University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 26

Direct Current Circuits - all with Video Answers

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Chapter Questions

Problem 1

A resistor and a capacitor are connected in series. If a second identical capacitor is connected in series in the same circuit, the time constant for the circuit will
a) decrease.
b) increase.
c) stay the same.

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

A resistor and a capacitor are connected in series. If a second identical resistor is connected in series in the same
circuit, the time constant for the circuit will
a) decrease.
b) increase.
c) stay the same.

Narayan Hari

Numerade Educator

Problem 3

A circuit consists of a source of emf, a resistor, and a capacitor, all connected in series. The capacitor is fully charged. How much current is flowing through it?
a) $i=V / R$
b) zero
c) neither (a) nor (b)

Narayan Hari

Numerade Educator

Problem 4

Which of the following will reduce the time constant in an RC circuit?
a) increasing the dielectric constant of the capacitor
b) adding an additional $20 \mathrm{~m}$ of wire between the capacitor and the resistor
c) increasing the voltage of the battery
d) adding an additional resistor in parallel with the first resistor
e) none of the above

Vishal Gupta

Numerade Educator

Problem 5

Kirchhoff's Junction Rule states that
a) the algebraic sum of the currents at any junction in a circuit must be zero.
b) the algebraic sum of the potential changes around any closed loop in a circuit must be zero.
c) the current in a circuit with a resistor and a capacitor varies exponentially with time.
d) the current at a junction is given by the product of the resistance and the capacitance.
e) the time for the current development at a junction is given by the product of the resistance and the capacitance.

Narayan Hari

Numerade Educator

Problem 6

How long would it take, in multiples of the time constant, $\tau,$ for the capacitor in an $\mathrm{RC}$ circuit to be $98 \%$ charged?
a) $9 \tau$
c) $90 \tau$
e) $0.98 \tau$
b) $0.9 \tau$
d) $4 \tau$

Vishal Gupta

Numerade Educator

Problem 7

A capacitor $C$ is initially uncharged. At time $t=0,$ the capacitor is attached through a resistor $R$ to a battery. The energy stored in the capacitor increases, eventually reaching a value $U$ as $t \rightarrow \infty$, After a time equal to the time constant $\tau=R C$, the energy stored in the capacitor is given by
a) $U / e$.
c) $U(1-1 / e)^{2}$
b) $U / e^{2}$
d) $U(1-1 / e)$.

Nishant Kumar

Numerade Educator

Problem 8

Which of the following has the same unit as the electromotive force (emf)?
a) current
b) electric potential
c) electric field
d) electric power
e) none of the above

Narayan Hari

Numerade Educator

Problem 9

The capacitor in each circuit in the figure is first charged by a $10-\mathrm{V}$ battery with no internal resistance. Then, the switch is flipped from position $A$ to position $B,$ and the capacitor is discharged through various resistors. For which circuit is the total energy dissipated by the resistor the largest?

Vishal Gupta

Numerade Educator

Problem 10

You want to measure simultaneously the potential difference across and the current through a resistor, $R$. As the circuit diagrams show, there are two ways to connect the two instruments-ammeter and voltmeter-in the circuit. Comment on the result of the measurement using each configuration.

Vishal Gupta

Numerade Educator

Problem 11

If the capacitor in an $\mathrm{RC}$ circuit is replaced with two identical capacitors connected in series, what happens to the time constant for the circuit?

Narayan Hari

Numerade Educator

Problem 12

You want to accurately measure the resistance, $R_{\text {device }}$ of a new device. The figure shows two ways to accomplish this task. On the left, an ohmmeter produces a current through the device and measures that current, $i$, and the potential difference, $\Delta V$, across the device. This potential difference includes the potential drops across the wires leading to and from the device and across the contacts that connect the wires to the device. These extra resistances cannot always be neglected, especially if the device has low resistance. This technique is called two-probe measurement since two probe wires are connected to the device. The resulting current, $i$, is measured with an ammeter. The total resistance is then determined by dividing $\Delta V$ by $i$. For this configuration, what is the resistance that the ohmmeter measures? In the alternative configuration, shown on the right, a similar current source is used to produce and measure the current through the device, but the potential difference, $\Delta V$, is measured directly across the device with a nearly ideal voltmeter with extremely large internal resistance. This technique is called a four-probe measurement since four probe wires are connected to the device. What resistance is being measured in this four-probe configuration? Is it different from that being assessed by the two-probe measurement? Why or why not? (Hint: Four-probe measurements are used extensively by scientists and engineers and are especially useful for accurate measurements of the resistance of materials or devices with low resistance.)

Dominador Tan

Numerade Educator

Problem 13

Explain why the time constant for an $\mathrm{RC}$ circuit increases with $R$ and with $C$. (The answer "That's what the formula says" is not sufficient.)

Vishal Gupta

Numerade Educator

Problem 14

A battery, a resistor, and a capacitor are connected in series in an RC circuit. What happens to the current through a resistor after a long time? Explain using Kirchhoff's rules.

Vishal Gupta

Numerade Educator

Problem 15

How can you light a $1.0-\mathrm{W}, 1.5-\mathrm{V}$ bulb with your $12.0-V$ car battery?

Vishal Gupta

Numerade Educator

Problem 16

A multiloop circuit contains a number of resistors and batteries. If the emf values of all the batteries are
doubled, what happens to the currents in all the components of the circuit?

Vishal Gupta

Numerade Educator

Problem 17

A multiloop circuit of resistors, capacitors, and batteries is switched on at $t=0$, at which time all the capacitors are uncharged. The initial distribution of currents and potential differences in the circuit can be analyzed by treating the capacitors as if they were connecting wires or closed switches. The final distribution of currents and potential differences, which occurs after a long time has passed, can be analyzed by treating the capacitors as open segments or open switches. Explain why these tricks work.

Vishal Gupta

Numerade Educator

Problem 18

Voltmeters are always connected in parallel with a circuit component, and ammeters are always connected in series. Explain why.

Vishal Gupta

Numerade Educator

Problem 19

You wish to measure both the current through and the potential difference across some component of a circuit. It is not possible to do this simultaneously and accurately with ordinary voltmeters and ammeters. Explain why not.

Vishal Gupta

Numerade Educator

Problem 20

Two light bulbs for use at $110 \mathrm{~V}$ are rated at $60 \mathrm{~W}$ and $100 \mathrm{~W}$, respectively. Which has the filament with lower resistance?

Narayan Hari

Numerade Educator

Problem 21

Two capacitors in series are charged through a resistor. Identical capacitors are instead connected in parallel and charged through the same resistor. How do the times required to fully charge the two sets of capacitors compare?

Vishal Gupta

Numerade Educator

Problem 22

The figure shows a circuit consisting of a battery connected to a resistor and a capacitor, which is fully discharged initially, in series with a switch.
a) What is the current in the circuit at any time $t ?$
b) Calculate the total energy provided by the battery from $t=0$ to $t=\infty$
c) Calculate the total energy dissipated from the resistor for the same time period.
d) Is energy conserved in this circuit?

Vishal Gupta

Numerade Educator

Problem 23

Two resistors, $R_{1}$ and $R_{2},$ are connected in series across a potential difference, $\Delta V_{0}$. Express the potential drop across each resistor individually, in terms of these quantities. What is the significance of this arrangement?

Vishal Gupta

Numerade Educator

Problem 24

A battery has $V_{\text {emf }}=12.0 \mathrm{~V}$ and internal resistance $r=1.00 \Omega$. What resistance, $R,$ can be put across the battery to extract $10.0 \mathrm{~W}$ of power from it?

Vishal Gupta

Numerade Educator

Problem 25

Three resistors are connected across a battery as shown in the figure. What values of $R$ and $V_{\text {emf }}$ will produce the indicated currents?

Narayan Hari

Numerade Educator

Problem 26

Find the equivalent resistance for the circuit in the figure,

Narayan Hari

Numerade Educator

Problem 27

The dead battery of your car provides a potential difference of $9.950 \mathrm{~V}$ and has an internal resistance of $1.100 \Omega$. You charge it by connecting it with jumper cables to the live battery of another car. The live battery provides a potential difference of $12.00 \mathrm{~V}$ and has an internal resistance of $0.0100 \Omega$, and the starter resistance is $0.0700 \Omega$.
a) Draw the circuit diagram for the connected batteries.
b) Determine the current in the live battery, in the dead battery, and in the starter immediately after you closed the circuit.

Vishal Gupta

Numerade Educator

Problem 28

In the circuit shown in the figure, $V_{1}=1.5 \mathrm{~V}, V_{2}=$ $2.5 \mathrm{~V}, R_{1}=4.0 \Omega,$ and $R_{2}=$
$5.0 \Omega$. What is the magnitude of the current, $i_{1},$ flowing through resistor $R_{1} ?$

Vishal Gupta

Numerade Educator

Problem 29

The circuit shown in the figure consists of two batteries with $V_{A}$ and $V_{B}$ and three light bulbs with resistances $R_{1}, R_{2},$ and $R_{3} .$ Calculate the magnitudes of the currents $i_{11}$ $i_{2},$ and $i_{3}$ flowing through the bulbs. Indicate the correct directions of current flow on the diagram. Calculate the power, $P_{A}$ and $P_{B}$, supplied by battery A and by battery B.

Prashant Bana

Numerade Educator

Problem 30

In the circuit shown in the figure, $R_{1}=5.00 \Omega$, $R_{2}=10.0 \Omega,$ and $R_{3}=15.0 \Omega$
$V_{\mathrm{emf}, 1}=10.0 \mathrm{~V},$ and $V_{\mathrm{emf}, 2}=$
$15.0 \mathrm{~V}$. Using Kirchhoff's Loop and Junction Rules, determine the currents $i_{1}, i_{2},$ and $i_{3}$ flowing through $R_{1}, R_{2}$ and $R_{3}$ respectively, in the direction indicated in the figure.

Vishal Gupta

Numerade Educator

Problem 31

For the circuit shown in the figure, find the magnitude and the direction of the current through each resistor and the power supplied by each battery, using the following values: $R_{1}=4.00 \Omega$,
$R_{2}=6.00 \Omega, R_{3}=8.00 \Omega, R_{4}=6.00 \Omega, R_{5}=5.00 \Omega, R_{6}=10.0 \Omega$
$R_{7}=3.0 \Omega, V_{\mathrm{emf}, 1}=6.00 \mathrm{~V},$ and $V_{\mathrm{emf} .2}=12.0 \mathrm{~V}$

Pritesh Ranjan

Numerade Educator

Problem 32

A Wheatstone bridge is constructed using a $1.00-\mathrm{m}-$ long Nichrome wire (the purple line in the figure) with a conducting contact that can slide along the wire. A resistor, $R_{1}=$ $100 . \Omega$, is placed on one side of the bridge, and another resistor, $R,$ of unknown resistance, is placed on the other side. The contact is moved along the Nichrome wire, and it is found that the ammeter reading is zero for $L=25.0 \mathrm{~cm} .$ Knowing that the wire has a uniform cross section throughout its length, determine the unknown resistance.

Vishal Gupta

Numerade Educator

Problem 33

A "resistive ladder" is constructed with identical resistors, $R$, making up its legs and rungs, as shown in the figure. The ladder has "infinite" height; that is, it extends very far in one direction. Find the equivalent resistance of the ladder, measured between its "feet" (points A and B).

Vishal Gupta

Numerade Educator

Problem 34

Consider an "infinite," that is, very large, two-dimensional square grid of identical resistors, $R$, as shown in the figure. Find the equivalent resistance of the grid, as measured across any individual resistor. (Hint: Symmetry and superposition are very helpful for solving this problem.)

Pritesh Ranjan

Numerade Educator

Problem 35

To extend the useful range of an ammeter, a shunt resistor, $R_{\text {shunt }}$, is placed in parallel with the ammeter as shown in the figure. If the internal resistance of the ammeter is $R_{\mathrm{i}, \mathrm{A}},$ determine the resistance that the shunt resistor has to have to extend the useful range of the ammeter by a factor $N$. Then, calculate the resistance the shunt resistor has to have to allow an ammeter with an internal resistance of $1.00 \Omega$ and a maximum range of $1.00 \mathrm{~A}$ to measure currents up to $100 .$ A. What fraction of the total $100 .-$ A current flows through the ammeter, and what fraction flows through the shunt resistor?

Vishal Gupta

Numerade Educator

Problem 36

To extend the useful range of a voltmeter, an additional resistor, $R_{\text {scric, }}$ is placed in series with the voltmeter as shown in the figure. If the internal resistance of the voltmeter is $R_{i, v}$ determine the resistance that the added series resistor has to have to extend the useful range of the voltmeter by a factor $N$. Then, calculate the resistance the series resistor has to have to allow a voltmeter with an internal resistance of $1.00 \mathrm{M} \Omega\left(10^{6} \Omega\right)$ and a maximum range of $1.00 \mathrm{~V}$ to measure potential differences up to $100 .$ V. What fraction of the total
100.-V potential drop occurs across the voltmeter, and what fraction of that drop occurs across the added series resistor?

Vishal Gupta

Numerade Educator

Problem 37

As shown in the figure, a $6.0000-\mathrm{V}$ battery is used to produce a current through two identical resistors, $R$, each having a resistance of $100.00 \mathrm{k} \Omega$. A digital multimeter (DMM) is used to measure the potential difference across the first resistor. DMMs typically have an internal resistance of $10.0 \mathrm{M} \Omega .$ Determine the potential differences $V_{a b}$ (the potential difference between points $a$ and $b$, which is the difference the DMM measures) and $V_{b c}$ (the potential difference between points $b$ and $c$, which is the difference across the second resistor). Nominally, $V_{a b}=V_{b}$, but this may not be the case here. How can this measurement error be reduced?

Vishal Gupta

Numerade Educator

Problem 38

You want to make an ohmmeter to measure the resistance of unknown resistors. You have a battery with voltage $\mathrm{V}_{\mathrm{emf}}=9.00 \mathrm{~V}$, a variable resistor, $R,$ and an ammeter that measures current on a linear scale from 0 to $10.0 \mathrm{~mA}$
a) What resistance should the variable resistor have so that the ammeter gives its full-scale (maximum) reading when the ohmmeter is shorted?
b) Using the resistance from part (a), what is the unknown resistance if the ammeter reads $\frac{1}{4}$ of its full scale?

Vishal Gupta

Numerade Educator

Problem 39

A circuit consists of two $1.00-\mathrm{k} \Omega$ resistors in series with an ideal $12.0-\mathrm{V}$ battery.
a) Calculate the current flowing through each resistor.
b) A student trying to measure the current flowing through one of the resistors inadvertently connects an ammeter in parallel with that resistor rather than in series with it. How much current will flow through the ammeter, assuming that it has an internal resistance of $1.0 \Omega ?$

Vishal Gupta

Numerade Educator

Problem 40

A circuit consists of two $100 .-\mathrm{k} \Omega$ resistors in series with an ideal $12.0-\mathrm{V}$ battery.
a) Calculate the potential drop across one of the resistors.
b) A voltmeter with internal resistance $10.0 \mathrm{M} \Omega$ is connected in parallel with one of the two resistors in order to measure the potential drop across the resistor. By what percentage will the voltmeter reading deviate from the value you determined in part (a)? (Hint: The difference is rather small so it is helpful to solve algebraically first to avoid a rounding error.)

Vishal Gupta

Numerade Educator

Problem 41

Initially, switches $S_{1}$ and $\mathrm{S}_{2}$ in the circuit shown in the figure are open and the capacitor has a charge of $100 . \mathrm{mC}$. About how long will it take after switch $\mathrm{S}_{1}$ is closed for the charge on the capacitor to drop to $5.00 \mathrm{mC} ?$

Vishal Gupta

Numerade Educator

Problem 42

What is the time constant for the discharging of the capacitors in the circuit shown in the figure? If the $2.00-\mu \mathrm{F}$ capacitor initially has a potential difference of $10.0 \mathrm{~V}$ across its plates, how much charge is left on it after the switch has been closed for a time equal to half of the time constant?

Vishal Gupta

Numerade Educator

Problem 43

The circuit shown in the figure has a switch, $S$, two resistors, $R_{1}=1.00 \Omega$ and $R_{2}=$ $2.00 \Omega,$ a $12.0-V$ battery, and a capacitor with $C=20.0 \mu \mathrm{F}$ After the switch is closed, what will the maximum charge on the capacitor be? How long after the switch has been closed will the capacitor have $50.0 \%$ of this maximum charge?

Narayan Hari

Numerade Educator

Problem 44

In the movie Back to the Future, time travel is made possible by a flux capacitor, which generates 1.21 GW of power. Assuming that a 1.00 - F capacitor is charged to its maximum capacity with a $12.0-\mathrm{V}$ car battery and is discharged through a resistor, what resistance is necessary to produce a peak power output of 1.21 GW in the resistor? How long would it take for a $12.0-\mathrm{V}$ car battery to charge the capacitor to $90.0 \%$ of its maximum capacity through this resistor?

Vishal Gupta

Numerade Educator

Problem 45

During a physics demonstration, a fully charged $90.0-\mu \mathrm{F}$ capacitor is discharged through a $60.0-\Omega$ resistor. How long will it take for the capacitor to lose $80.0 \%$ of its initial energy?

Vishal Gupta

Numerade Educator

Problem 46

Two parallel plate capacitors, $C_{1}$ and $C_{2},$ are con nected in series with a $60.0-\mathrm{V}$ battery and a $300 .-\mathrm{k} \Omega$ resistor, as shown in the figure. Both capacitors have plates with an area of $2.00 \mathrm{~cm}^{2}$ and a separation of $0.100 \mathrm{~mm}$. Capacitor $C_{1}$ has air between its plates, and capacitor $C_{2}$ has the gap filled with a certain porcelain (dielec-

Dominador Tan

Numerade Educator

Problem 47

A parallel plate capacitor with $C=0.050 \mu \mathrm{F}$ has a separation between its plates of $d=50.0 \mu \mathrm{m} .$ The dielectric that fills the space between the plates has dielectric constant $\kappa=2.5$ and resistivity $\rho=4.0 \cdot 10^{12} \Omega \mathrm{m} .$ What is the time constant for this capacitor? (Hint: First calculate the area of the plates for the given $C$ and $\kappa$, and then determine the resistance of the dielectric between the plates.)

Vishal Gupta

Numerade Educator

Problem 48

A $12.0-V$ battery is attached to a $2.00-\mathrm{mF}$ capacitor and a $100 .-\Omega$ resistor. Once the capacitor is fully charged, what is the energy stored in it? What is the energy dissipated as heat by the resistor as the capacitor is charging?

Vishal Gupta

Numerade Educator

Problem 49

A capacitor bank is designed to discharge 5.0 J of energy through a $10.0-\mathrm{k} \Omega$ resistor array in under $2.0 \mathrm{~ms}$ To what potential difference must the bank be charged, and what must the capacitance of the bank be?

Narayan Hari

Numerade Educator

Problem 50

The circuit in the figure has a capacitor connected to a battery, two switches, and three resistors. Initially, the capacitor is uncharged and both of the switches
are open.
a) Switch $S_{1}$ is closed. What is the current flowing out of the battery immediately after switch $S_{1}$ is closed?
b) After about $10.0 \mathrm{~min},$ switch $\mathrm{S}_{2}$ is closed. What is the current flowing out of the battery immediately after switch $\mathrm{S}_{2}$ is closed?
c) What is the current flowing out of the battery about 10.0 min after switch $S_{2}$ has been closed?
d) After another $10.0 \mathrm{~min},$ switch $\mathrm{S}_{1}$ is opened. How long will it take until the current in the $200 .-\Omega$ resistor is below $1.00 \mathrm{~mA} ?$

Dominador Tan

Numerade Educator

Problem 51

In the circuit shown in the figure, $R_{1}=10.0 \Omega$ $R_{2}=4.00 \Omega,$ and $R_{3}=10.0 \Omega,$ and the capacitor has capacitance $C=2.00 \mu \mathrm{F}$
a) Determine the potential difference, $\Delta V_{C}$, across the capacitor after switch S has been closed for a long time.
b) Determine the energy stored in the capacitor when switch S has been closed for a long time.
c) After switch S is opened, how much energy is dissipated through $R_{3} ?$

Vishal Gupta

Numerade Educator

Problem 52

A cube of gold that is $2.5 \mathrm{~mm}$ on a side is connected across the terminals of a $15-\mu \mathrm{F}$ capacitor that initially has a potential difference of $100.0 \mathrm{~V}$ between its plates.
a) What time is required to fully discharge the capacitor?
b) When the capacitor is fully discharged, what is the temperature of the gold cube?

Dominador Tan

Numerade Educator

Problem 53

A "capacitive ladder" is constructed with identical capacitors, $C$, making up its legs and rungs, as shown in the figure. The ladder has "infinite" height; that is, it extends very far in one direction. Calculate the equivalent capacitance of the ladder, measured between its "feet" (points $\mathrm{A}$ and $\mathrm{B}$ ).

Vishal Gupta

Numerade Educator

Problem 54

In the circuit in the figure, the capacitors are completely uncharged. The switch is then closed for a long time.
a) Calculate the current through the $4.0-\Omega$ resistor.
b) Find the potential difference across the $4.0-\Omega$. $6.0-\Omega,$ and $8.0-\Omega$ resistors.
c) Find the potential difference across the $1.0-\mu \mathrm{F}$ capacitor.

Vishal Gupta

Numerade Educator

Problem 55

The ammeter your physics instructor uses for in-class demonstrations has internal resistance $R_{\mathrm{i}}=75 \Omega$ and measures a maximum current of $1.5 \mathrm{~mA}$. The same ammeter can be used to measure currents of much greater magnitudes by wiring a shunt resistor of relatively small resistance, $R_{\text {shunt }}$, in parallel with the ammeter. (a) Sketch the circuit diagram, and explain why the shunt resistor connected in parallel with the ammeter allows it to measure larger currents. (b) Calculate the resistance the shunt resistor has to have to allow the ammeter to measure a maximum current of 15 A.

Vishal Gupta

Numerade Educator

Problem 56

Many electronics devices can be dangerous even after they are shut off. Consider an RC circuit with a $150 .-\mu \mathrm{F}$ capacitor and a $1.00-\mathrm{M} \Omega$ resistor connected to a $200 .-\mathrm{V}$ power source for a long time and then disconnected and shorted, as shown in the figure. How long will it be until the potential difference across the capacitor drops to below $50.0 \mathrm{~V} ?$

Vishal Gupta

Numerade Educator

Problem 57

Design a circuit like that shown in the figure to operate a strobe light. The capacitor discharges power through the light bulb filament (resistance of $2.5 \mathrm{k} \Omega$ ) in $0.20 \mathrm{~ms}$ and charges through a resistor $R$, with a repeat cycle of $1000 \mathrm{~Hz}$. What capacitor and resistor should be used?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 58

An ammeter with an internal resistance of $53 \Omega$ measures a current of $5.25 \mathrm{~mA}$ in a circuit containing a battery and a total resistance of $1130 \Omega$. The insertion of the ammeter alters the resistance of the circuit, and thus the measurement does not give the actual value of the current in the circuit without the ammeter. Determine the actual value of the current.

Narayan Hari

Numerade Educator

Problem 59

In the circuit shown in the figure, a $10.0-\mu \mathrm{F}$ capacitor is charged by a $9.00-V$ battery with the two-way switch kept in position $X$ for a long time. Then the switch is suddenly flicked to position Y. What current flows through the $40.0-\Omega$ resistor
a) immediately after the switch moves to position Y?
b) $1.00 \mathrm{~ms}$ after the switch moves to position $\mathrm{Y}$ ?

Vishal Gupta

Numerade Educator

Problem 60

How long will it take for the current in a circuit to drop from its initial value to $1.50 \mathrm{~mA}$ if the circuit contains two $3.8-\mu \mathrm{F}$ capacitors that are initially uncharged, two $2.2-\mathrm{k} \Omega$ resistors, and a $12.0-\mathrm{V}$ battery all connected in series?

Vishal Gupta

Numerade Educator

Problem 61

An RC circuit has a time constant of 3.1 s. At $t=0$, the process of charging the capacitor begins. At what time will the energy stored in the capacitor reach half of its maximum value?

Vishal Gupta

Numerade Educator

Problem 62

For the circuit shown in the figure, determine the charge on each capacitor when (a) switch S has been closed for a long time and (b) switch S has been open for a long time.

Vishal Gupta

Numerade Educator

Problem 63

Three resistors, $R_{1}=10.0 \Omega, R_{2}=20.0 \Omega,$ and $R_{3}=30.0 \Omega,$ are connected in a multiloop circuit, as shown in the figure. Determine the amount of power dissipated in the three resistors.

Vishal Gupta

Numerade Educator

Problem 64

The figure shows a circuit containing two batteries and three resistors. The batteries provide $V_{\text {emf. }}=12.0 \mathrm{~V}$ and $V_{\mathrm{emf}, 2}=16.0 \mathrm{~V}$ and have no
internal resistance. The resistors have resistances of $R_{1}=30.0 \Omega, R_{2}=40.0 \Omega,$ and
$R_{3}=20.0 \Omega$. Find the magnitude of the potential drop across $R_{2}$

Vishal Gupta

Numerade Educator

Problem 65

The figure shows a spherical capacitor. The inner sphere has radius $a=1.00 \mathrm{~cm}$ and the outer sphere has radius $b=1.10 \mathrm{~cm} .$ The battery has $V_{\mathrm{cmf}}=10.0 \mathrm{~V},$ and the resistor
has a value of $R=10.0 \mathrm{M} \Omega$
a) Determine the time constant of the RC circuit.
b) Determine how much charge has accumulated on the capacitor after switch S has been closed for $0.1 \mathrm{~ms}$

Vishal Gupta

Numerade Educator

Problem 66

Write the set of equations that determines the three currents
in the circuit shown in the figure. (Assume that the capacitor is initially uncharged.)

Pritesh Ranjan

Numerade Educator

Problem 67

Consider a series $\mathrm{RC}$ circuit with $R=10.0 \Omega$ $C=10.0 \mu \mathrm{F}$ and $V=10.0 \mathrm{~V}$ a) How much time, expressed as a multiple of the time constant, does it take for the capacitor to be charged to half of its maximum value?
b) At this instant, what is the ratio of the energy stored in the capacitor to its maximum possible value?
c) Now suppose the capacitor is fully charged. At time $t=$
0 , the original circuit is opened and the capacitor is allowed to discharge across another resistor, $R^{\prime}=1.00 \Omega$, that is connected across the capacitor. What is the time constant for the discharging of the capacitor?
d) How many seconds does it take for the capacitor to discharge half of its maximum stored charge, $Q$ ?

Vishal Gupta

Numerade Educator

Problem 68

a) What is the current in shown in the figure?
b) What is the power dissipated in the $5.00-\Omega$ resistor?? $?$

Vishal Gupta

Numerade Educator

Problem 69

In the Wheatstone bridge shown in the figure, the known resistances are $R_{1}=8.00 \Omega, R_{4}=2.00 \Omega,$ and
$R_{5}=6.00 \Omega,$ and the battery has $V_{\mathrm{emf}}=15.0 \mathrm{~V}$. The variable resistance $R_{2}$ is adjusted until the potential difference across $R_{3}$ is zero $(V=0)$. Find $i_{2}$ (the current through resistor $R_{2}$ ) at this point.

Vishal Gupta

Numerade Educator

Problem 70

Consider the circuit with five resistors and two batteries (with no internal resistance) shown in the figure.
a) Write a set of equations that will allow you to solve for the current in each of the resistors.
b) Solve the equations from part (a) for the current in the $4.00-\Omega$ resistor.

Pritesh Ranjan

Numerade Educator

Problem 71

Consider "infinite," that is, very large, two-dimensional square grid of identical capacitors, $C$, shown in the figure. Find the effective capacitance of the grid, as measured across any individual capacitor.

Pritesh Ranjan

Numerade Educator