Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 18

Electric Current and Circuits - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

A battery charger delivers a current of $3.0 \mathrm{A}$ for $4.0 \mathrm{h}$ to a $12 \mathrm{V}$ storage battery. What is the total charge that passes through the battery in that time?

Sarah Chapman

Sarah Chapman

Numerade Educator

Problem 2

The current in a wire is 0.500 A. (a) How much charge flows through a cross section of the wire in 10.0 s? (b) How many electrons move through the same cross section in $10.0 \mathrm{s} ?$

TH

Thaxter Hensley

Numerade Educator

Problem 3

(a) What is the direction of the current in the vacuum tube shown in the figure? (b) Electrons hit the anode at a rate of $6.0 \times 10^{12}$ per second. What is the current in the tube?

Sarah Chapman

Numerade Educator

Problem 4

In an ion accelerator, $3.0 \times 10^{13}$ helium-4 nuclein (charge $+2 e$ ) per second strike a target. What is the beam current?

Ivan Kochetkov

Numerade Educator

Problem 5

The current in the electron beam of a computer monitor is $320 \mu$ A. How many electrons per second hit the screen?

Sarah Chapman

Numerade Educator

Problem 6

A potential difference is applied between the electrodes in a gas discharge tube. In $1.0 \mathrm{s}, 3.8 \times 10^{16}$ electrons and $1.2 \times 10^{16}$ singly charged positive ions move in opposite directions through a surface perpendicular to the length of the tube. What is the current in the tube?

TH

Thaxter Hensley

Numerade Educator

Problem 7

Two electrodes are placed in a calcium chloride solution, and a potential difference is maintained between them. If $3.8 \times 10^{16} \mathrm{Ca}^{2+}$ ions and $6.2 \times 10^{16} \mathrm{Cl}^{-}$ ions per second move in opposite directions through an imaginary area between the electrodes, what is the current in the solution?

Sarah Chapman

Numerade Educator

Problem 8

A Vespa scooter and a Toyota automobile might both use a $12 \mathrm{V}$ battery, but the two batteries are of different sizes and can pump different amounts of charge. Suppose the scooter battery can pump $4.0 \mathrm{kC}$ of charge and the automobile battery can pump $30.0 \mathrm{kC}$ of charge. How much energy can each battery deliver, assuming the batteries are ideal?

TH

Thaxter Hensley

Numerade Educator

Problem 9

What is the energy stored in a small battery if it can move $675 \mathrm{C}$ through a potential difference of $1.20 \mathrm{V} ?$

Sarah Chapman

Numerade Educator

Problem 10

The label on a $12.0 \mathrm{V}$ truck battery states that it is rated at $180.0 \mathrm{A} \cdot \mathrm{h}$ (ampere-hours). Treat the battery as ideal.
(a) How much charge in coulombs can be pumped by the battery? [Hint: Convert A.h to A.s.] (b) How much electric energy can the battery supply?
(c) Suppose the radio in the truck is left on when the engine is not running. The radio draws a current of 3.30 A. How long does it take to drain the battery if it starts out fully charged?

Shahab Ullah

Numerade Educator

Problem 11

The starter motor in a car draws 220.0 A of current from the $12.0 \mathrm{V}$ battery for $1.20 \mathrm{s}$. (a) How much charge is pumped by the battery? (b) How much electric energy is supplied by the battery?

Sarah Chapman

Numerade Educator

Problem 12

A solar cell provides an emf of 0.45 V. (a) If the cell supplies a constant current of $18.0 \mathrm{mA}$ for $9.0 \mathrm{h},$ how much electric energy does it supply? (b) What is the power-the rate at which it supplies electric energy?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 13

Six copper wires are characterized by their dimensions and by the current they carry. Rank the wires in order of decreasing drift velocity.
(a) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $80 \mathrm{mA}$
(b) diameter $1 \mathrm{mm}$, length $1 \mathrm{m}$, current $80 \mathrm{mA}$
(c) diameter $4 \mathrm{mm}$, length $16 \mathrm{m}$, current $40 \mathrm{mA}$
(d) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $160 \mathrm{mA}$
(e) diameter $1 \mathrm{mm}$, length $4 \mathrm{m}$, current $20 \mathrm{mA}$
(f) diameter $2 \mathrm{mm}$, length $1 \mathrm{m}$, current $40 \mathrm{mA}$

Vishal Gupta

Numerade Educator

Problem 14

Two copper wires, one double the diameter of the other, have the same current flowing through them. If the thinner wire has a drift speed $v_{1}$ and the thicker wire has a drift speed $v_{2},$ how do the drift speeds of the charge carriers compare?

Mayukh Banik

Numerade Educator

Problem 15

A current of $2.50 \mathrm{A}$ is carried by a copper wire of radius $1.00 \mathrm{mm} .$ If the density of the conduction electrons is $8.47 \times 10^{28} \mathrm{m}^{-3},$ what is the drift speed of the conduction electrons?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 16

A current of $10.0 \mathrm{A}$ is carried by a copper wire of diameter $1.00 \mathrm{mm}$. If the density of the conduction electrons is $8.47 \times 10^{28} \mathrm{m}^{-3},$ how long does it take for a conduction electron to move $1.00 \mathrm{m}$ along the wire?

Mayukh Banik

Numerade Educator

Problem 17

A silver wire of diameter $1.0 \mathrm{mm}$ carries a current of $150 \mathrm{mA}$. The density of conduction electrons in silver is $5.8 \times 10^{28} \mathrm{m}^{-3} .$ How long (on average) does it take for a conduction electron to move $1.0 \mathrm{cm}$ along the wire?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 18

A strip of doped silicon $260 \mu \mathrm{m}$ wide contains $8.8 \times 10^{22}$ conduction electrons per cubic meter and an insignificant number of holes. When the strip carries a current of $130 \mu \mathrm{A},$ the drift speed of the electrons is $44 \mathrm{cm} / \mathrm{s} .$ What is the thickness of the strip?

Mayukh Banik

Numerade Educator

Problem 19

A gold wire of 0.50 mm diameter has $5.90 \times 10^{28}$ conduction electrons per cubic meter. If the drift speed is $6.5 \mu \mathrm{m} / \mathrm{s},$ what is the current in the wire?

Ivan Kochetkov

Numerade Educator

Problem 20

A copper wire of cross-sectional area $1.00 \mathrm{mm}^{2}$ has a current of $2.0 \mathrm{A}$ flowing along its length. What is the drift speed of the conduction electrons? Assume 1.3 conduction electrons per copper atom. The mass density of copper is $9.0 \mathrm{g} / \mathrm{cm}^{3}$ and its molar mass is $64 \mathrm{g} / \mathrm{mol}$.

Narayan Hari

Numerade Educator

Problem 21

An aluminum wire of diameter $2.6 \mathrm{mm}$ carries a current of 12 A. How long on average does it take an electron to move $12 \mathrm{m}$ along the wire? Assume 3.5 conduction electrons per aluminum atom. The mass density of aluminum is $2.7 \mathrm{g} / \mathrm{cm}^{3},$ and its molar mass is $27 \mathrm{g} / \mathrm{mol}$.

Narayan Hari

Numerade Educator

Problem 22

Six wires are characterized by their dimensions and by the metal they are made from. Assume the tungsten alloy has exactly twice the resistivity of aluminum. Rank the wires in order of decreasing resistance.
(a) diameter $2 \mathrm{mm},$ length $1 \mathrm{m},$ tungsten alloy
(b) diameter $4 \mathrm{mm},$ length $2 \mathrm{m},$ tungsten alloy
(c) diameter $2 \mathrm{mm}$, length $1 \mathrm{m}$, aluminum
(d) diameter $1 \mathrm{mm},$ length $1 \mathrm{m},$ aluminum
(e) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, tungsten alloy
(f) diameter $4 \mathrm{mm},$ length $4 \mathrm{m},$ aluminum

Shahab Ullah

Numerade Educator

Problem 23

A $12 \Omega$ resistor has a potential difference of $16 \mathrm{V}$ across it. What current flows through the resistor?

Mayukh Banik

Numerade Educator

Problem 24

Current of 83 mA flows through the resistor in the diagram. (a) What is the resistance of the resistor?
(b) In what direction does the current flow through the resistor?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 25

A copper wire and an aluminum wire of the same length have the same resistance. What is the ratio of the diameter of the copper wire to that of the aluminum wire?

Mayukh Banik

Numerade Educator

Problem 26

A bird sits on a high-voltage power line with its feet $2.0 \mathrm{cm}$ apart. The wire is made from aluminum, is $2.0 \mathrm{cm}$ in diameter, and carries a current of $150 \mathrm{A}$. What is the potential difference between the bird's feet?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 27

A person can be killed if a current as small as $50 \mathrm{mA}$ passes near the heart. An electrician is working on a humid day with hands damp from perspiration. Suppose his resistance from one hand to the other is $1 \mathrm{k} \Omega$ and he is touching two wires, one with each hand.
(a) What potential difference between the two wires would cause a $50 \mathrm{mA}$ current from one hand to the other? (b) An electrician working on a "live" circuit keeps one hand behind his or her back. Why?

Mayukh Banik

Numerade Educator

Problem 28

Some digital thermometers measure the current through a semiconductor to determine a patient's temperature. If a thermometer uses a germanium wire that has a resistance of $R$ at $37.0^{\circ} \mathrm{C}$ (normal body temperature), what is its resistance at $40.0^{\circ} \mathrm{C} ?$

Narayan Hari

Numerade Educator

Problem 29

Pure water has very few ions (about $1.2 \times 10^{14}$ ions per cubic centimeter), giving it a high resistivity, about $1 \times 10^{5} \Omega \cdot \mathrm{m}$ at $37^{\circ} \mathrm{C} .$ Blood plasma has a much lower resistivity of roughly $0.6 \Omega \cdot \mathrm{m}$ at $37^{\circ} \mathrm{C}$ due to the ions dissolved in the plasma. Assuming the resistivity depends only on the concentration of ions, how many ions per cubic centimeter are in blood plasma?

Narayan Hari

Numerade Educator

Problem 30

An electric device has the current-voltage ( $I-V$ ) graph shown. What is its resistance at
(a) point 1 and
(b) point $2 ?[$ Hint: Use the definition of resistance. $]$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 31

If $46 \mathrm{m}$ of nichrome wire is to have a resistance of $10.0 \Omega$ at $20^{\circ} \mathrm{C},$ what diameter wire should be used?

Mayukh Banik

Numerade Educator

Problem 32

The resistance of a conductor is $19.8 \Omega$ at $15.0^{\circ} \mathrm{C}$ and $25.0 \Omega$ at $85.0^{\circ} \mathrm{C} .$ What is the temperature coefficient of resistance of the material?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 33

A common flashlight bulb is rated at $0.300 \mathrm{A}$ and $2.90 \mathrm{V}$ (the values of current and voltage under operating conditions). If the resistance of the bulb's tungsten filament at room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ is $1.10 \Omega,$ estimate the temperature of the tungsten filament when the bulb is turned on.

Mayukh Banik

Numerade Educator

Problem 34

Find the maximum current that a fully charged D-cell can supply - if only briefly - such that its terminal voltage is at least $1.0 \mathrm{V}$. Assume an emf of $1.5 \mathrm{V}$ and an internal resistance of $0.10 \Omega$.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 35

A battery has a terminal voltage of $12.0 \mathrm{V}$ when no current flows. Its internal resistance is $2.0 \Omega .$ If a $1.0 \Omega$ resistor is connected across the battery terminals, what is the terminal voltage and what is the current through the $1.0 \Omega$ resistor?

Mayukh Banik

Numerade Educator

Problem 36

(a) What are the ratios of the resistances of (a) silver and (b) aluminum wire to the resistance of copper wire $\left(R_{\mathrm{Ag}} / R_{\mathrm{Cu}}\right.$ and $\left.R_{\mathrm{Al}} / R_{\mathrm{Cu}}\right)$ for wires of the same length and the same diameter?
(c) Which material is the best conductor, for wires of equal length and diameter?

Narayan Hari

Numerade Educator

Problem 37

What are the ratios of the resistances of (a) silver and (b) aluminum wire to the resistance of copper wire $\left(R_{\mathrm{Ag}} / R_{\mathrm{Cu}}\right.$ and $\left.R_{\mathrm{Al}} / R_{\mathrm{Cu}}\right)$ for wires of the same length and the same mass (not the same diameter)?
(c) Which material is the best conductor, for wires of equal length and equal mass? The densities are: silver $10.1 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ;$ copper $8.9 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ;$ aluminum $2.7 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$.

Narayan Hari

Numerade Educator

Problem 38

A wire with cross-sectional area $A$ carries a current I. Assuming the wire is ohmic, show that the electric field strength $E$ in the wire is proportional to the current per unit area (I/A) and identify the constant of proportionality.

Narayan Hari

Numerade Educator

Problem 39

A copper wire is connected to an ideal battery at room temperature. The current increases by a factor of 78 when the wire is immersed in liquid nitrogen (temperature 77 K ). Ignoring changes in the wire's dimensions, and assuming that the number of conduction electrons per unit volume $(n)$ does not change, find the change in each of the following quantities: the resistance, the resistivity, the electric field in the wire, the drift speed, and the power dissipated.

Vishal Gupta

Numerade Educator

Problem 40

Suppose a collection of five batteries is connected as shown below. (a) What is the equivalent emf of the collection? Treat them as ideal sources of emf.
(b) What is the current through the resistor if its value is $3.2 \Omega ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 41

Suppose four batteries are connected in series as shown below. (a) What is the equivalent emf of the set of four batteries? Treat them as ideal sources of emf.
(b) If the current in the circuit is $0.40 \mathrm{A},$ what is the value of the resistor $R ?$

Mayukh Banik

Numerade Educator

Problem 42

(a) Find the equivalent capacitance between points $A$ and $B$ for the three capacitors. (b) What is the charge on the $6.0 \mu \mathrm{F}$ capacitor if a $44.0 \quad V$ emf is connected to the terminals $A$ and $B$ for a long time?

Narayan Hari

Numerade Educator

Problem 43

(a) Find the equivalent capacitance between points $A$ and $B$ for the five capacitors. (b) If a $16.0 \mathrm{V}$ emf is connected to the terminals $A$ and $B,$ what is the charge on a single equivalent capacitor that replaces all five?
(c) What is the charge on the $3.0 \mu \mathrm{F}$ capacitor?

Narayan Hari

Numerade Educator

Problem 44

(a) What is the equivalent resistance between points $A$ and $B ?$ (b) $A 276 \mathrm{V}$ emf is connected to the terminals $A$ and $B .$ What is the current in the $12 \Omega$ resistor?

Ivan Kochetkov

Numerade Educator

Problem 45

(a) What is the equivalent resistance between points $A$ and $B$ if $R=1.0 \Omega ?$ (b) If a $20 \mathrm{V}$ emf is connected to the terminals $A$ and $B,$ what is the current in the $2.0 \Omega$ resistor?

Ivan Kochetkov

Numerade Educator

Problem 46

If a $93.5 \mathrm{V}$ emf is connected to the terminals $A$ and $B$ and the current in the $4.0 \Omega$ resistor is $17 \mathrm{A},$ what is the value of the unknown resistor $R ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 47

(a) What is the equivalent capacitance between points $A$ and $B$ if $C=1.0 \mu \mathrm{F} ?$
(b) What is the charge on the $4.0 \mu \mathrm{F}$ capacitor when it is fully charged?

Ivan Kochetkov

Numerade Educator

Problem 48

The equivalent capacitance between points $A$ and $B$ is $1.63 \mu \mathrm{F}$
(a) What is the capacitance of the unknown capacitor $C ?$
(b) What is the charge on the $4.0 \mu \mathrm{F}$ capacitor when it is fully charged?

Ivan Kochetkov

Numerade Educator

Problem 49

(a) Find the value of a single capacitor to replace the three capacitors in the diagram.
(b) What is the potential difference across the $12 \mu \mathrm{F}$ capacitor at the left side of the diagram?
(c) What is the charge on the $12 \mu \mathrm{F}$ capacitor to the far right side of the circuit?

Shahab Ullah

Numerade Educator

Problem 50

A $6.0 \mathrm{pF}$ capacitor is needed to construct a circuit. The only capacitors available are rated as $9.0 \mathrm{pF}$. How can a combination of three 9.0 pF capacitors be assembled so that the equivalent capacitance of the combination is $6.0 \mathrm{pF} ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 51

A $24 \mathrm{V}$ emf is connected to terminals $A$ and $B$ in the following circuit. Find the current in each of the resistors.

Ivan Kochetkov

Numerade Educator

Problem 52

(a) Find the equivalent resistance between points $A$ and $B$ for the combination of resistors shown.
(b) An $18 \mathrm{V}$ emf is connected to the terminals $A$ and $B .$ What is the current through the $1.0 \Omega$ resistor connected directly to point $A ?(\mathrm{c})$ What is the current in the $8.0 \Omega$ resistor?

Shahab Ullah

Numerade Educator

Problem 53

(a) What is the equivalent resistance between points $A$ and $B ?$ Each resistor has the same resistance $R$.
(b) What is the equivalent resistance between points $B$ and $C ?$
(c) If a $32 \mathrm{V}$ emf is connected to terminals $A$ and $B$ and if $R=2.0 \Omega,$ what is the current in one of the resistors?

Narayan Hari

Numerade Educator

Problem 54

(a) Find the equivalent resistance between points $A$ and $B$ for the combination of resistors shown.
(b) What is the potential difference across each of the $4.0 \Omega$ resistors?
(c) What is the current in the $3.0 \Omega$ resistor?

Vishal Gupta

Numerade Educator

Problem 55

(a) Find the equivalent resistance between terminals
$A$ and $B$ to replace all of the resistors in the diagram.
(b) What current flows through the emf? (c) What is the current through the $4.00 \Omega$ resistor at the bottom?

Mayukh Banik

Numerade Educator

Problem 56

Find the unknown emf and the current in each branch of the circuit.

Vishal Gupta

Numerade Educator

Problem 57

Find the unknown resistances in this circuit.

Narayan Hari

Numerade Educator

Problem 58

Find the unknown emfs in the circuit.

Narayan Hari

Numerade Educator

Problem 59

Find the unknown emf and the unknown resistor in the circuit.

Vishal Gupta

Numerade Educator

Problem 60

Consider the circuit in the diagram. Given: $I_{1}=2.50 \mathrm{A}$, $\mathscr{E}_{1}=30.0 \mathrm{V}, \mathscr{E}_{2}=9.00 \mathrm{V}, R_{1}=8.00 \Omega,$ and $R_{2}=5.00 \Omega$.
Find the values of $I_{2}, I_{3},$ and $R_{3}$.

Narayan Hari

Numerade Educator

Problem 61

The figure shows a simplified circuit diagram for an automobile. The equivalent resistor $R$ represents the total electrical load due to spark plugs, lights, radio, fans, starter, rear window defroster, and the like in parallel. If $R=0.850 \Omega,$ find the current in each branch. What is the terminal voltage of the battery? Is the battery charging or discharging?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 62

What is the power dissipated by the resistor in the circuit if the emf is $2.00 \mathrm{V}$ ?

Ivan Kochetkov

Numerade Educator

Problem 63

What is the power dissipated by the resistor in the circuit if $R=5.00 \Omega ?$

Ivan Kochetkov

Numerade Educator

Problem 64

What is the current in a $60.0 \mathrm{W}$ bulb when connected to
a $120 \mathrm{V}$ emf?

Mayukh Banik

Numerade Educator

Problem 65

What is the resistance of a $40.0 \mathrm{W}, 120 \mathrm{V}$ incandescent lightbulb?

Ivan Kochetkov

Numerade Educator

Problem 66

If a chandelier has a label stating $120 \mathrm{V}, 5.0 \mathrm{A},$ can its power rating be determined? If so, what is it?

Mayukh Banik

Numerade Educator

Problem 67

An automatic cat feeder does not have a power rating listed, but it has a label stating that it draws a maximum current of $250.0 \mathrm{mA} .$ The feeder uses three $1.50 \mathrm{V}$ batteries connected in series. What is the maximum power consumed?

Ivan Kochetkov

Numerade Educator

Problem 68

How much work are the batteries in the circuit doing in every $10.0 \mathrm{s}$ time interval?

Ivan Kochetkov

Numerade Educator

Problem 69

Show that $\mathrm{A}^{2} \times \Omega=\mathrm{W}$ (amperes squared times ohms $=$ watts $)$ .

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 70

Consider the circuit in the diagram. (a) What current flows from the battery? (b) What is the potential difference between points $A$ and $B ?$ (c) What current flows through each branch between points $A$ and $B ?$
(d) Determine the power dissipated in the $40.0 \Omega$ resistor.

Vishal Gupta

Numerade Educator

Problem 71

(a) What is the equivalent resistance of this circuit if $R_{1}=10.0 \Omega$ and $R_{2}=15.0 \Omega ?$
(b) What current flows through $R_{1} ?$ (c) What is the voltage drop across $R_{2} ?$
(d) What current flows through $R_{2} ?$
(e) How much power is dissipated in $R_{2} ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 72

At what rate is energy dissipated in the $4.00 \Omega$ and $5.00 \Omega$ resistors in the circuit shown?

Vishal Gupta

Numerade Educator

Problem 73

In the circuit shown, $R_{1}=15.0 \Omega, R_{2}=R_{4}=40.0 \Omega$
$R_{3}=20.0 \Omega,$ and $R_{5}=10.0 \Omega$. (a) What is the equivalent resistance of this circuit?
(b) What current flows through resistor $R_{1} ?$
(c) What is the total power dissipated by this circuit?
(d) What is the potential difference across $R_{3} ?$ (e) What current flows through $R_{3} ?(\mathrm{f})$ What is the power dissipated in $R_{3} ?$

Vishal Gupta

Numerade Educator

Problem 74

A battery has a $6.00 \mathrm{V}$ emf and an internal resistance of $0.600 \Omega$
(a) What is the voltage across its terminals when the current drawn from the battery is $1.20 \mathrm{A} ?$
(b) What is the power supplied by the battery?

Mayukh Banik

Numerade Educator

Problem 75

During a "brownout," which occurs when the power companies cannot keep up with high demand, the voltage of the household circuits drops below its normal $120 \mathrm{V}$. (a) If the voltage drops to $108 \mathrm{V}$, what would be the power consumed by a " $100 \mathrm{W}$ " incandescent lightbulb (i.e., a lightbulb that consumes $100.0 \mathrm{W}$. when connected to $120 \mathrm{V}$ )? Ignore (for now) changes in the resistance of the lightbulb filament.
(b) More realistically, the lightbulb filament will not be as hot as usual during the brownout. Does this make the power drop more or less than that you calculated in part (a)? Explain.

Vishal Gupta

Numerade Educator

Problem 76

A source of emf $\mathscr{E}$ has internal resistance $r$. (a) What is the terminal voltage when the source supplies a current $I ?$ (b) The net power supplied is the terminal voltage times the current. Starting with $P=I \Delta V$, derive Eq. $(18-39)$ for the net power supplied by the source. Interpret each of the two terms.
(c) Suppose that a battery of emf $\mathscr{E}$ and internal resistance $r$ is being recharged: another emf sends a current $I$ through the battery in the reverse direction (from positive terminal to negative). At what rate is electric energy converted to chemical energy in the recharging battery? (d) What is the power supplied by the recharging circuit to the battery?

Kemuel Roberts

Numerade Educator

Problem 77

Redraw the circuit in Problem 44 to show how an ammeter would be connected to measure (a) the current through the $15 \Omega$ resistor and (b) the current through the $24 \Omega$ resistor.

Ivan Kochetkov

Numerade Educator

Problem 78

Redraw the circuit in Problem 44 to show how a voltmeter would be connected to measure (a) the potential drop across the $15 \Omega$ resistor and (b) the potential drop across the $24 \Omega$ resistor.

Ivan Kochetkov

Numerade Educator

Problem 79

(a) Redraw the following circuit to show how an ammeter would be connected to measure the current through the $1.40 \mathrm{k} \Omega$ resistor. (b) Assuming the ammeter to be ideal, what is its reading? (c) If the ammeter has a resistance of $120 \Omega,$ what is its reading?

Vishal Gupta

Numerade Educator

Problem 80

(a) Redraw the circuit to show how a voltmeter would be connected to measure the voltage across the $83.0 \mathrm{k} \Omega$ resistor. (b) Assuming the voltmeter to be ideal, what is its reading? (c) If the voltmeter has a resistance of $1.00 \mathrm{M} \Omega,$ what is its reading?

Ivan Kochetkov

Numerade Educator

Problem 81

An ammeter with a full-scale deflection for $I=10.0 \mathrm{A}$ has an internal resistance of $24 \Omega .$ We need to use this
ammeter to measure currents up to 12.0 A. The lab instructor advises that we get a resistor and use it to protect the ammeter. (a) What size resistor do we need and how should it be connected to the ammeter, in series or in parallel? (b) How do we interpret the ammeter readings?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 82

In the circuit shown, assume the battery emf is $20.0 \mathrm{V}, R=$ $1.00 \mathrm{M} \Omega,$ and $C=2.00 \mu \mathrm{F}$.
The switch is closed at $t=0$. At what time $t$ will the voltage across the capacitor be $15.0 \mathrm{V} ?$

Ivan Kochetkov

Numerade Educator

Problem 83

In the circuit, $R=30.0 \mathrm{k} \Omega$ and $C=0.10 \mu \mathrm{F}$. The capacitor is allowed to charge fully, and then the switch is changed from position $a$ to position $b$ What will the voltage across the resistor be $8.4 \mathrm{ms}$ later?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 84

A capacitor is charged to an initial voltage $V_{0}=9.0 \mathrm{V}$. The capacitor is then discharged by connecting its terminals through a resistor. The current $I(t)$ through this resistor, determined by measuring the voltage $\Delta V_{\mathrm{R}}(t)=I(t) R$ with
an oscilloscope, is shown in the graph.
(a) Find the capacitance $C,$ the resistance $R,$ and the total energy dissipated in the resistor. (b) At what time is the energy in the capacitor half its initial value? (c) Graph the voltage across the capacitor, $\Delta V_{\mathrm{C}}(t),$ as a function of time.

Dominador Tan

Numerade Educator

Problem 85

A charging $R C$ circuit controls the intermittent windshield wipers in a car. The emf is $12.0 \mathrm{V}$. The wipers are triggered when the voltage across the $125 \mu \mathrm{F}$ capacitor reaches $10.0 \mathrm{V} ;$ then the capacitor is quickly discharged (through a much smaller resistor) and the cycle repeats. What resistance should be used in the charging circuit if the wipers are to operate once every $1.80 \mathrm{s} ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 86

A defibrillator passes a brief burst of current through the heart to restore normal beating. In one such defibrillator, a $50.0 \mu \mathrm{F}$ capacitor is charged to $6.0 \mathrm{kV}$. Paddles are used to make an electric connection to the patient's chest. A pulse of current lasting $1.0 \mathrm{ms}$ partially discharges the capacitor through the patient. The electrical resistance of the patient (from paddle to paddle) is $240 \Omega$
(a) What is the initial energy stored in the capacitor?
(b) What is the initial current through the patient?
(c) How much energy is dissipated in the patient during the $1.0 \mathrm{ms} ?$
(d) If it takes $2.0 \mathrm{s}$ to recharge the capacitor, compare the average power supplied by the power source with the average power delivered to the patient.
(e) Referring to your answer to part (d), explain one reason a capacitor is used in a defibrillator.

Mayukh Banik

Numerade Educator

Problem 87

Capacitors are used in many applications where one needs to supply a short burst of relatively large current. A $100.0 \mu \mathrm{F}$ capacitor in an electronic flash lamp supplies a burst of current that dissipates $20.0 \mathrm{J}$ of energy (as light and heat) in the lamp.
(a) To what potential difference must the capacitor initially be charged?
(b) What is its initial charge?
(c) Approximately what is the resistance of the lamp if the current reaches $5.0 \%$ of its original value in $2.0 \mathrm{ms} ?$

Mayukh Banik

Numerade Educator

Problem 88

Consider the circuit shown with $R_{1}=25 \Omega, R_{2}=33 \Omega$ $C_{1}=12 \mu \mathrm{F}, C_{2}=23 \mu \mathrm{F}, C_{3}=46 \mu \mathrm{F},$ and $V=6.0 \mathrm{V}$.
(a) Draw an equivalent circuit with one resistor and one capacitor and label it with the values of the
equivalent resistor and capacitor.
(b) $\mathrm{A}$ long time after switch $S$ is
closed, what are the charge on capacitor $C_{1}$ and the current in resistor $R_{1} ?$ (c) What is the time constant of the circuit?

Vishal Gupta

Numerade Educator

Problem 89

In the circuit of Problem $88,$ at what time after switch $S$ is closed is the voltage across the combination of three capacitors $50 \%$ of its final value?

Ivan Kochetkov

Numerade Educator

Problem 90

In a defibrillator (see Example 17.12 ), a charged capacitor is connected to paddles that make electrical contact with the patient's skin. If gel is applied to the patient's chest to make a good connection between the paddles and the skin, the effective resistance through which the capacitor discharges is 52.0 \Omega. (a) To what voltage must the capacitor be charged to generate a maximum current of $40.0 \mathrm{A} ?$
(b) If the current $1.00 \mathrm{ms}$ later is $10.0 \mathrm{A},$ what is the capacitance?
(c) Why does a paramedic shout "Clear!" before administering the shock?

Vishal Gupta

Numerade Educator

Problem 91

In the circuit, the capacitor is initially uncharged. At $t=0$, switch $S$ is closed. Find the currents $I_{1}$ and $I_{2}$ and voltages $V_{1}$ and $V_{2}$ (assuming $V_{3}=0$ ) at points 1 and 2 at $(a) t=0$ (i.e., just after the switch is closed ) and at (b) $t=1.0 \mathrm{ms}$.

Vishal Gupta

Numerade Educator

Problem 92

In the circuit, the initial energy stored in the capacitor is 25 J. At $t=0$ the
switch is closed.
(a) Sketch a graph of the voltage
across the resistor $\left(V_{\mathrm{R}}\right)$ as a function of $t .$ Label the vertical axis with key numerical value(s) and units.
(b) At what time is the energy stored in the capacitor $1.25 \mathrm{J} ?$

Mayukh Banik

Numerade Educator

Problem 93

(a) In a charging $R C$ circuit, how many time constants have elapsed when the capacitor has $99.0 \%$ of its final charge?
(b) How many time constants have elapsed when the capacitor has $99.90 \%$ of its final charge?
(c) How many time constants have elapsed when the current has $1.0 \%$ of its initial value?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 94

A $20 \mu \mathrm{F}$ capacitor is discharged through a $5.0 \mathrm{k} \Omega$ resistor. The initial charge on the capacitor is $200 \mu \mathrm{C}$.
(a) Sketch a graph of the current through the resistor as a function of time. Label both axes with numbers and
units. (b) What is the initial power dissipated in the resistor? (c) What is the total energy dissipated?

Deborah Israel

Numerade Educator

Problem 95

Consider the circuit in the diagram. After the switch $S$ has been closed for a long time, what are the current through the $12 \Omega$ resistor and the voltage across the capacitor?

Ivan Kochetkov

Numerade Educator

Problem 96

A parallel plate capacitor used in a flash for a camera must be able to store 32 J of energy when connected to $300 \mathrm{V} .$ (Most electronic flashes actually use a $1.5 \mathrm{to}$ $6.0 \mathrm{V}$ battery, but increase the effective voltage using a dc-dc inverter.) (a) What should be the capacitance of this capacitor? (b) If this capacitor has an area of $9.0 \mathrm{m}^{2}$ and a distance between the plates of $1.1 \times 10^{-6} \mathrm{m},$ what is the dielectric constant of the material between the plates? (The large effective area can be put into a small volume by rolling the capacitor tightly in a cylinder.)
(c) Assuming the capacitor completely discharges to produce a flash in $4.0 \times 10^{-3}$ s, what average power is dissipated in the flashbulb during this time?
(d) If the distance between the plates of the capacitor could be reduced to half its value, how much energy would the capacitor store if charged to the same voltage?

Vishal Gupta

Numerade Educator

Problem 97

Consider the camera flash in Problem $96 .$ If the flash really discharges according to Eq. $(18-48),$ then it takes an infinite amount of time to discharge. When Problem 96 assumes that the capacitor discharges in $4.0 \times 10^{-3} \mathrm{s},$ we mean that the capacitor has almost no charge stored on it after that amount of time. Suppose that after $4.0 \times 10^{-3} \mathrm{s}$ the capacitor has only $1.0 \%$ of the original charge still on it. (a) What is the time constant of this $R C$ circuit? (b) What is the resistance of the flashbulb in this case?
(c) What is the maximum power dissipated in the flashbulb?

Vishal Gupta

Numerade Educator

Problem 98

A capacitor is charged by a $9.0 \mathrm{V}$ battery. The charging current $I(t)$ is shown. (a) Find the capacitance $C$ of the capacitor and the total resistance $R$ in the circuit. (b) At what time is the stored energy in the capacitor half of its maximum value?

Dominador Tan

Numerade Educator

Problem 99

A charged capacitor is discharged through a resistor. The current $I(t)$ through this resistor, determined by measuring the voltage $\Delta V_{\mathrm{R}}(t)=I(t) R$ with an oscilloscope, is shown in the graph. The total energy dissipated in the resistor is $2.0 \times 10^{-4}$ J. (a) Find the capacitance $C$, the resistance $R$, and the initial charge on the capacitor. (b) At what time is the stored energy in the capacitor $5.0 \times 10^{-5} \mathrm{J} ?$

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator

Problem 100

A person in bare feet is standing under a tree during a thunderstorm, seeking shelter from the rain. A lightning strike hits the tree. A burst of current lasting $40 \mu$ s passes through the ground; during this time the potential difference between his feet is $20 \mathrm{kV}$. If the resistance between one foot and the other is $500 \Omega,$ (a) what is the current through his body and (b) how much energy is dissipated in his body by the lightning?

Narayan Hari

Numerade Educator

Problem 101

In the physics laboratory, Oscar measured the resistance between his hands to be $2.0 \mathrm{k} \Omega .$ Being curious by nature, he then took hold of two conducting wires that were connected to the terminals of an emf
with a terminal voltage of $100.0 \mathrm{V} .$ (a) What current passes through Oscar?
(b) If one of the conducting wires is grounded and the other has an alternative path to ground through a $15 \Omega$ resistor (so that Oscar and the resistor are in parallel), how much current would pass through Oscar if the maximum current that can be drawn from the emf is $1.00 \mathrm{A} ?$

Vishal Gupta

Numerade Educator

Problem 102

Chelsea inadvertently bumps into a set of batteries with an emf of $100.0 \mathrm{V}$ that can supply a maximum power of $5.0 \mathrm{W} .$ If the resistance between the points where she contacts the batteries is $1.0 \mathrm{k} \Omega,$ how much current passes through her?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 103

In her bathroom, Mindy has an overhead heater that consists of a coiled wire made of nichrome that gets hot when turned on. The wire has a length of $3.0 \mathrm{m}$ when it is uncoiled. The heating element is attached to the normal $120 \mathrm{V}$ wiring, and when the wire is glowing red hot, it has a temperature of about $420^{\circ} \mathrm{C}$ and dissipates $2200 \mathrm{W}$ of power. Nichrome has a resistivity of $108 \times$ $10^{-8} \Omega \cdot \mathrm{m}$ at $20^{\circ} \mathrm{C}$ and a temperature coefficient of $\mathrm{re}-$ sistivity of $0.00040^{\circ} \mathrm{C}^{-1}$. (a) What is the resistance of the heater when it is turned on? (b) What current does the wire carry? (c) If the wire has a circular cross section, what is its diameter? Ignore the small changes in the wire's diameter and length due to changes in temperature. (d) When the heater is first turned on, it has not yet heated up, so it is operating at $20^{\circ} \mathrm{C}$. What is the current through the wire when it is first turned on?

Vishal Gupta

Numerade Educator

Problem 104

Che wiring circuit for a typical room is shown schematically. (a) Of the six locations for a circuit breaker indicated by $A, B, C, D, E,$ and $F,$ which one would best protect the wiring against a short circuit in any one of the three appliances? Explain.
(b) The potential difference between hot and neutral is $120 \mathrm{V}$. Suppose the heater draws $1500 \mathrm{W}$, the lamp draws $300 \mathrm{W}$, and the microwave draws 1200 W. The circuit breaker is rated at 20.0 A. Can all three devices be operated simultaneously without tripping the breaker? Explain.

Vishal Gupta

Numerade Educator

Problem 105

We can model some of the electrical properties of an unmyelinated axon as an electric cable covered with defective insulation so that current leaks out of the axon
to the surrounding fluid. We assume the axon consists of a cylindrical membrane filled with conducting fluid. A current of ions can travel along the axon in this fluid and can also leak out through the membrane. The inner radius of the cylinder is $5.0 \mu \mathrm{m} ;$ the membrane thickness is $8.0 \mathrm{nm}$. (a) If the resistivity of the axon fluid is 2.0 $\Omega \cdot \mathrm{m}$, calculate the resistance of a $1.0 \mathrm{cm}$ length of axon to current flow along its length. (b) If the resistivity of the porous membrane is $2.5 \times 10^{7} \Omega \cdot \mathrm{m},$ calculate the resistance of the wall of a $1.0 \mathrm{cm}$ length of axon to current flow across the membrane.
(c) Find the length of axon for which the two resistances are equal. This length is a rough measure of the distance a signal can travel without amplification.

Mayukh Banik

Numerade Educator

Problem 106

(a) Given two identical, ideal batteries $(\mathrm{emf}=\mathscr{E})$ and two identical incandescent lightbulbs (resistance $=R$ assumed constant), design a circuit to make both bulbs glow as brightly as possible. (b) What is the power dissipated by each bulb? (c) Design a circuit to make both bulbs glow, but one more brightly than the other. Identify the brighter bulb.

DM

Debra Mangion

Numerade Educator

Problem 107

Copper and aluminum are being considered for the cables in a high-voltage transmission line where each must carry a current of 50 A. The resistance of each cable is to be $0.15 \Omega$ per kilometer.
(a) If this line carries power from Niagara Falls to New York City (approximately $500 \mathrm{km}$ ), how much power is lost along the way in the cable? Compute for each choice of cable material (b) the necessary cable diameter and (c) the mass per meter of the cable. The electrical resistivities for copper and aluminum are given in Table $18.1 ;$ the mass density of copper is $8920 \mathrm{kg} / \mathrm{m}^{3}$ and that of aluminum is $2702 \mathrm{kg} / \mathrm{m}^{3}$.

Mayukh Banik

Numerade Educator

Problem 108

About $5.0 \times 10^{4} \mathrm{m}$ above Earth's surface, the atmosphere is sufficiently ionized that it behaves as a conductor. Earth and the ionosphere form a giant spherical capacitor, with the lower atmosphere acting as a leaky dielectric. (a) Find the capacitance $C$ of the Earthionosphere system by treating it as a parallel plate capacitor. Why is it OK to do that? [Hint: Compare Earth's radius to the distance between the "plates."]
(b) The fair-weather electric field is about $150 \mathrm{V} / \mathrm{m},$ downward. How much energy is stored in this capacitor? (c) Due to radioactivity and cosmic rays, some air molecules are ionized even in fair weather. The resistivity of air is roughly $3.0 \times 10^{14} \Omega \cdot \mathrm{m} .$ Find the resistance of the lower atmosphere and the total current that flows between Earth's surface and the ionosphere. [Hint: since we treat the system as a parallel plate capacitor, treat the atmosphere as a dielectric of uniform thickness between the plates.] (d) If there were no lightning, the capacitor would discharge. In this model, how much time would elapse before Earth's charge were reduced to $1 \%$ of its normal value? (Thunderstorms are the sources of emf that maintain the charge on this leaky capacitor.)

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 109

A $1.5 \mathrm{V}$ flashlight battery can maintain a current of 0.30 A for $4.0 \mathrm{h}$ before it is exhausted. How much
chemical energy is converted to electrical energy in this process? (Assume zero internal resistance of the battery.)

Mayukh Banik

Numerade Educator

Problem 110

In the diagram, the positive terminal of the $12 \mathrm{V}$ battery is grounded-it is at zero potential. At what potential is point $X ?$

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 111

$\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ represent ammeters with negligible resistance. What are the values of the currents
(a) in $\mathrm{A}_{1}$ and
(b) in $\mathrm{A}_{2} ?$

Vishal Gupta

Numerade Educator

Problem 112

Repeat Problem 111 if each of the ammeters has resistance $0.200 \Omega$.

Mayukh Banik

Numerade Educator

Problem 113

A 1.5 hp motor operates on 120 V. Ignoring $I^{2} R$ losses, how much current does it draw?

Narayan Hari

Numerade Educator

Problem 114

A certain electric device has the current-voltage $(I-V)$ graph shown with Problem $30 .$ What is the power dissipated at points 1 and $2 ?$

Vishal Gupta

Numerade Educator

Problem 115

Given two identical, ideal batteries of emf $\mathscr{E}$ and two identical incandescent lightbulbs of resistance $R$ (assumed constant), find the total power dissipated in the circuit in terms of $\mathscr{E}$ and $R$.

Narayan Hari

Numerade Educator

Problem 116

Two circuits are constructed using identical, ideal batteries $(\mathrm{emf}=\mathscr{E})$ and identical incandescent lightbulbs (resistance $=R$ ). If each bulb in circuit 1 dissipates $5.0 \mathrm{W}$ of power, how much power does each bulb in circuit 2 dissipate? Ignore changes in the resistance of the bulbs due to temperature changes.

Narayan Hari

Numerade Educator

Problem 117

A $500 \mathrm{W}$ electric heater unit is designed to operate with an applied potential difference of $120 \mathrm{V}$. (a) If the local power company imposes a voltage reduction to lighten its load, dropping the voltage to $110 \mathrm{V},$ by what percentage does the heat output of the heater drop? (Assume the resistance does not change.) (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than calculated in part (a)?

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 118

Consider a $60.0 \mathrm{W}$ incandescent lightbulb and a $100.0 \mathrm{W}$ incandescent lightbulb designed for use in a household lamp socket at $120 \mathrm{V}$. (a) What are the resistances of these two bulbs? (b) If they are wired together in a series circuit, which bulb shines brighter (dissipates more power)? Explain. (c) If they are connected in parallel in a circuit, which bulb shines brighter? Explain.

Narayan Hari

Numerade Educator

Problem 119

The Wheatstone bridge is a circuit used to measure unknown resistances. The bridge in the figure is balanced-no current flows through the galvanometer $\mathrm{G}$ (a sensitive detector of current whose operation is based on magnetic forces).
(a) What is the unknown resistance $R_{x} ?[$ Hint: What is the potential difference between points $A$ and $B ?]$ (b) Do the resistance of the galvanometer or the internal resistance of the emf affect the measurement? Explain.

Vishal Gupta

Numerade Educator

Problem 120

The filament of an incandescent lightbulb is made of tungsten. At room temperature of $20.0^{\circ} \mathrm{C}$ the filament has a resistance of $10.0 \Omega$. (a) What is the power dissipated in the lightbulb immediately after it is connected to a $120 \mathrm{V}$ emf (when the filament is still at $\left.20.0^{\circ} \mathrm{C}\right) ?$ (b) After a brief time, the lightbulb filament has changed temperature and it glows brightly. The current is now 0.833 A. What is the resistance of the lightbulb now? (c) What is the power dissipated in the lightbulb when it is glowing brightly as in part (b)?
(d) What is the temperature of the filament when it is glowing brightly? (e) Explain why incandescent lightbulbs usually burn out when they are first turned on rather than after they have been glowing for a long time.

Vishal Gupta

Numerade Educator

Problem 121

(a) What is the resistance of the heater element in a $1500 \mathrm{W}$ hair dryer that plugs into a $120 \mathrm{V}$ outlet?
(b) What is the current through the hair dryer when it is turned on? (c) At a cost of $\$ 0.10$ per $\mathrm{kW} \cdot \mathrm{h},$ how much does it cost to run the hair dryer for 5.00 min?
(d) If you were to take the hair dryer to Europe where the voltage is $240 \mathrm{V}$, how much power would your hair dryer be using in the brief time before it is ruined?
(e) What current would be flowing through the hair dryer during this time?

Narayan Hari

Numerade Educator

Problem 122

In the circuit shown, an emf of $150 \mathrm{V}$ is connected across a resistance network. What is the current
through $R_{2}$ ? Each of the resistors has a value of $10 \Omega$.

Mederic Rodriguez

Mederic Rodriguez

Numerade Educator

Problem 123

A $2.00 \mu \mathrm{F}$ capacitor is charged using a $5.00 \mathrm{V}$ battery, and a $3.00 \mu \mathrm{F}$ capacitor is charged using a $10.0 \mathrm{V}$ battery. (a) What is the total energy stored in the two capacitors? (b) The batteries are disconnected, and the two capacitors are connected together $(+\mathrm{to}+\mathrm{and}-$ to $-$ ). Find the charge on each capacitor and the total energy in the two capacitors after they are connected.
(c) Explain what happened to the "missing" energy. [Hint: The wires that connect the two have some resistance.]

Vishal Gupta

Numerade Educator

Problem 124

A string of 25 decorative lights has bulbs rated at 9.0 W, and the bulbs are connected in parallel. The string is connected to a $120 \mathrm{V}$ power supply. (a) What is the resistance of each of these lights? (b) What is the current through each bulb? (c) What is the total current coming from the power supply? (d) The string of bulbs has a fuse that will blow if the current is greater than 2.0 A. How many of the bulbs can you replace with $10.4 \mathrm{W}$ bulbs without blowing the fuse?

Mayukh Banik

Numerade Educator

Problem 125

A portable radio requires an emf of 4.5 V. Olivia has only two nonrechargeable $1.5 \mathrm{V}$ batteries, but she finds a larger $6.0 \mathrm{V}$ battery.
(a) How can she arrange the batteries to produce an emf of $4.5 \mathrm{V} ?$ Draw a circuit diagram. (b) Is it advisable to use this combination with her radio? Explain.

Ivan Kochetkov

Numerade Educator

Problem 126

Three identical incandescent lightbulbs are connected with wires to an ideal battery. The two terminals on each socket connect to the two terminals of its
lightbulb. Wires do not connect with one another where they appear to cross in the picture. Ignore the change of the resistances of the filaments due to temperature changes. (a) Which of the schematic circuit diagrams correctly represent(s) the circuit? (List more than one choice if more than one diagram is correct.) (b) Which bulb(s) is/are the brightest? Which is/are the dimmest? Or are they all the same? Explain. (c) Find the current through each bulb if the filament resistances are each $24.0 \Omega$ and the emf is $6.0 \mathrm{V}$.

Vishal Gupta

Numerade Educator

Problem 127

A voltmeter with a resistance of $670 \mathrm{k} \Omega$ is used to mea- $-$
sure the voltage across the $83.0 \mathrm{k} \Omega$ resistor in the figure with Problems 79 and $80 .$ What is the voltmeter reading?

Vishal Gupta

Numerade Educator

Problem 128

A piece of gold wire of length $L$ has a resistance $R_{0}$. Suppose the wire is drawn out so that its length increases by a factor of $3 .$ What is the new resistance $R$ in terms of the original resistance?

Ivan Kochetkov

Numerade Educator

Problem 129

The circuit is used to study the charging of a capacitor.
(a) At $t=0,$ the switch is closed. What initial charging current is measured by the ammeter? (b) After the current has decayed to zero, what are the voltages at points $A, B,$ and $C ?$

Mayukh Banik

Numerade Educator

Problem 130

A gold wire and an aluminum wire have the same dimensions and carry the same current. The electron density (in electrons/cm $^{3}$ ) in aluminum is three times larger than the density in gold. How do the drift speeds of the electrons in the two wires, $v_{\mathrm{Au}}$ and $v_{\mathrm{Al}},$ compare?

Mayukh Banik

Numerade Educator

Problem 131

A potentiometer is a resistor with a sliding contact. It can be used to measure emfs accurately (Problem 131$)$ or to supply a variable voltage to a circuit (Problem 132 ). In the diagram with switch $S_{1}$ closed and $S_{2}$ open, there is no current through the galvanometer G (a sensitive detector of current whose operation is based on magnetic forces) for $R_{1}=20.0 \Omega$ with a standard cell $\mathscr{E}_{\mathrm{s}}$ of $2.00 \mathrm{V}$. With switch $S_{2}$ closed and $S_{1}$ open, there is no current through the galvanometer G for $R_{2}=80.0 \Omega$. (a) What is the unknown emf $\mathscr{E}_{x} ?$
(b) Explain why the potentiometer accurately measures the emf even for a source with substantial internal resistance.

Narayan Hari

Numerade Educator

Problem 132

In the circuit, $\mathscr{E}=45.0 \mathrm{V}$ and $R=100.0 \Omega$. Assume
the emf is ideal. If a voltage $V_{x}=30.0 \mathrm{V}$ is needed for a circuit, what should resistance $R_{x}$ be?

Narayan Hari

Numerade Educator

Problem 133

Near Earth's surface the air contains both negative and positive ions due to radioactivity in the soil and cosmic rays from space. As a simplified model, assume there are 600.0 singly charged positive ions per cubic centimeter and 500.0 singly charged negative ions per cubic centimeter. Ignore the presence of multiplycharged ions. The electric field is $100.0 \mathrm{V} / \mathrm{m},$ directed downward. (a) In which direction do the positive ions move? The negative ions? (b) What is the direction of the current due to these ions? (c) The measured resistivity of the air in the region is $4.0 \times 10^{13} \Omega \cdot \mathrm{m} .$ Calculate the drift speed of the ions, assuming it to be the same for positive and negative ions. [Hint: Consider a vertical tube of air of length $L$ and cross-sectional area $A .$ How is the potential difference across the tube related to the electric field strength?] (d) If these conditions existed simultaneously over the entire surface, what would be the total current due to the movement of ions in the air?

Dominador Tan

Numerade Educator

Problem 134

A parallel plate capacitor is constructed from two square conducting plates of length $L=0.10 \mathrm{m}$ on a side. There is air between the plates, which are separated by a distance $d=89 \mu \mathrm{m} .$ The capacitor is connected to a $10.0 \mathrm{V}$ battery.
(a) After the capacitor is fully charged, what is the charge on the upper plate? (b) The battery is disconnected from the plates, and the capacitor is discharged through a resistor $R=0.100 \mathrm{M} \Omega .$ Sketch the current through the resistor as a function of time $t(t=0$ corresponds to the time when $R$ is connected to the capacitor).
(c) How much energy is dissipated in $R$ over the whole discharging process?

Mayukh Banik

Numerade Educator

Problem 135

A coffee maker can be modeled as a heating element (resistance $R$ ) connected to the outlet voltage of $120 \mathrm{V}$ (assumed to be dc). The heating element boils small amounts of water at a time as it brews the coffee. When bubbles of water vapor form, they carry liquid water up through the tubing. Because of this, the coffee maker boils $5.0 \%$ of the water that passes through it; the rest is heated to $100^{\circ} \mathrm{C}$ but remains liquid. Starting with water at $10^{\circ} \mathrm{C},$ the coffee maker can brew $1.0 \mathrm{L}$ of coffee in 8.0 min. Find the resistance $R$.

Aishwarya Krishnakumar

Aishwarya Krishnakumar

Numerade Educator

Problem 136

Two immersion heaters, $A$ and $B$, are both connected to a $120 \mathrm{V}$ supply. Heater $A$ can raise the temperature of $1.0 \mathrm{L}$ of water from $20.0^{\circ} \mathrm{C}$ to $90.0^{\circ} \mathrm{C}$ in $2.0 \mathrm{min},$ whereas heater $B$ can raise the temperature of $5.0 \mathrm{L}$ of water from $20.0^{\circ} \mathrm{C}$ to $90.0^{\circ} \mathrm{C}$ in $5.0 \mathrm{min} .$ What is the ratio of the resistance of heater $A$ to the resistance of heater $B ?$

Narayan Hari

Numerade Educator

Problem 137

A copper wire has a resistance of $24 \Omega$ at $20^{\circ} \mathrm{C}$. An aluminum wire has 3.0 times the length and 2.0 times the radius of the copper wire. (a) What is the resistance of the aluminum wire at $20^{\circ} \mathrm{C} ?$
(b) The graph shows a $V-I$ plot for the copper wire. What is the resistance of the wire when operating steadily at a current of $10 \mathrm{A} ?$ (c) What was the temperature of the copper wire when the current was 10 A? Ignore changes in the wire's dimensions. (d) Would your answer to (c) change significantly if you took into account the thermal expansion of the wire? Explain.

Vishal Gupta

Numerade Educator

Problem 138

The field between the plates of a parallel plate capacitor, $E=Q /\left(\epsilon_{0} A\right),$ is due to the superposition of equal contributions from the charges on the two plates. Therefore, each plate exerts an electric force on the other. (a) Find the magnitude of this force in terms of
(b) Suppose the plates have no other $Q, \epsilon_{0},$ and $A$ forces acting on them and they start a distance $d$ apart. Find the kinetic energy of each plate when they collide. [Hint: Two different methods are possible.]

Narayan Hari

Numerade Educator

Problem 139

Many home heating systems operate by pumping hot water through radiator pipes. The flow of the water to different "zones" in the house is controlled by zone valves that open in response to thermostats. The opening and closing of a zone valve is commonly performed by a wax actuator, as shown in the diagram. When the thermostat signals the valve to open, a dc voltage of $24 \mathrm{V}$ is applied across a heating element (resistance $R=200 \Omega$ ) in the actuator. As the wax melts, it expands and pushes a cylindrical rod (radius $2.0 \mathrm{mm}$ ) out a distance $1.0 \mathrm{cm}$ to open the zone switch. The actuator contains $2.0 \mathrm{mL}$ of solid wax of density $0.90 \mathrm{g} / \mathrm{cm}^{3}$ at room temperature $\left(20^{\circ} \mathrm{C}\right) .$ The specific heat of the wax is $0.80 \mathrm{J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)$, its
latent heat of fusion is $60 \mathrm{J} / \mathrm{g},$ and its melting point is $90^{\circ} \mathrm{C} .$ When the wax melts its volume expands by $15 \%$ How long does it take until the valve is fully open?

Dominador Tan

Numerade Educator

Problem 140

Poiseuille’s law [Eq. (9-41)] gives the volume flow rate of a viscous fluid through a pipe. (a) Show that Poiseuille's law can be written in the form $\Delta P=I R$, where $I=\Delta V / \Delta t$ represents the volume flow rate and $R$ is a constant of proportionality called the fluid flow resistance. (b) Find $R$ in terms of the viscosity of the fluid and the length and radius of the pipe. (c) If two or more pipes are connected in series so that the volume flow rate through them is the same, do the resistances of the pipes add as for electrical resistors $\left(R_{\mathrm{eq}}=R_{1}+R_{2}+\cdots\right) ?$ Explain. (d) If two or more pipes are connected in parallel, so the pressure drop across them is the same, do the reciprocals of the resistances add as for electrical resistors $\left(1 / R_{\mathrm{eq}}=1 / R_{1}+1 / R_{2}+\cdots\right) ?$ Explain.

Narayan Hari

Numerade Educator