}$ .

Sophie S

Numerade Educator

Problem 60

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 61

In Figure $\mathrm{P} 28.61$ , 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 $(\mathrm{b})$ the charge $Q$ on the capacitor. (c) The switch is now opened at $t=0 .$ Write an equation for the current $I_{R_{2}}$ 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.

Sophie S

Numerade Educator

Problem 62

The circuit shown in Figure $\mathrm{P} 28.62$ 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 $\ln (\boldsymbol{E} / \Delta V)$ 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.

Artemisa Mazón

Numerade Educator

Problem 63

The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P28.63). 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) 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 $R_{2}=6.00 \Omega ?$

Sophie S

Numerade Educator

Problem 64

The switch in Figure $P 28.64$ a closes when $\Delta V_{c}>2 \Delta V / 3$ and opens when $\Delta V_{c}<\Delta V / 3 .$ The voltmeter reads a potential difference as plotted in Figure $\mathrm{P} 28.64 \mathrm{b}$ . What is the period $T$ of the waveform in terms of $R_{1}, R_{2},$ and $C ?$

Sheh Lit Chang

University of Washington

Problem 65

An electric teakettle has a multiposition 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.

Sophie S

Numerade Educator

Problem 66

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 $\mathrm{pF}$ , 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 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 5000 M\Omega. A pair of shoes with special static-dissipative soles can have an equivalent resistance of 1.00 $\mathrm{MD}$ . 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 person's potential from 3000 $\mathrm{V}$ to 100 $\mathrm{V}$ ? (b) How long does it take the static-dissipative shoes to do the same thing?

Sophie S

Numerade Educator