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University Physics with Modern Physics

Hugh D. Young

Chapter 25

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

Educators

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Chapter Questions

01:18

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?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
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
02:11

Problem 3

A 5.00 -A current runs through a 12 -gauge copper wire (diameter 2.05 $\mathrm{mm}$ ) 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 of twice the
diameter, which of the above answers would change? Would they increase or decrease?

Penny Riley
Penny Riley
Numerade Educator
03:22

Problem 4

An 18 -gauge copper wire (diameter 1.02 $\mathrm{mm} )$ carries a current with a current density of $1.50 \times 10^{6} \mathrm{A} / \mathrm{m}^{2} .$ 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 drift velocity of electrons in the wire.

Dading Chen
Dading Chen
Numerade Educator
02:20

Problem 5

Copper has $8.5 \times 10^{28}$ free electrons per cubic meter. A 71.0 -cm length of 12 -gauge copper wire that is 2.05 $\mathrm{mm}$ in diameter carries 4.85 A of current. (a) How much time does it take for an electron to travel the length of the wire? (b) Repeat part (a) for 6-gauge copper wire (diameter 4.12 $\mathrm{mm}$ ) of the same length that carries the same current.(c) Generally speaking, how does changing the diameter of a wire that carries a given amount of current affect the drift velocity of the electrons in the wire?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:12

Problem 6

Consider the 18 -gauge wire in Example $25.1 .$ How many atoms are in 1.00 $\mathrm{m}^{3}$ of copper? With the density of free electrons given in the example, how many free electrons are there per copper atom?

Dading Chen
Dading Chen
Numerade Educator
01:59

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 $t=8.0 \mathrm{s} ?$ (b) What constant current would
transport the same charge in the same time interval?

Dading Chen
Dading Chen
Numerade Educator
05:25

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 current passing between the electrodes? (b) What is the direction of the current?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:28

Problem 9

Transmission of Nerve Impulses. 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?

Yaqub Khan
Yaqub Khan
Numerade Educator
02:32

Problem 10

(a) At room temperature what is the strength of the electric field in a 12 -gauge copper wire (diameter 2.05 $\mathrm{mm}$ ) that is needed to cause a 2.75 -A current to flow? (b) What field would be
needed if the wire were made of silver instead?

Dading Chen
Dading Chen
Numerade Educator
02:50

Problem 11

A 1.50 -m cylindrical rod of diameter 0.500 $\mathrm{cm}$ is connected to a power supply that maintains a constant potential difference of 15.0 $\mathrm{V}$ 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 $18.5 \mathrm{A},$ while at $92.0^{\circ} \mathrm{C}$ it reads 17.2 $\mathrm{A} .$ 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.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
08:52

Problem 12

A copper wire has a square cross section 2.3 $\mathrm{mm}$ on a side. The wire is 4.0 $\mathrm{m}$ 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?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
01:37

Problem 13

A 14 -gauge copper wire of diameter 1.628 $\mathrm{mm}$ carries a current of 12.5 $\mathrm{mA}$ . (a) What is the potential difference across a 2.00 -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?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:36

Problem 14

A wire 6.50 $\mathrm{m}$ long with diameter of 2.05 $\mathrm{mm}$ has a resistance of 0.0290$\Omega .$ What material is the wire most likely made of?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
04:21

Problem 15

A cylindrical tungsten filament 15.0 $\mathrm{cm}$ long with a diameter of 1.00 $\mathrm{mm}$ is to be used in a machine for which the temperature will range from room temperature $\left(20^{\circ} \mathrm{C}\right)$ up to $120^{\circ} \mathrm{C}$ . It will carry a current of 12.5 $\mathrm{A}$ at all temperatures (consult Tables 25.1 and 25.2$)$ (a) What will be the maximum electric field in this filament, and (b) what will be its resistance with that field? (c) What will be the maximum potential drop over the full length of the filament?

Dading Chen
Dading Chen
Numerade Educator
04:22

Problem 16

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
01:02

Problem 17

In household wiring, copper wire 2.05 $\mathrm{mm}$ in diameter is often used. Find the resistance of a $24.0-\mathrm{m}$ length of this wire.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:21

Problem 18

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 3.26 $\mathrm{mm}$ ?

Prashant Bana
Prashant Bana
Numerade Educator
01:55

Problem 19

You need to produce a set of cylindrical copper wires 3.50 $\mathrm{m}$ long that will have a resistance of 0.125$\Omega$ each. What will be the mass of each of these wires?

Dading Chen
Dading Chen
Numerade Educator
06:00

Problem 20

A tightly coiled spring having 75 coils, each 3.50 $\mathrm{cm}$ in diameter, is made of insulated metal wire 3.25 $\mathrm{mm}$ in diameter. An ohmmeter connected across its opposite ends reads 1.74$\Omega$ ? What is the resistivity of the metal?

Meghan Miholics
Meghan Miholics
Numerade Educator
01:18

Problem 21

An aluminum cube has sides of length 1.80 $\mathrm{m} .$ What is the resistance between two opposite faces of the cube?

Dading Chen
Dading Chen
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
02:17

Problem 23

A current-carrying gold wire has diameter 0.84 $\mathrm{mm}$ . The electric field in the wire is 0.49 $\mathrm{V} / \mathrm{m} .$ What are (a) the current carried by the wire; (b) the potential difference between two points in the wire 6.4 $\mathrm{m}$ apart; $(\mathrm{c})$ the resistance of a $6.4-\mathrm{m}$ length of this wire?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:11

Problem 24

A hollow aluminum cylinder is 2.50 $\mathrm{m}$ long and has an inner radius of 3.20 $\mathrm{cm}$ and an outer radius of 4.60 $\mathrm{cm} .$ Treat each surface (inner, outer, and the two end faces) as an equipotential surface. At room temperature, what will an ohmmeter read if it is connected between (a) the opposite faces and (b) the inner and outer surfaces?

Dading Chen
Dading Chen
Numerade Educator
02:37

Problem 25

(a) What is the resistance of a Nichrome wire at $0.0^{\circ} \mathrm{C}$ if its resistance is 100.00$\Omega$ 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 0.0160$\Omega$ at $0.0^{\circ} \mathrm{C} ?$

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:46

Problem 26

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 217.3$\Omega .$ S. What is the temperature on a spring day when the resistance is 215.8$\Omega ?$ (Take the reference temperature $T_{0}$ to be $4.0^{\circ} \mathrm{C}$ .

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
01:24

Problem 27

A strand of wire has resistance 5.60$\mu \Omega .$ Find the net resistance of 120 such strands if they are (a) placed side by side to form a cable of the same length as a single strand, and (b) connected end to end to form a wire 120 times as long as a single strand.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:26

Problem 28

Consider the circuit shown in Fig. E25. $28 .$ 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?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:33

Problem 29

A copper transmission cable $100 \mathrm{~km}$ long and $10.0 \mathrm{~cm}$ in diameter carries a current of $125 \mathrm{~A}$.
(a) What is the potential drop across the cable?
(b) How much electrical energy is dissipated as thermal energy every hour?

Dading Chen
Dading Chen
Numerade Educator
02:27

Problem 30

An idealized ammeter is connected to a battery as shown in Fig. $\mathrm{E} 25.30$ . 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
07:53

Problem 31

An ideal voltmeter $V$ 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. $\mathrm{E} 25.31 .$ (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
03:36

Problem 32

The circuit shown in Fig. E25. 22 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; (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.

Dading Chen
Dading Chen
Numerade Educator
02:31

Problem 33

When switch $S$ in Fig. $E 25.33$ is open, the voltmeter $V$ of the battery reads 3.08 V. When the switch is closed, the voltmeter reading drops to 2.97 $\mathrm{V}$ and the ammeter $\mathrm{A}$ reads 1.65 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.

Justin Hameline
Justin Hameline
Numerade Educator
07:49

Problem 34

In the circuit of Fig. $\mathrm{E} 25.32$ , the $5.0-\Omega$ resistor is removed and replaced by a resistor of unknown resistance $R .$ When this is done, an ideal voltmeter connected across the points $b$ and $c$ reads 1.9 $\mathrm{V}$ . Find (a) the current in the circuit and (b) the resistance $R .$ (c) Graph the potential rises and drops in this circuit (see Fig. 25.20$) .$

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:53

Problem 35

In the circuit shown in Fig. E25.32, the 16.0 -V battery is removed and reinserted with the opposite polarity, so that its negative terminal is now next to point $a .$ Find (a) the current in the circuit (magnitude and direction); (b) the terminal voltage $V_{b a}$ of the 16.0 - $\mathrm{V}$ battery; (c) the potential difference $V_{a c}$ of point $a$ with respect to point $c .$ (d) Graph the potential rises and drops in this circuit (see Fig. 25.20$)$ .

Dading Chen
Dading Chen
Numerade Educator
00:30

Problem 36

The following measurements were made on a Thyrite resistor: $$\begin{array}{llll}{I(\mathbf{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\ {V_{a b}(\mathbf{V})} & {2.55} & {3.11} & {3.77} & {4.58}\end{array}$$
(a) Graph $V_{a b}$ as a function of $I .(\mathrm{b})$ Does Thyrite obey Ohm's law? How can you tell? (c) Graph the resistance $R=V_{a b} / I$ as a function of $I$ .

Dading Chen
Dading Chen
Numerade Educator
00:32

Problem 37

The following measurements of current and potential difference were made on a resistor constructed of Nichrome wire: $$\begin{array}{llll}{\boldsymbol{I}(\mathbf{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\ {\boldsymbol{V}_{a b}(\mathbf{V})} & {1.94} & {3.88} & {7.76} & {15.52}\end{array}$$ (a) Graph $V_{a b}$ as a function of $I$ (b) Does Nichrome obey Ohm's law? How can you tell? (c) What is the resistance of the resistor in ohms?

Dading Chen
Dading Chen
Numerade Educator
02:02

Problem 38

The circuit shown in Fig. E25. 38 contains two batteries, each with an emf and an internal resistance, and two resistors. Find (a) the current in the circuit (magnitude and direction) and (b) the terminal voltage $V_{a b}$ of the 16.0 -V battery.

Dading Chen
Dading Chen
Numerade Educator
02:43

Problem 39

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 $120-\mathrm{V}$ 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?

Dading Chen
Dading Chen
Numerade Educator
01:04

Problem 40

If a $75-\mathrm{W}^{\prime \prime}$ bulb (see Problem 25.39$)$ is connected across a 220 -V potential difference (as is used in Europe), how much power does it dissipate?

Dading Chen
Dading Chen
Numerade Educator
01:56

Problem 41

European Light Bulb. In Europe the standard voltage in homes is 220 $\mathrm{V}$ instead of the 120 $\mathrm{V}$ used in the United States. Therefore a "100-W" European bulb would be intended for use with a $220-\mathrm{V}$ potential difference (see Problem 25.40$)$ . (a) If you bring a "100-W" European bulb home to the United States, what should be its U.S. power rating? (b) How much current will the 100 -W European bulb draw in normal use in the United States?

Dading Chen
Dading Chen
Numerade Educator
00:54

Problem 42

A battery-powered global positioning system (GPS)
receiver operating on 9.0 $V$ draws a current of 0.13 $A$. How much
electrical energy does it consume during $1.5 \mathrm{~h} ?$

Dading Chen
Dading Chen
Numerade Educator
02:04

Problem 43

Consider a resistor with length $L,$ uniform cross-sectional area $A,$ and uniform resistivity $\rho$ that is carrying a current with uniform current density $J .$ Use Eq. $(25.18)$ to find the electrical power dissipated per unit volume, $p$ . Express your result in terms of (a)E and $J ;$ (b) $J$ and $\rho ;(c) E$ and $\rho$

Dading Chen
Dading Chen
Numerade Educator
02:17

Problem 44

Electric Eels. 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 500 $\mathrm{V}$ and produce currents of 80 $\mathrm{mA}$ (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?

Yaqub Khan
Yaqub Khan
Numerade Educator
00:50

Problem 45

Treatment of Heart Failure. 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?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
04:15

Problem 46

Consider the circuit of Fig. E25.32. (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 the power output of the 16.0 -V battery equals the overall rate of dissipation of electrical energy in the rest of the circuit.

Dading Chen
Dading Chen
Numerade Educator
02:15

Problem 47

The capacity of a storage battery, such as those used in automobile electrical systems, is rated in ampere-hours $(\mathrm{A} \cdot \mathrm{h}) . \mathrm{A}$ $50-\mathrm{A} \cdot \mathrm{h}$ 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 gasoline has a total heat of combustion equal to the energy obtained in part (a)? (See Section 17.6 ; the density of gasoline 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?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:25

Problem 48

In the circuit analyzed in Example 25.8 the $4.0-\Omega$ resistor is replaced by a $8.0-\Omega$ resistor, as in Example 25.9 . (a) Calculate the rate of conversion of chemical energy to electrical energy in the battery. How does your answer compare to the result calculated in Example 25.8$?$ (b) Calculate the rate of electrical energy dissipation in the internal resistance of the battery. How does your answer compare to the result calculated in Example 25.8$?$ (c) Use the results of parts (a) and (b) to calculate the net power output of the battery. How does your result compare to the electrical power dissipated in
the 8.0 - $\Omega$ resistor as calculated for this circuit in Example 25.9$?$

Dading Chen
Dading Chen
Numerade Educator
01:51

Problem 49

A $25.0-\Omega$ bulb is connected across the terminals of a $12.0-\mathrm{V}$ battery having 3.50$\Omega$ 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?

Narayan Hari
Narayan Hari
Numerade Educator
03:49

Problem 50

An idealized voltmeter is connected across the terminals of a $15.0-\mathrm{V}$ battery, and a $75.0-\Omega$ appliance is also connected across its terminals. If the voltmeter reads $11.3 \mathrm{V} :$ (a) how much power is being dissipated by the appliance, and (b) what is the internal resistance of the battery?

Yaqub Khan
Yaqub Khan
Numerade Educator
02:11

Problem 51

In the circuit in Fig. $\mathrm{E} 25.51$ 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.

Dading Chen
Dading Chen
Numerade Educator
08:05

Problem 52

A typical small flashlight contains two batteries, each having an emf of 1.5 $\mathrm{V}$ , connected in series with a bulb having resistance 17$\Omega$. (a) If the internal resistance of the batteries is negligible, what power is delivered to the bulb? (b) If the batteries last for 5.0 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.)

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:11

Problem 53

$\mathrm{A}^{4} 540-\mathrm{W}^{\prime \prime}$ electric heater is designed to operate from $120-\mathrm{V}$ lines. (a) What is its resistance? (b) What current does it draw? (c) If the line line voltage drops to 110 $\mathrm{V}$ , 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.

Dading Chen
Dading Chen
Numerade Educator
04:31

Problem 54

Pure silicon 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
02:33

Problem 55

An electrical conductor designed to carry large currents has a circular cross section 2.50 $\mathrm{mm}$ in diameter and is 14.0 $\mathrm{m}$ long. The resistance between its ends is 0.104$\Omega .$ (a) What is the resistivity of the material? (b) If the electric-field magnitude in the
conductor is $1.28 \mathrm{V} / \mathrm{m},$ 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).

Dading Chen
Dading Chen
Numerade Educator
01:53

Problem 56

A plastic tube 25.0 $\mathrm{m}$ long and 3.00 $\mathrm{cm}$ in diameter is dipped into a silver solution, depositing a layer of silver 0.100 $\mathrm{mm}$ thick uniformly over the outer surface of the tube. If this coated tube is then connected across a $12.0-\mathrm{V}$ battery, what will be the current?

Dading Chen
Dading Chen
Numerade Educator
03:43

Problem 57

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 12.6 $\mathrm{V} .$ 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 7.00 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 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?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
06:01

Problem 58

A 2.0 -mm length of wire is made by welding the end of a 120 -cm-long silver wire to the end of an 80 -cm-long copper wire. Each piece of wire is 0.60 $\mathrm{mm}$ in diameter. The wire is at room
temperature, so the resistivities are as given in Table $25.1 .$ A potential difference of 5.0 $\mathrm{V}$ is maintained between the ends of the 2.0 -m composite wire. (a) What is the current in the copper section? (b) What is the current in the silver section? (c) What is the magnitude of $\vec{\boldsymbol{E}}$ in the copper? (d) What is the magnitude of $\vec{\boldsymbol{E}}$ in the silver? (e) What is the potential difference between the ends of the silver section of wire?

Dading Chen
Dading Chen
Numerade Educator
05:08

Problem 59

A 3.00 -m length of copper wire at $20^{\circ} \mathrm{C}$ has a 1.20 -m-long section with diameter 1.60 $\mathrm{mm}$ and a 1.80 -m-long section with diameter 0.80 $\mathrm{mm}$ . There is a current of 2.5 $\mathrm{mA}$ in the $1.60-$ mm-diameter section. (a) What is the current in the $0.80 \mathrm{mm}-$ diameter section? (b) What is the magnitude of $\vec{E}$ in the 1.60 -mm-diameter section? (c) What is the magnitude of $\vec{E}$ in the 0.80 -mm-diameter section? (d) What is the potential difference between the ends of the $3.00-\mathrm{m}$ length of wire?

Dading Chen
Dading Chen
Numerade Educator
02:13

Problem 60

Critical Current Density in Superconductors. One problem with some of the newer high-temperature superconductors is getting a large enough current density for practical use without causing the resistance to reappear. The maximum current density for which the material will remain a superconductor is called the critical current density of the material. In $1987,$ IBM research labs had produced thin films with critical current densities of $1.0 \times 10^{5} \mathrm{A} / \mathrm{cm}^{2} .($ a) How much current could an 18 -gauge wire (see Example 25.1 in Section 25.1 ) of this material carry and still remain superconducting? (b) Researchers are trying to develop superconductors with critical current densities of $1.0 \times 10^{6} \mathrm{A} / \mathrm{cm}^{2}$ What diameter cylindrical wire of such a material would be needed to carry 1000 A without losing its superconductivity?

Dading Chen
Dading Chen
Numerade Educator
03:41

Problem 61

A Nichrome heating element that has resistance 28.0$\Omega$ is connected to a battery that has emf 96.0 $\mathrm{V}$ and internal resistance 1.2$\Omega$ . An aluminum cup with mass 0.130 kg contains 0.200 $\mathrm{kg}$ of water. The heating element is placed in the water and the electrical energy dissipated in the resistance of the heating element all goes into the cup and water. The element itself has very small mass. How much time does it take for the temperature of the cup and water to rise from $21.2^{\circ} \mathrm{C}$ to $34.5^{\circ} \mathrm{C}$ ? (The change of the resistance of the Nichrome due to its temperature change can be neglected.)

Dading Chen
Dading Chen
Numerade Educator
02:39

Problem 62

A resistor with resistance $R$ is connected to a battery that has emf 12.0 $\mathrm{V}$ and internal resistance $r=0.40 \Omega .$ For what two values of $R$ will the power dissipated in the resistor be 80.0 $\mathrm{W} ?$

Pawan Yadav
Pawan Yadav
Numerade Educator
02:54

Problem 63

Struck by Lightning. Lightning strikes can involve currents as high as $25,000$ A that last for about 40$\mu$ 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 75 $\mathrm{kg}$ , that he is wet (after all, he is in a rainstorm) and therefore has a resistance of $1.0 \mathrm{k} \Omega,$ 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 is about $37^{\circ} \mathrm{C},$ would the person's temperature actually increase that much? Why not? What would happen first?

Dading Chen
Dading Chen
Numerade Educator
01:07

Problem 64

In the Bohr model of the hydrogen atom, the electron makes $6.0 \times 10^{15} \mathrm{rev} / \mathrm{s}$ around the nucleus. What is the average current at a point on the orbit of the electron?

Dading Chen
Dading Chen
Numerade Educator
06:29

Problem 65

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. $P 25.65 )$ . (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 $r_{1}=r_{2}$ .

Keshav Singh
Keshav Singh
Numerade Educator
02:41

Problem 66

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
09:45

Problem 67

The temperature coefficient of resistance $\alpha$ in Eq. $(25.12)$ equals the temperature coefficient of resistivity $\alpha$ in Eq. $(25.6)$ only if the coefficient of thermal expansion is small. A cylindrical column of mercury is in a vertical glass tube. At $20^{\circ} \mathrm{C}$ , the length of the mercury column is 12.0 $\mathrm{cm} .$ The diameter of the mercury column is 1.6 $\mathrm{mm}$ and doesn't change with temperature because glass has a small coefficient of thermal expansion. The coefficient of volume expansion of the mercury is given in Table 17.2 ,its resistivity at $20^{\circ} \mathrm{C}$ is given in Table $25.1,$ and its temperature coefficient of resistivity is given in Table 25.2 (a) At $20^{\circ} \mathrm{C},$ what is the resistance between the ends of the mercury column? (b) The mercury column is heated to $60^{\circ} \mathrm{C}$ . What is the change in its resistivity? (c) What is the change in its length? Explain why the coefficient of volume expansion, rather than the coefficient of linear expansion, determines the change in length. (d) What is the change in its resistance? (Hint: Since the percentage changes in $\rho$ and $L$
are small, you may find it helpful to derive from Eq. $(25.10)$ anequation for $\Delta R$ in terms of $\Delta \rho$ and $\Delta L . )($ e) $W$ hat is the temperature coefficient of resistance $\alpha$ for the mercury column, as defined in Eq. $(25.12) ?$ How does this value compare with the temperature
coefficient of resistivity? Is the effect of the change in length important?

Dading Chen
Dading Chen
Numerade Educator
13:59

Problem 68

(a) What is the potential difference $V_{a d}$ in the circuit of Fig. $P 25.68 ?$ (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
02:27

Problem 69

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
05:33

Problem 70

A person with body resistance between his hands of 10 $\mathrm{k} \Omega$ accidentally grasps the terminals of a $14-\mathrm{kV}$ power supply. (a) If the internal resistance of the power supply is $2000 \Omega,$ 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?

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

Problem 71

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 m long and 0.10 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
02:15

Problem 72

A typical cost for electric power is $\$ 0.120$ per kilowatt- hour. (a) Some people leave their porch light on all the time. What is the yearly cost to keep a $75-$ W bulb buming day and night? (b) Suppose your refrigerator uses 400 $\mathrm{W}$ of power when it's running, and it runs 8 hours a day. What is the yearly cost of operating your refrigerator?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
02:23

Problem 73

A $12.6-\mathrm{V}$ car battery with negligible internal resistance is connected to a series combination of a $3.2-\Omega$ resistor that obeys Ohm's law and a thermistor that does not obey Ohm's law but
instead has a current-voltage relationship $V=\alpha I+\beta I^{2},$ with $\alpha=$ 3.8$\Omega$ and $\beta=1.3 \Omega / \mathrm{A} .$ What is the current through the $3.2-\Omega$ resistor?

Dading Chen
Dading Chen
Numerade Educator
02:00

Problem 74

A cylindrical copper cable 1.50 $\mathrm{km}$ long is connected across a 220.0 -V potential difference. (a) What should be its diameter so that it produces heat at a rate of 75.0 $\mathrm{W}^{\prime}$ (b) What is the electric field inside the cable under these conditions?

Anand Jangid
Anand Jangid
Numerade Educator
04:10

Problem 75

A Nonideal Ammeter. Unlike the idealized ammeter described in Section $25.4,$ any real ammeter has a nonzero resistance. (a) An ammeter with resistance $R_{\mathrm{A}}$ is connected in series with a resistor $R$ and a battery of emf $\mathcal{E}$ and internal resistance $r .$ The current measured by the ammeter is $I_{\mathrm{A}}$ . Find the current through the circuit if the ammeter is removed so that the battery and the resistor form a complete circuit. Express your answer in terms of $I_{A}, r, R_{\mathrm{A}},$ and $R .$ The more "ideal" the ammeter, the smaller the difference between this current and the current $I_{\mathrm{A}}$ . (b) If $R=3.80 \Omega, \mathcal{E}=7.50 \mathrm{V},$ and $r=0.45 \Omega,$ find the maximum value of the ammeter resistance $R_{\mathrm{A}}$ so that $l_{\mathrm{A}}$ is within 1.0$\%$ of the current in the circuit when the ammeter is absent. (c) Explain why your answer in part (b) represents a maximum value.

Dading Chen
Dading Chen
Numerade Educator
07:15

Problem 76

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 $1.75-$ A current? (c) If we cut the rod into two 75.0 -cm halves, what is the resistance of each half?

Dading Chen
Dading Chen
Numerade Educator
04:26

Problem 77

According to the U.S. National Electrical Code, copper wire used for interior wiring of houses, hotels, office buildings, and industrial plants is permitted to carry no more than a specified maximum amount of current. The table below shows the maximum current $I_{\text { max }}$ for several common sizes of wire with varnished cambric insulation. The "wire gauge" is a standard used to
describe the diameter of wires. Note that the larger the diameter of the wire, the smaller the wire gauge. $$\begin{array}{ccc}{\text { Wire gauge }} & {\text { Diameter (cm) }} & {I_{\max }(\mathbf{A})} \\ {14} & {0.163} & {18} \\ {12} & {0.205} & {25} \\ {10} & {0.259} & {30} \\ {8} & {0.326} & {40} \\ {6} & {0.412} & {60} \\ {5} & {0.462} & {65} \\ {4} & {0.519} & {85}\end{array}$$
(a) What considerations determine the maximum current-carrying capacity of household wiring? (b) A total of 4200 $\mathrm{W}$ of power is to be supplied through the wires of a house to the household electrical appliances. If the potential difference across the group of appliances is 120 $\mathrm{V}$ , determine the gauge of the thinnest permissible wire that can be used. (c) Suppose the wire used in this house is of the gauge found in part (b) and has total length 42.0 $\mathrm{m}$ . At what rate is energy dissipated in the wires? (d) The house is built in a community where the consumer cost of electric energy is $\$ 0.11$ per kilowatt-hour. If the house were built with wire of the next larger diameter than that found in part (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.

Dading Chen
Dading Chen
Numerade Educator
02:36

Problem 78

Compact fluorescent bulbs are much more efficient at producing light than are ordinary incandescent bulbs. They initially cost much more, but they last far longer and use much less electricity. According to one study of these bulbs, a compact bulb that produces as much light as a 100-W incandescent bulb uses only $23 \mathrm{~W}$ of power. The compact bulb lasts 10,000 hours, on the average, and costs $\$ 11.00,$ whereas the incandescent bulb costs only $\$ 0.75,$ but lasts just 750 hours. The study assumed that electricity costs $\$ 0.080$ per kilowatt-hour and that the bulbs are on for $4.0 \mathrm{~h}$ per day. (a) What is the total cost (including the price of the bulbs) to run each bulb for 3.0 years? (b) How much do you save over 3.0 years if you use a compact fluorescent bulb instead of an incandescent bulb? (c) What is the resistance of $\mathrm{a}^{*} 100-\mathrm{W}^{* *}$ fluorescent bulb? (Remember, it actually uses only $23 \mathrm{~W}$ of power and operates across $120 \mathrm{~V}$.)

Dading Chen
Dading Chen
Numerade Educator
03:55

Problem 79

In the circuit of Fig. P25.79, find (a) the current through the $8.0-\Omega$ - resistor and (b) the total rate of dissipation of electrical energy in the $8.0-\Omega$ resistor and in the internal resistance of the batteries. (c) In one of the batteries, chemical energy is being converted into electrical energy. In which one is this happening, and at what rate? (d) In one of the batteries, electrical energy is being converted into chemical energy. In which one is this happening, and at what rate? (e) Show that the overall rate of production of electrical energy equals the overall rate of consumption of electrical energy in the circuit.

Dading Chen
Dading Chen
Numerade Educator
02:15

Problem 80

A lightning bolt strikes one end of a steel lightning rod, producing a $15,000-$ A current burst that lasts for 65 \mus. The rod is 2.0 $\mathrm{m}$ long and 1.8 $\mathrm{cm}$ in diameter, and its other end is connected to the ground by 35 $\mathrm{m}$ of $8.0-\mathrm{mm}$ -diameter copper wire.
(a) Find the potential difference between the top of the steel rod and the lower end of the copper wire during the current burst. (b) Find the total energy deposited in the rod and wire by the current burst.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:43

Problem 81

A $12.0-\mathrm{V}$ battery has an internal resistance of 0.24 $\mathrm{s}$ and a capacity of 50.0 $\mathrm{A} \cdot \mathrm{h}$ (see Exercise 25.47$) .$ The battery is charged by passing a 10 -A current through it for 5.0 $\mathrm{h}$ . (a) What is the terminal voltage during charging? (b) What total electricalenergy is supplied to the battery during charging? (c) What electrical energy is dissipated in the internal resistance during charging? (d) The battery is now completely discharged through a resistor, again with a constant current of 10 $\mathrm{A}$ . What is the external circuit resistance? (e) What total electrical energy is supplied to the external resistor? (f) What total electrical energy is dissipated in the internal resistance? (g) Why are the answers to parts (b) and (e) not the same?

Dading Chen
Dading Chen
Numerade Educator
04:03

Problem 82

Repeat Problem 25.81 with charge and discharge currents of 30 A. The charging and discharging times will now be 1.7 $\mathrm{h}$ rather than 5.0 $\mathrm{h}$ . What differences in performance do you see?

Dading Chen
Dading Chen
Numerade Educator
02:22

Problem 83

Consider the circuit shown in Fig. $\mathrm{P} 25.83 .$ The emf source has negligible internal resistance. The resistors have resistances $R_{1}=6.00 \Omega$ and $R_{2}=4.00 \Omega .$ The capacitor has capacitance $C=9.00 \mu \mathrm{F}$ When the capacitor is fully charged, the magnitude of the charge on its plates is $Q=36.0 \mu \mathrm{C} .$ Calculate the emf $\mathcal{E} .$

Dading Chen
Dading Chen
Numerade Educator
03:45

Problem 84

Consider the circuit shown in Fig. $\mathrm{P} 25.84 .$ The battery has emf 60.0 $\mathrm{V}$ and negligible internal resistance. $R_{2}=2.00 \Omega$ $C_{1}=3.00 \mu \mathrm{F},$ and $C_{2}=6.00 \mu \mathrm{F}$ . After the capacitors have attained their final charges, the charge on $C_{1}$ is $Q_{1}=18.0 \mu C$ . (a) What is the final charge on $C_{2} ?$ (b) What is the resistance $R_{1} ?$

Dading Chen
Dading Chen
Numerade Educator
03:04

Problem 85

The Tolman-Stewart experiment in 1916 demonstrated that the free charges in a metal have negative charge and provided a quantitative measurement of their charge-to-mass ratio, $|q| / m .$ The experiment consisted of abruptly stopping a rapidly rotating spool of wire and measuring the potential difference that this produced between the ends of the wire. In a simplified model
of this experiment, consider a metal rod of length $L$ that is given a uniform acceleration $\vec{a}$ to the right. Initially the free charges in the metal lag behind the rod's motion, thus setting up an electric field $\vec{E}$ in the rod. In the steady state this field exerts a force on the free charges that makes them accelerate along with the rod. (a) Apply $\sum \vec{\boldsymbol{F}}=m \vec{\boldsymbol{a}}$ to the free charges to obtain an expression for $|q| / m$ in terms of the magnitudes of the induced electric field $\vec{\boldsymbol{E}}$ and the acceleration $\vec{\boldsymbol{a}}$ . (b) If all the free charges in the metal rod have the same acceleration, the electric field $\vec{\boldsymbol{E}}$ is the same at all points in the rod. Use this fact to rewrite the expression for $|q| / m$ in terms of the potential $V_{b c}$ between the ends of the rod (Fig. P25.85). (c) If the free charges have negative charge, which end of the rod, $b$ or $c,$ is at higher potential? (d) If the rod is 0.50 $\mathrm{m}$ long and the free charges are electrons (charge $q=-1.60 \times 10^{-19} \mathrm{C},$ mass $9.11 \times 10^{-31} \mathrm{kg} ),$ what magnitude of acceleration is required to produce a potential difference of 1.0 $\mathrm{mV}$ between the ends of the
rod? (e) Discuss why the actual experiment used a rotating spool of thin wire rather than a moving bar as in our simplified analysis.

Dading Chen
Dading Chen
Numerade Educator
02:13

Problem 86

A source with emf $\mathcal{E}$ and internal resistance $r$ is connected to an external circuit. (a) Show that the power output of the source is maximum when the current in the circuit is one-half the short-circuit current of the source. (b) If the external circuit consists of a resistance $R,$ show that the power output is maximum when $R=r$ and that the maximum power is $\mathcal{E}^{2} / 4 r_{1}$

Dading Chen
Dading Chen
Numerade Educator
03:29

Problem 87

The resistivity of a semiconductor can be modified by 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 $x=L$ .

Dading Chen
Dading Chen
Numerade Educator