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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 25

Current, Resistance, and Electromotive Force - all with Video Answers

Educators

+ 7 more educators

Chapter Questions

02:15

Problem 1

Lightning Strikes. During lightning strikes from a cloud to the ground, currents as high as 25,000 A can occur and last for about $40 \mu$ s. How much charge is transferred from the cloud to the earth during such a strike?

Mark J
Mark J
Numerade Educator
04:39

Problem 2

A silver wire $2.6 \mathrm{~mm}$ in diameter transfers a charge of $420 \mathrm{C}$ in $80 \mathrm{~min}$. Silver contains $5.8 \times 10^{28}$ free electrons per cubic meter.
(a) What is the current in the wire? (b) What is the magnitude of the drift velocity of the electrons in the wire?

JW
Jiaxuan Wang
Numerade Educator
05:19

Problem 3

A 0.40 A current runs through a copper wire of cross sectional area $1.5 \mathrm{~mm}^{2}$ and through a light bulb. Copper has $8.5 \times 10^{28}$ free electrons per cubic meter.
(a) How many electrons pass through the light bulb each second? (b) What is the current density in the wire?
(c) At what speed does a typical electron pass by any given point in the wire?
(d) If you were to use wire with a larger cross sectional area, which of the above answers would change? Would they increase or decrease?

Melissa Walsh
Melissa Walsh
Numerade Educator
02:25

Problem 4

An copper wire of cross sectional area $2.5 \mathrm{~mm}^{2}$ carries a current with a current density of $2.40 \times 10^{6} \mathrm{~A} / \mathrm{m}^{2}$. The density of free electrons for copper is $8.5 \times 10^{28}$ electrons per cubic meter. Calculate
(a) the current in the wire and (b) the magnitude of the drift velocity of electrons in the wire.

Melissa Walsh
Melissa Walsh
Numerade Educator
04:05

Problem 5

The free-electron density in a copper wire is $8.5 \times 10^{28}$ electrons $/ \mathrm{m}^{3}$. The electric field in the wire is $0.0600 \mathrm{~N} / \mathrm{C}$ and the temperature of the wire is $20.0^{\circ} \mathrm{C}$. (a) What is the drift speed $v_{\mathrm{d}}$ of the electrons in the wire? (b) What is the potential difference between two points in the wire that are separated by $20.0 \mathrm{~cm} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
10:14

Problem 6

You want to produce three 1.00 -mm-diameter cylindrical wires, each with a resistance of $1.00 \Omega$ at room temperature. One wire is gold, one is copper, and one is aluminum. Refer to Table 25.1 for the resistivity values.
(a) What will be the length of each wire?
(b) Gold has a density of $1.93 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}$. What will be the mass of the gold wire? If you consider the current price of gold, is this wire very expensive?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
05:23

Problem 7

The current in a wire varies with time according to the relationship $I=55 \mathrm{~A}-\left(0.65 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}$. (a) How many coulombs of charge pass a cross section of the wire in the time interval between $t=0$ and $t=8.2 \mathrm{~s} ?$ (b) What constant current would transport the same charge in the same time interval?

Aparna Shakti
Aparna Shakti
Numerade Educator
02:21

Problem 8

Current passes through a solution of sodium chloride. In $1.00 \mathrm{~s}, 2.68 \times 10^{16} \mathrm{Na}^{+}$ ions arrive at the negative electrode and $3.92 \times 10^{16} \mathrm{Cl}^{-}$ ions arrive at the positive electrode. (a) What is the cur-
(b) What is the direction of the current? rent passing between the electrodes?

Melissa Walsh
Melissa Walsh
Numerade Educator
01:03

Problem 9

Nerve cells transmit electric signals through their long tubular axons. These signals propagate due to a sudden rush of $\mathrm{Na}^{+}$ ions, each with charge $+e$, into the axon. Measurements have revealed that typically about $5.6 \times 10^{11} \mathrm{Na}^{+}$ ions enter each meter of the axon during a time of $10 \mathrm{~ms}$. What is the current during this inflow of charge in a meter of axon?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:02

Problem 10

The mass density of silver at room temperature is $10.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and its atomic mass is $108 \mathrm{~g} / \mathrm{mol}$. If we assume there is one free electron per silver atom, what is the free-electron density for silver, in electrons $/ \mathrm{m}^{3}$ ? How does your answer compare to the free-electron density for copper given in Example $25.1 ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
07:55

Problem 11

A metal wire has a circular cross section with radius $0.800 \mathrm{~mm}$. You measure the resistivity of the wire in the following way: You connect one end of the wire to one terminal of a battery that has emf $12.0 \mathrm{~V}$ and negligible internal resistance. To the other terminal of the battery you connect a point along the wire so that the length of wire between the battery terminals is $d$. You measure the current in the wire as a function of $d$. The currents are small, so the temperature change of the wire is very small. You plot your results as $I$ versus $1 / d$ and find that the data lie close to a straight line that has slope $600 \mathrm{~A} \cdot \mathrm{m}$. What is the resistivity of the material of which the wire is made?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:55

Problem 12

A metal wire has a circular cross section with radius $0.800 \mathrm{~mm}$. You measure the resistivity of the wire in the following way: You connect one end of the wire to one terminal of a battery that has emf $12.0 \mathrm{~V}$ and negligible internal resistance. To the other terminal of the battery you connect a point along the wire so that the length of wire between the battery terminals is $d$. You measure the current in the wire as a function of $d$. The currents are small, so the temperature change of the wire is very small. You plot your results as $I$ versus $1 / d$ and find that the data lie close to a straight line that has slope $600 \mathrm{~A} \cdot \mathrm{m}$. What is the resistivity of the material of which the wire is made?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:46

Problem 13

A $1.50 \mathrm{~m}$ cylindrical rod of diameter $0.500 \mathrm{~cm}$ is connected to a power supply that maintains a constant potential difference of $17.0 \mathrm{~V}$ across its ends, while an ammeter measures the current through it. You observe that at room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ the ammeter reads $18.8 \mathrm{~A}$, while at $92.0^{\circ} \mathrm{C}$ it reads $17.1 \mathrm{~A}$. You can ignore any thermal expansion of the rod. Find (a) the resistivity at $20.0^{\circ} \mathrm{C}$ and $(\mathrm{b})$ the temperature coefficient of resistivity at $20^{\circ} \mathrm{C}$ for the material of the rod.

Melissa Walsh
Melissa Walsh
Numerade Educator
03:36

Problem 14

A copper wire has a square cross section $2.1 \mathrm{~mm}$ on a side. The wire is $4.3 \mathrm{~m}$ long and carries a current of $3.6 \mathrm{~A} .$ The density of free electrons is $8.5 \times 10^{28} / \mathrm{m}^{3}$. Find the magnitudes of (a) the current density in the wire and (b) the electric field in the wire.
(c) How much time is required for an electron to travel the length of the wire?

Melissa Walsh
Melissa Walsh
Numerade Educator
02:41

Problem 15

A copper wire of cross sectional area $2.5 \mathrm{~mm}^{2}$ carries a current of $13.5 \mathrm{~mA}$. (a) What is the potential difference across a $2.00 \mathrm{~m}$ length of the wire? (b) What would the potential difference in part (a) be if the wire were silver instead of copper, but all else were the same?

Melissa Walsh
Melissa Walsh
Numerade Educator
04:46

Problem 16

A copper wire of cross sectional area $2.5 \mathrm{~mm}^{2}$ carries a current of $13.5 \mathrm{~mA}$. (a) What is the potential difference across a $2.00 \mathrm{~m}$ length of the wire? (b) What would the potential difference in part (a) be if the wire were silver instead of copper, but all else were the same?

Supratim Pal
Supratim Pal
Numerade Educator
03:56

Problem 17

A copper wire has radius $0.800 \mathrm{~mm}$ and carries current $I$ at $20.0^{\circ} \mathrm{C}$. A silver wire with radius $0.500 \mathrm{~mm}$ carries the same current and is also at $20.0^{\circ} \mathrm{C}$. Let $E_{\mathrm{Cu}}$ be the electric field in the copper wire and $E_{\mathrm{Ag}}$ be the electric field in the silver wire. What is the ratio $E_{\mathrm{Cu}} / E_{\mathrm{Ag}} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
04:22

Problem 18

A ductile metal wire has resistance $R$. What will be the resistance of this wire in terms of $R$ if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched? (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

Yaqub Khan
Yaqub Khan
Numerade Educator
00:58

Problem 19

In household wiring, copper wire with cross sectional area of $2.5 \mathrm{~mm}^{2}$ is often used. Find the resistance of a $24.0 \mathrm{~m}$ length of this wire.

Melissa Walsh
Melissa Walsh
Numerade Educator
03:45

Problem 20

What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminum wire with diameter $3.84 \mathrm{~mm}$ ?

Melissa Walsh
Melissa Walsh
Numerade Educator
02:56

Problem 21

A current-carrying gold wire has diameter $0.87 \mathrm{~mm}$. The electric field in the wire is $0.54 \mathrm{~V} / \mathrm{m}$. What are (a) the current carried by the wire;
(b) the potential difference between two points in the wire $7.0 \mathrm{~m}$ apart;
(c) the resistance of a $7.0 \mathrm{~m}$ length of this wire?

Melissa Walsh
Melissa Walsh
Numerade Educator
02:38

Problem 22

You apply a potential difference of $4.50 \mathrm{~V}$ between the ends of a wire that is $2.50 \mathrm{~m}$ in length and $0.654 \mathrm{~mm}$ in radius. The resulting current through the wire is $17.6 \mathrm{~A}$. What is the resistivity of the wire?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:45

Problem 23

(a) What is the resistance of a Nichrome wire at $0.0^{\circ} \mathrm{C}$ if its resistance is $180.0 \Omega$ at $11.5^{\circ} \mathrm{C} ?$ (b) What is the resistance of a carbon rod at $25.8^{\circ} \mathrm{C}$ if its resistance is $0.0140 \Omega$ at $0.0^{\circ} \mathrm{C} ?$

Melissa Walsh
Melissa Walsh
Numerade Educator
02:17

Problem 24

A carbon resistor is to be used as a thermometer. On a winter day when the temperature is $4.0^{\circ} \mathrm{C},$ the resistance of the carbon resistor is $217.8 \Omega$. What is the temperature on a spring day when the resistance is $215.5 \Omega$ ? (Take the reference temperature $T_{0}$ to be $4.0^{\circ} \mathrm{C}$.)

Melissa Walsh
Melissa Walsh
Numerade Educator
03:03

Problem 25

A copper transmission cable $110 \mathrm{~km}$ long and $11.0 \mathrm{~cm}$ in diameter carries a current of $150 \mathrm{~A}$. (a) What is the potential drop across the cable? (b) How much electrical energy is dissipated as thermal energy every hour?

Melissa Walsh
Melissa Walsh
Numerade Educator
03:22

Problem 26

Consider the circuit shown in Fig. E25.26. The terminal voltage of the $24.0 \mathrm{~V}$ battery is $21.2 \mathrm{~V}$. What are (a) the internal resistance $r$ of the battery and (b) the resistance $R$ of the circuit resistor?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
07:53

Problem 27

An ideal voltmeter $\mathrm{V}$ is connected to a $2.0 \Omega$ resistor and a battery with emf $5.0 \mathrm{~V}$ and internal resistance $0.5 \Omega$ as shown in Fig. E25.27. (a) What is the current in the $2.0 \Omega$ resistor? (b) What is the terminal voltage of the battery? (c) What is the reading on the voltmeter? Explain your answers.

Shital Rijal
Shital Rijal
Numerade Educator
02:27

Problem 28

An idealized ammeter is connected to a battery as shown in Fig. E25.28. Find (a) the reading of the ammeter, (b) the current through the $4.00 \Omega$ resistor, (c) the terminal voltage of the battery.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
View

Problem 29

When switch $S$ in Fig. $\mathbf{E} 25.29$ is open, the voltmeter $\mathrm{V}$ reads $3.14 \mathrm{~V}$. When the switch is closed, the voltmeter reading drops to $2.93 \mathrm{~V}$, and the ammeter A reads 1.67 A. Find the emf, the internal resistance of the battery, and the circuit resistance $R$. Assume that the two meters are ideal, so they don't affect the circuit.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
15:58

Problem 30

The circuit shown in Fig. E25.30 contains two batteries, each with an emf and an internal resistance, and two resistors. Find (a) the current in the circuit (magnitude and direction); (b) the terminal voltage $V_{a b}$ of the $16.0 \mathrm{~V}$ battery; $(\mathrm{c})$ the potential difference $V_{a c}$ of point $a$ with respect to point $c$. (d) Using Fig. 25.20 as a model, graph the potential rises and drops in this circuit.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
04:03

Problem 31

In the circuit shown in Fig. E25.30, the $16.0 \mathrm{~V}$ battery is removed and reinserted with the opposite polarity, so its negative terminal is now next to point $a$. Find
(a) the current in the circuit (magnitude and direction) and (b) the terminal voltage $V_{a b}$ of the $16.0 \mathrm{~V}$ battery.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:19

Problem 32

A battery has emf $30.0 \mathrm{~V}$ and internal resistance $r$. A $9.00 \Omega$ resistor is connected to the terminals of the battery, and the voltage drop across the resistor is $27.0 \mathrm{~V}$. What is the internal resistance of the battery?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:29

Problem 33

A battery has emf $24.0 \mathrm{~V}$ and internal resistance $3.00 \Omega$. A resistor of resistance $R$ is connected to the battery. What are the two values of $R$ for which $21.0 \mathrm{~W}$ of electrical power is consumed in the resistor?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:32

Problem 34

When a resistor with resistance $R$ is connected to a $1.50 \mathrm{~V}$ flashlight battery, the resistor consumes $0.0625 \mathrm{~W}$ of electrical power. (Throughout, assume that each battery has negligible internal resistance.) (a) What power does the resistor consume if it is connected to a $12.6 \mathrm{~V}$ car battery? Assume that $R$ remains constant when the power consumption changes. (b) The resistor is connected to a battery and consumes $5.00 \mathrm{~W}$. What is the voltage of this battery?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:08

Problem 35

Light Bulbs. The power rating of a light bulb (such as a $100 \mathrm{~W}$ bulb) is the power it dissipates when connected across a $230 \mathrm{~V}$ potential difference. What is the resistance of (a) a $100 \mathrm{~W}$ bulb and (b) a $60 \mathrm{~W}$ bulb? (c) How much current does each bulb draw in normal use?

Melissa Walsh
Melissa Walsh
Numerade Educator
01:28

Problem 36

American Light Bulb. In the United States the standard voltage in homes is $120 \mathrm{~V}$. Therefore, a " $75 \mathrm{~W}$ " American bulb would be intended for use with a $120 \mathrm{~V}$ potential difference. If this bulb is connected across a $230 \mathrm{~V}$ potential difference (as is used, for example, in Europe), how much power does it dissipate? Ignore the temperature dependence of the bulb's resistance.

Melissa Walsh
Melissa Walsh
Numerade Educator
01:47

Problem 37

European Light Bulb. In Europe the standard voltage in homes is $230 \mathrm{~V}$. Therefore, a "100 W" European bulb would be intended for use with a $230 \mathrm{~V}$ potential difference (a) If you use a "100 W" European bulb in the United States, where the standard voltage is $120 \mathrm{~V}$, what should be its U.S. power rating?
(b) How much current will the $100 \mathrm{~W}$ European bulb draw in normal use in the United States?

Melissa Walsh
Melissa Walsh
Numerade Educator
01:40

Problem 38

A battery-powered global positioning system (GPS) receiver operating on $9.5 \mathrm{~V}$ draws a current of $0.14 \mathrm{~A}$. How much electrical energy does it consume during $1.7 \mathrm{~h}$ ?

Melissa Walsh
Melissa Walsh
Numerade Educator
03:46

Problem 39

Consider the circuit of Fig. E25.30. (a) What is the total rate at which electrical energy is dissipated in the $5.0 \Omega$ and $9.0 \Omega$ resistors? (b) What is the power output of the $16.0 \mathrm{~V}$ battery?
(c) At what rate is electrical energy being converted to other forms in the $8.0 \mathrm{~V}$ battery? (d) Show that the power output of the $16.0 \mathrm{~V}$ battery equals the overall rate of consumption of electrical energy in the rest of the circuit.

Shoukat Ali
Shoukat Ali
Other Schools
01:29

Problem 40

Electric eels generate electric pulses along their skin that can be used to stun an enemy when they come into contact with it. Tests have shown that these pulses can be up to $502 \mathrm{~V}$ and produce currents of $79 \mathrm{~mA}$ (or even larger). A typical pulse lasts for $10 \mathrm{~ms}$. What power and how much energy are delivered to the unfortunate enemy with a single pulse, assuming a steady current?

Melissa Walsh
Melissa Walsh
Numerade Educator
00:38

Problem 41

A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at $25 \mathrm{~V}$ for a very short time, usually about $3.0 \mathrm{~ms}$. (a) What power does the defibrillator deliver to the body, and
(b) how much energy is transferred?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
00:38

Problem 42

A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at $25 \mathrm{~V}$ for a very short time, usually about $3.0 \mathrm{~ms}$. (a) What power does the defibrillator deliver to the body, and (b) how much energy is transferred?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
03:49

Problem 43

The capacity of a storage battery, such as those used in automobile electrical systems, is rated in ampere-hours $(\mathrm{A} \cdot \mathrm{h}) .$ A $50 \mathrm{~A} \cdot \mathrm{h}$ battery can supply a current of $50 \mathrm{~A}$ for $1.0 \mathrm{~h}$, or $25 \mathrm{~A}$ for $2.0 \mathrm{~h},$ and so on. (a) What total energy can be supplied by a $12 \mathrm{~V}, 60 \mathrm{~A} \cdot \mathrm{h}$ battery if its internal resistance is negligible? (b) What volume (in liters) of petrol has a total heat of combustion equal to the energy obtained in part (a)? (See Section 17.5 ; the density of petrol is $900 \mathrm{~kg} / \mathrm{m}^{3} .$ ) (c) If a generator with an average electrical power output of $0.45 \mathrm{~kW}$ is connected to the battery, how much time will be required for it to charge the battery fully?

Melissa Walsh
Melissa Walsh
Numerade Educator
02:49

Problem 44

A battery has emf $\mathcal{E}$ and internal resistance $r=2.00 \Omega$. A $12.0 \Omega$ resistor is connected to the battery, and the resistor consumes electrical power at a rate of $96.0 \mathrm{~J} / \mathrm{s}$. What is the emf of the battery?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:44

Problem 45

A $28.0 \Omega$ bulb is connected across the terminals of a $12.0 \mathrm{~V}$ battery having $2.50 \Omega$ of internal resistance. What percentage of the power of the battery is dissipated across the internal resistance and hence is not available to the bulb?

Melissa Walsh
Melissa Walsh
Numerade Educator
03:37

Problem 46

A typical small flashlight contains two batteries, each having an emf of $1.5 \mathrm{~V},$ connected in series with a bulb having resistance $15 \Omega$. (a) If the internal resistance of the batteries is negligible, what power is delivered to the bulb?
(b) If the batteries last for $6.0 \mathrm{~h}$, what is the total energy delivered to the bulb?
(c) The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value? (Assume that the resistance of the bulb is constant. Actually, it will change somewhat when the current through the filament changes, because this changes the temperature of the filament and hence the resistivity of the filament wire.)

Supratim Pal
Supratim Pal
Numerade Educator
01:31

Problem 47

In the circuit in Fig. E25.47, find
(a) the rate of conversion of internal (chemical) energy to electrical energy within the battery; (b) the rate of dissipation of electrical energy in the battery;
(c) the rate of dissipation of electrical energy in the external resistor.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:51

Problem 48

A "2000 W" electric heater is designed to operate on a $230 \mathrm{~V}$ supply. (a) What is its operating resistance? (b) What current does it draw? (c) If the voltage drops to $220 \mathrm{~V}$, what power does the heater take? (Assume that the resistance is constant. Actually, it will change because of the change in temperature.) (d) The heater coils are metallic, so that the resistance of the heater decreases with decreasing temperature. If the change of resistance with temperature is taken into account, will the electrical power consumed by the heater be larger or smaller than what you calculated in part (c)? Explain.

Melissa Walsh
Melissa Walsh
Numerade Educator
04:31

Problem 49

Pure silicon at room temperature contains approximately $1.0 \times 10^{16}$ free electrons per cubic meter. (a) Referring to Table 25.1, calculate the mean free time $\tau$ for silicon at room temperature. (b) Your answer in part (a) is much greater than the mean free time for copper given in Example 25.11 . Why, then, does pure silicon have such a high resistivity compared to copper?

Naresh Adhikari
Naresh Adhikari
Numerade Educator
09:52

Problem 50

A cell phone or computer battery has three ratings marked on it: a charge capacity listed in mAh (milliamp-hours), an energy capacity in Wh (watt-hours), and a potential rating in volts.
(a) What are these three values for your cell phone? (b) Convert the charge capacity $Q$ into
coulombs. (c) Convert the energy capacity $U$ into joules. (d) Multiply the charge rating $Q$ by the potential rating $V,$ and verify that this is equivalent to the energy capacity $U$. (e) If the charge $Q$ were stored on a parallel-plate capacitor with air as the dielectric, at the potential $V$, what would be the corresponding capacitance? (f) If the energy in the battery were used to heat $1 \mathrm{~L}$ of water, estimate the corresponding change in the water temperature? (The heat capacity of water is $4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} .$ )

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:07

Problem 51

An electrical conductor designed to carry large currents has a circular cross section $2.00 \mathrm{~mm}$ in diameter and is $14.6 \mathrm{~m}$ long. The resistance between its ends is $0.110 \Omega$. (a) What is the resistivity of the material? (b) If the electric-field magnitude in the conductor is $1.22 \mathrm{~V} / \mathrm{m},$ what is the total current?
(c) If the material has $8.5 \times 10^{28}$ free electrons per cubic meter, find the average drift speed under the conditions of part (b).

Ajay Singhal
Ajay Singhal
Numerade Educator
View

Problem 52

(a) Estimate the maximum volume of an immersion heater in a single family house. (b) How much heat would be required to raise the temperature of that water from $20^{\circ} \mathrm{C}$ to a standard household hotwater temperature of $45^{\circ} \mathrm{C} ?$ (Water has a density of $1.00 \mathrm{~kg} / \mathrm{L}$ and a heat capacity of $4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} .$ ) (c) Suppose the water should be fully heated in $1.5 \mathrm{~h}$. To what power output does this correspond? (d) If the element has a potential difference of $230 \mathrm{~V},$ what current is required?
(e) What should be the resistance of the element?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
View

Problem 53

On your first day at work as an electrical technician, you are asked to determine the resistance per meter of a long piece of wire. The company you work for is poorly equipped. You find a battery, a voltmeter, and an ammeter, but no meter for directly measuring resistance (an ohmmeter). You put the leads from the voltmeter across the terminals of the battery, and the meter reads $12.9 \mathrm{~V}$. You cut off a $20.0 \mathrm{~m}$ length of wire and connect it to the battery, with an ammeter in series with it to measure the current in the wire. The ammeter reads $6.90 \mathrm{~A}$. You then cut off a $40.0 \mathrm{~m}$ length of wire and connect it to the battery, again with the ammeter in series to measure the current. The ammeter reads $4.20 \mathrm{~A}$. Even though the equipment you have available to you is limited, your boss assures you of its high quality: The ammeter has very small resistance, and the voltmeter has very large resistance. What is the resistance of 1 meter of wire?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
View

Problem 54

In the circuit shown in Fig. $\mathbf{P} 2 \mathbf{5 . 5 4}, R$ is a variable resistor whose value ranges from 0 to $\infty,$ and $a$ and $b$ are the terminals of a battery that has an emf $\mathcal{E}=15.0 \mathrm{~V}$ and an internal resistance of $4.00 \Omega$. The ammeter and voltmeter are idealized meters. As $R$ varies over its full range of values, what will be the largest and smallest readings of (a) the voltmeter and (b) the ammeter? (c) Sketch qualitative graphs of the readings of both meters as functions of $R$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
09:34

Problem 55

A $3.00 \mathrm{~m}$ length of copper wire at $20^{\circ} \mathrm{C}$ has a 1.20 -m-long section with diameter $1.20 \mathrm{~mm}$ and a 1.80 -m-long section with diameter $0.60 \mathrm{~mm}$. There is a current of $2.0 \mathrm{~mA}$ in the 1.20 -mm-diameter section.
(a) What is the current in the 0.60 -mm-diameter section? (h) What is the magnitude of $\overrightarrow{\boldsymbol{E}}$ in the 1.20 -mm-diameter section? (c) What is the magnitude of $\overrightarrow{\boldsymbol{E}}$ in the 0.60 -mm-diameter section? (d) What is the potential difference between the ends of the $3.00-\mathrm{m}$ length of wire?

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
01:52

Problem 56

A heating element made of tungsten wire is connected to a large battery that has negligible internal resistance. When the heating element reaches $80.0^{\circ} \mathrm{C}$, it consumes electrical energy at a rate of $480 \mathrm{~W}$. What is its power consumption when its temperature is $150.0^{\circ} \mathrm{C} ?$ Assume that the temperature coefficient of resistivity has the value given in Table 25.2 and that it is constant over the temperature range in this problem. In Eq. (25.12) take $T_{0}$ to be $20.0^{\circ} \mathrm{C}$.

James Kiss
James Kiss
Numerade Educator
01:34

Problem 57

Lightning strikes can involve currents as high as 25,000 A that last for about $40 \mu \mathrm{s}$. If a person is struck by a bolt of lightning with these properties, the current will pass through his body. We shall assume that his mass is $71.0 \mathrm{~kg},$ that he is wet (after all, he is in a rainstorm) and therefore has a resistance of $1.50 \mathrm{k} \Omega$, and that his body is all water (which is reasonable for a rough, but plausible, approximation). (a) By how many degrees Celsius would this lightning bolt increase the temperature of $71 \mathrm{~kg}$ of water? (b) Given that the internal body temperature is about $37^{\circ} \mathrm{C}$, would the person's temperature actually increase that much? Why not? What would happen first?

Penny Riley
Penny Riley
Numerade Educator
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Problem 58

A resistor with resistance $R$ is connected to a battery that has emf $16.0 \mathrm{~V}$ and internal resistance $r=0.370 \Omega$. For what two values of $R$ will the power dissipated in the resistor be $81.0 \mathrm{~W} ?$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:40

Problem 59

A material of resistivity $\rho$ is formed into a solid, truncated cone of height $h$ and radii $r_{1}$ and $r_{2}$ at either end (Fig. P25.59).
(a) Calculate the resistance of the cone between the two flat end faces. (Hint: Imagine slicing the cone into very many thin disks, and calculate the resistance of one such disk.)
(b) Show that your result agrees with Eq. ( 25.10 ) when $r_{1}=r_{2}$.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:41

Problem 60

The region between two concentric conducting spheres with radii $a$ and $b$ is filled with a conducting material with resistivity $\rho$.
(a) Show that the resistance between the spheres is given by
$$
R=\frac{\rho}{4 \pi}\left(\frac{1}{a}-\frac{1}{b}\right)
$$
(b) Derive an expression for the current density as a function of radius, in terms of the potential difference $V_{a b}$ between the spheres. (c) Show that the result in part (a) reduces to Eq. (25.10) when the separation $L=b-a$ between the spheres is small.

Dading Chen
Dading Chen
Numerade Educator
02:27

Problem 61

The potential difference across the terminals of a battery is $8.40 \mathrm{~V}$ when there is a current of $1.50 \mathrm{~A}$ in the battery from the negative to the positive terminal. When the current is $3.50 \mathrm{~A}$ in the reverse direction, the potential difference becomes $10.20 \mathrm{~V}$. (a) What is the internal resistance of the battery? (b) What is the emf of the battery?

Salamat Ali
Salamat Ali
Numerade Educator
13:59

Problem 62

(a) What is the potential difference $V_{a d}$ in the circuit of Fig. P25.62? (b) What is the terminal voltage of the $4.00 \mathrm{~V}$ battery?
(c) A battery with emf $10.30 \mathrm{~V}$ and internal resistance $0.50 \Omega$ is inserted in the circuit at $d$, with its negative terminal connected to the negative terminal of the $8.00 \mathrm{~V}$ battery. What is the difference of potential $V_{b c}$ between the terminals of the $4.00 \mathrm{~V}$ battery now?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
01:41

Problem 63

The average bulk resistivity of the human body (apart from surface resistance of the skin) is about $5.0 \Omega \cdot \mathrm{m}$. The conducting path between the hands can be represented approximately as a cylinder $1.6 \mathrm{~m}$ long and $0.10 \mathrm{~m}$ in diameter. The skin resistance can be made negligible by soaking the hands in salt water. (a) What is the resistance between the hands if the skin resistance is negligible? (b) What potential difference between the hands is needed for a lethal shock current of $100 \mathrm{~mA}$ ? (Note that your result shows that small potential differences produce dangerous currents when the skin is damp.) (c) With the current in part (b), what power is dissipated in the body?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:25

Problem 64

A person with body resistance between his hands of $11 \mathrm{k} \Omega$ accidentally grasps the terminals of a $13 \mathrm{kV}$ power supply. (a) If the internal resistance of the power supply is $1500 \Omega,$ what is the current through the person's body? (b) What is the power dissipated in his body? (c) If the power supply is to be made safe by increasing its internal resistance, what should the internal resistance be for the maximum current in the above situation to be $1.00 \mathrm{~mA}$ or less?

Supratim Pal
Supratim Pal
Numerade Educator
04:25

Problem 65

A typical cost for electrical power in Europe is $€ 0.219$ per kilowatt-hour. (a) Some people leave their porch light on all the time. What is the yearly cost to keep a $75 \mathrm{~W}$ bulb burning day and night? (b) Suppose your refrigerator uses $200 \mathrm{~W}$ of power when it's running, and it runs 8 hours a day. What is the yearly cost of operating your refrigerator? 25.66 -o A cylindrical copper cable $1.50 \mathrm{~km}$ long is connected across a $230.0 \mathrm{~V}$ potential difference. (a) What should be its diameter so that it produces heat at a rate of $90.0 \mathrm{~W} ?$ (b) What is the electric field inside the cable under these conditions?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:52

Problem 66

A cylindrical copper cable $1.50 \mathrm{~km}$ long is connected across a $230.0 \mathrm{~V}$ potential difference.
(a) What should be its diameter so that
(b) What is the electric field inside it produces heat at a rate of $90.0 \mathrm{~W} ?$ the cable under these conditions?

Supratim Pal
Supratim Pal
Numerade Educator
07:15

Problem 67

A $1.50 \mathrm{~m}$ cylinder of radius $1.10 \mathrm{~cm}$ is made of a complicated mixture of materials. Its resistivity depends on the distance $x$ from the left end and obeys the formula $\rho(x)=a+b x^{2},$ where $a$ and $b$ are constants. At the left end, the resistivity is $2.25 \times 10^{-8} \Omega \cdot \mathrm{m},$ while at the right end it is $8.50 \times 10^{-8} \Omega \cdot \mathrm{m}$. (a) What is the resistance of this rod? (b) What is the electric field at its midpoint if it carries a $1.75 \mathrm{~A}$ current? (c) If we cut the rod into two $75.0 \mathrm{~cm}$ halves, what is the resistance of each half?

Dading Chen
Dading Chen
Numerade Educator
05:23

Problem 68

LED Bulbs. LED bulbs are much more efficient at producing light than are traditional incandescent bulbs. They initially cost much more, but they last far longer and use much less electricity. An LED bulb that produces as much light as a $100 \mathrm{~W}$ incandescent bulb uses only $12.5 \mathrm{~W}$ of power. The LED bulb lasts 15,000 hours, on the average, and costs $£ 8.00$, whereas the cost of an incandescent bulb was only $£ 0.80$, but it lasted just 750 hours. Assuming that electricity costs $£ 0.125$ per kilowatt-hour and that the bulbs are on for $4.0 \mathrm{~h}$ per day,
(a) compare the total cost (including the price of the bulbs) of running each bulb for 3.0 years. (b) How much do you save over 3.0 years if you use an LED bulb instead of an incandescent bulb? (c) What is the resistance of a "100 W" LED bulb? (Remember, it actually uses only $12.5 \mathrm{~W}$ of power and operates across $230 \mathrm{~V}$.)

Stephen Zaffke
Stephen Zaffke
Numerade Educator
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Problem 69

Two cylindrical cans with insulating sides and conducting end caps are filled with water, attached to the circuitry shown in Fig. $\mathbf{P 2 5 . 6 9},$ and used to determine salinity levels. The cans are identical, with radius $r=5.00 \mathrm{~cm}$ and length $L=3.00 \mathrm{~cm}$. The 25.69 . 00 CP Two cylindrical cans with insulating sides and conducting end caps are filled with water, attached to the circuitry shown in Fig. P25.69, and used to determine salinity levels. The cans are identical, with radius $r=5.00 \mathrm{~cm}$ and length $L=3.00 \mathrm{~cm}$. The the resistivity of the saltwater solution is determined by the relationship $\rho=\left(s_{0} / s\right) \Omega \cdot \mathrm{m},$ where $s$ is the salinity in parts per thousand (ppt) and $s_{0}=6.30$ ppt. (a) The ammeter registers a current of 484 mA. What is the salinity of the saltwater solution? (b) The left cylinder acts as a capacitor. Use Eq. (24.19) for its capacitance. How much charge is present on its upper plate? Note that pure water has a dielectric constant of $80.4 .$ (c) At what rate is energy dissipated by the saltwater? (d) For what salinity level would the $15.0 \Omega$ resistor dissipate half the power supplied by the battery?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:49

Problem 70

Consider the circuit shown in Fig. P25.70. The battery has emf $52.0 \mathrm{~V}$ and negligible internal resistance. $R_{2}=3.00 \Omega$, $C_{1}=4.00 \mu \mathrm{F},$ and $C_{2}=5.00 \mu \mathrm{F}$
After the capacitors have attained their final charges, the charge on $C_{1}$ is $Q_{1}=18.0 \mu \mathrm{C}$. What is
(a) the final charge on $C_{2} ;(b)$ the resistance $R_{1} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
01:51

Problem 71

Consider the circuit shown in Fig. P25.71. The emf source has negligible internal resistance. The resistors have resistances $R_{1}=7.00 \Omega$ and $R_{2}=4.00 \Omega$ The capacitor has capacitance $C=6.00 \mu \mathrm{F}$. When the capacitor is fully charged, the magnitude of the charge on its plates is $Q=35.0 \mu \mathrm{C}$. Calculate the emf $\mathcal{E}$.

Justin Swantek
Justin Swantek
Numerade Educator
06:44

Problem 72

An external resistor $R$ is connected between the terminals of a battery. The value of $R$ varies. For each $R$ value, the current $I$ in the circuit and the terminal voltage $V_{a b}$ of the battery are measured. The results are plotted in Fig. P25.72, a graph of $V_{a b}$ versus $I$ that shows the best straight-line fit to the data. (a) Use the graph in Fig. P25.72 to calculate the battery's emf and internal resistance. (b) For what value of $R$ is $V_{a b}$ equal to $80.0 \%$ of the battery emf?

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 73

The voltage drop $V_{a b}$ across each of resistors $A$ and $B$ was measured as a function of the current $I$ in the resistor. The results are shown in the table:
(a) For each resistor, graph $V_{a b}$ as a function of $I$ and graph the resistance $R=V_{a b} / I$ as a function of $I$. (b) Does resistor $A$ obey Ohm's law? Explain. (c) Does resistor $B$ obey Ohm's law? Explain.
(d) What is the power dissipated in $A$ if it is connected to a $4.00 \mathrm{~V}$ battery that has negligible internal resistance? (e) What is the power dissipated in $B$ if it is connected to the battery?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 74

According to the UK wiring regulations, copper wire used for interior wiring of houses is permitted to carry no more than a specified maximum amount of current. The table shows values of the maximum current $I_{\max }$ for several common sizes of PVC-insulated and sheathed wire.
(a) What considerations determine the maximum current-carrying capacity of household wiring? (b) A total of $3900 \mathrm{~W}$ of power is to be supplied through the wires of a flat to the household electrical appliances. If the potential difference across the group of appliances is $230 \mathrm{~V}$, determine the cross-sectional area of the thinnest permissible wire that can be used. (c) Suppose the wire used in this flat is of the size found in part (b) and has total length $42.0 \mathrm{~m}$. At what rate is energy dissipated in the wires? (d) The consumer cost of electrical energy is $£ 0.125$ per kilowatt-hour. If the flat were built with wire of the next larger crosssectional area than that found in part $(\mathrm{b})$, what would be the savings in electricity costs in one year? Assume that the appliances are kept on for an average of 12 hours a day.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
14:24

Problem 75

A material with resistivity $\rho$ is formed into a cylinder of length $L$ and outer radius $r_{\text {outer }}$. A cylindrical core with radius $r_{\text {inner }}$ is removed from the axis of this cylinder and filled with a conducting material, which is attached to a wire. The outer surface of the cylinder is coated with a conducting material and attached to another wire. (a) If the second wire has potential $V$ greater than the first wire, in what direction does the local electric field point inside of the cylinder? (b) The magnitude of this electric field is $c / r,$ where $c$ is a constant and $r$ is the distance from the axis of the cylinder. Use the relationship $V=\int \vec{E} \cdot d \vec{l}$ to determine the constant $c .$ (c) What is the resistance of this device? (d) A $1.00-\mathrm{cm}$ -long hollow cylindrical resistor has an inner radius of $1.50 \mathrm{~mm}$ and an outer radius of 3.00 $\mathrm{mm}$. The material is a blend of powdered carbon and ceramic whose resistivity $\rho$ may be altered by changing the amount of carbon. If this device should have a resistance of $6.80 \mathrm{k} \Omega,$ what value of $\rho$ should be selected?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
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Problem 76

An incandescent light bulb uses a coiled filament of tungsten that is $533 \mathrm{~mm}$ long with a diameter of $16.0 \mu \mathrm{m}$. At $20.0^{\circ} \mathrm{C}$ tungsten has a resistivity of $5.25 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Its temperature $\mathrm{co}-$ efficient of resistivity is $0.0045\left({ }^{\circ} \mathrm{C}\right)^{-1},$ and this remains accurate even at high temperatures. The temperature of the filament increases linearly with current, from $20^{\circ} \mathrm{C}$ when no current flows to $2520^{\circ} \mathrm{C}$ at $1.00 \mathrm{~A}$ of current. (a) What is the resistance of the light bulb at $20^{\circ} \mathrm{C} ?$ (b) What is the current through the light bulb when the potential difference across its terminals is $230 \mathrm{~V}$ ? (Hint: First determine the temperature as a function of the current; then use this to determine the resistance as a function of the current. Substitute this result into the equation $V=I R$ and solve for the current $I .$ ) (c) What is the resistance when the potential is $230 \mathrm{~V} ?$ (d) How much energy does the light bulb dissipate in 1 min when $230 \mathrm{~V}$ is supplied across its terminals? (e) How much energy does the light bulb dissipate in 1 min when half that voltage is supplied?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 77

Adding different amounts of impurities. A rod of semiconducting material of length $L$ and cross-sectional area $A$ lies along the $x$ -axis between $x=0$ and $x=L$. The material obeys Ohm's law, and its resistivity varies along the rod according to $\rho(x)=\rho_{0} \exp (-x / L) .$ The end of the rod at $x=0$ is at a potential $V_{0}$ greater than the end at $x=L$. (a) Find the total resistance of the rod and the current in the rod. (b) Find the electric-field magnitude $E(x)$ in the rod as a function of $x$.
(c) Find the electric potential $V(x)$ in the rod as a function of $x$. (d) Graph the functions $\rho(x), E(x),$ and $V(x)$ for values of $x$ between $x=0$ and $x=L$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
11:45

Problem 78

An external resistor with resistance $R$ is connected to a battery that has emf $\mathcal{E}$ and internal resistance $r$. Let $P$ be the electrical power output of the source. By conservation of energy, $P$ is equal to the power consumed by $R$. What is the value of $P$ in the limit that $R$ is (a) very small; (b) very large?
(c) Show that the power output of the battery is a maximum when $R=r$. What is this maximum $P$ in terms of $\mathcal{E}$ and $r ?$ (d) A battery has $\mathcal{E}=64.0 \mathrm{~V}$ and $r=4.00 \Omega$. What is the power output of this battery when it is connected to a resistor $R,$ for $R=2.00 \Omega, R=4.00 \Omega,$ and $R=6.00 \Omega ?$ Are your results consistent with the general result that you derived in part (b)?

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 79

What is the best explanation for the behavior exhibited in the data? (a) Longer threads can carry more current than shorter threads do and so make better electrical conductors. (b) The thread stops being a conductor when it is stretched to $13 \mathrm{~mm}$, due to breaks that occur in the thin coating. (c) As the thread is stretched, the coating thins and its resistance increases; as the thread is relaxed, the coating returns nearly to its original state.
(d) The resistance of the thread increases with distance from the end of the thread.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 80

If the conductivity of the thread results from the aqueous coating only, how does the cross-sectional area $A$ of the coating compare when the thread is $13 \mathrm{~mm}$ long versus the starting length of $5 \mathrm{~mm}$ ? Assume that the resistivity of the coating remains constant and the coat-
(a) $\frac{1}{10} A_{5} \mathrm{~mm}$;
ing is uniform along the thread. $A_{13} \mathrm{~mm}$ is about
(b) $\frac{1}{4} A_{5 \mathrm{~mm}}$
(c) $\frac{2}{5} A_{5} \mathrm{~mm} ;$ (d) the same as $A_{5} \mathrm{~mm}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
00:38

Problem 81

What is the maximum current that flows in the thread during this experiment if the voltage source is a $9 \mathrm{~V}$ battery? (a) About $1 \mathrm{~A}$; (b) about $0.1 \mathrm{~A} ;(\mathrm{c})$ about $1 \mu \mathrm{A} ;(\mathrm{d})$ about $1 \mathrm{nA}$.

Salamat Ali
Salamat Ali
Numerade Educator
01:26

Problem 82

In another experiment, a piece of the web is suspended so that it can move freely. When either a positively charged object or a negatively charged object is brought near the web, the thread is observed to move toward the charged object. What is the best interpretation of this observation? The web is (a) a negatively charged conductor; (b) a positively charged conductor; (c) either a positively or negatively charged conductor; (d) an electrically neutral conductor.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University