Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 27

Current and Resistance - all with Video Answers

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Chapter Questions

Problem 1

A proton beam in an accelerator carries a current of 125$\mu A$ . If the beam is incident on a target, how many protons strike the target in a period of 23.0 $\mathrm{s}$ ?

Prabhat Tyagi

Prabhat Tyagi

Numerade Educator

Problem 2

A copper wire has a circular cross section with a radius of 1.25 $\mathrm{mm}$ . ( a) If the wire carries a current of 3.70 $\mathrm{A}$ , find the drift speed of the electrons in this wire. (b) All other things being equal, what happens to the drift speed in wires made of metal having a larger number of conduction electrons per atom than copper? Explain.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 3

An aluminum wire having a cross-sectional area equal to $4.00 \times 10^{-6} \mathrm{m}^{2}$ carries a current of 5.00 $\mathrm{A}$ . The density of aluminum is $2.70 \mathrm{g} / \mathrm{cm}^{3} .$ Assume each aluminum atom supplies one conduction electron per atom. Find the drift speed of the electrons in the wire.

Kayla Gephart

Numerade Educator

Problem 4

In the Bohr model of the hydrogen atom (which will be covered in detail in Chapter $42 ),$ an electron in the lowest energy state moves at a speed of $2.19 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a circular path of radius $5.29 \times 10^{-11} \mathrm{m}$ . What is the effective current associated with this orbiting electron?

Kayla Gephart

Numerade Educator

Problem 5

Suppose the current in a conductor decreases exponentially with time according to the equation $I(t)=I_{0} e^{-t / \tau}$ where $I_{0}$ is the initial current (at $t=0 )$ and $\tau$ is a constant having dimensions of time. Consider a fixed observation point within the conductor. (a) How much charge passes this point between $t=0$ and $t=\tau ?$ (b) How much charge passes this point between $t=0$ and $t=10 \tau ?(\mathrm{c})$ What If? How much charge passes this point between $t=0$ and $t=\infty$ ?

Kayla Gephart

Numerade Educator

Problem 6

Figure P27.6 represents a section of a conductor of nonuniform diameter carrying a current of $I=5.00 \mathrm{A}$ . The radius of cross-section $A_{1}$ is $r_{1}=0.400 \mathrm{cm} .$ (a) What is the magnitude of the current density across $A_{1} ?$ The radius $r_{2}$ at $A_{2}$ is larger than the radius $r_{1}$ at $A_{1} .$ (b) Is the current at $A_{2}$ larger, smaller, or the same? (c) Is the current density at $A_{2}$ larger, smaller, or the same? Assume $A_{2}=4 A_{1} .$ Specify the (d) radius, (e) current, and (f) current density at $A_{2}$ .

Kayla Gephart

Numerade Educator

Problem 7

The quantity of charge $q$ (in coulombs) that has passed through a surface of area 2.00 $\mathrm{cm}^{2}$ varies with time according to the equation $q=4 t^{3}+5 t+6,$ where $t$ is in seconds. (a) What is the instantaneous current through the surface at $t=1.00 \mathrm{s}^{2}$ (b) What is the value of the current density?

Kayla Gephart

Numerade Educator

Problem 8

An electric current in a conductor varies with time according to the expression $I(t)=100 \sin (120 \pi t),$ where $I$ is in amperes and $t$ is in seconds. What is the total charge passing a given point in the conductor from $t=0$ to $t=\frac{1}{240} \mathrm{s}$ ?

Kayla Gephart

Numerade Educator

Problem 9

A teapot with a surface area of 700 $\mathrm{cm}^{2}$ is to be plated with silver. It is attached to the negative electrode of an electrolytic cell containing silver nitrate $\left(\mathrm{Ag}^{+} \mathrm{NO}_{3}^{-}\right) .$ The cell is powered by a $12.0-\mathrm{V}$ battery and has a resistance of 1.80$\Omega$ . If the density of silver is $10.5 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3},$ over what time interval does a 0.133 -mm layer of silver build up on the teapot?

Kayla Gephart

Numerade Educator

Problem 10

A Van de Graaff generator produces a beam of 2.00-MeV deuterons, which are heavy hydrogen nuclei containing a proton and a neutron. (a) If the beam current is 10.0$\mu \mathrm{A}$ , what is the average separation of the deuterons? (b) Is the electrical force of repulsion among them a significant factor in beam stability? Explain.

Kayla Gephart

Numerade Educator

Problem 11

The electron beam emerging from a certain high-energy electron accelerator has a circular cross section of radius $1.00 \mathrm{mm} .$ (a) The beam current is 8.00$\mu \mathrm{A}$ . Find the current density in the beam assuming it is uniform throughout. (b) The speed of the electrons is so close to the speed of light that their speed can be taken as 300 $\mathrm{Mm} / \mathrm{s}$ with negligible error. Find the electron density in the beam. (c) Over what time interval does Avogadro's number of electrons emerge from the accelerator?

Kayla Gephart

Numerade Educator

Problem 12

A lightbulb has a resistance of 240$\Omega$ when operating with a potential difference of 120 $\mathrm{V}$ across it. What is the current in the lightbulb?

Kayla Gephart

Numerade Educator

Problem 13

An electric heater carries a current of 13.5 $\mathrm{A}$ when operating at a voltage of 120 $\mathrm{V}$ . What is the resistance of the heater?

Kayla Gephart

Numerade Educator

Problem 14

A $0.900-\mathrm{V}$ potential difference is maintained across a 1.50 -m length of tungsten wire that has a cross-sectional area of $0.600 \mathrm{mm}^{2} .$ What is the current in the wire?

Kayla Gephart

Numerade Educator

Problem 15

Suppose you wish to fabricate a uniform wire from 1.00 $\mathrm{g}$ of copper. If the wire is to have a resistance of $R=$ 0.500$\Omega$ and all the copper is to be used, what must be (a) the length and (b) the diameter of this wire?

Kayla Gephart

Numerade Educator

Problem 16

Suppose you wish to fabricate a uniform wire from a mass $m$ of a metal with density $\rho_{m}$ and resistivity $\rho .$ If the wire is to have a resistance of $R$ and all the metal is to be used, what must be (a) the length and (b) the diameter of this wire?

Kayla Gephart

Numerade Educator

Problem 17

A portion of Nichrome wire of radius 2.50 $\mathrm{mm}$ is to be used in winding a heating coil. If the coil must draw a current of 9.25 $\mathrm{A}$ when a voltage of 120 $\mathrm{V}$ is applied across its ends, find (a) the required resistance of the coil and (b) the length of wire you must use to wind the coil.

Kayla Gephart

Numerade Educator

Problem 18

Aluminum and copper wires of equal length are found to have the same resistance. What is the ratio of their radii?

Kayla Gephart

Numerade Educator

Problem 19

If the magnitude of the drift velocity of free electrons in a copper wire is $7.84 \times 10^{-4} \mathrm{m} / \mathrm{s}$ , what is the electric field in the conductor?

Kayla Gephart

Numerade Educator

Problem 20

An iron wire has a cross-sectional area equal to $5.00 \times 10^{-6} \mathrm{m}^{2}$ . Carry out the following steps to determine the drift speed of the conduction electrons in the wire if it carries a current of 30.0 $\mathrm{A}$ (a) How many kilograms are there in 1.00 mole of iron? (b) Starting with the density of iron and the result of part (a), compute the molar density of iron (the number of moles of iron per cubic meter). (c) Calculate the number density of iron atoms using Avogadro's number. (d) Obtain the number density of conduction electrons given that there are two conduction electrons per iron atom. (e) Calculate the drift speed of conduction electrons in this wire.

Kayla Gephart

Numerade Educator

Problem 21

If a certain silver wire has a resistance of 6.00$\Omega$ at $20.0^{\circ} \mathrm{C},$ what resistance will it have at $34.0^{\circ} \mathrm{C} ?$

Kayla Gephart

Numerade Educator

Problem 22

A certain lightbulb has a tungsten filament with a resistance of 19.0$\Omega$ when at $20.0^{\circ} \mathrm{C}$ and 140$\Omega$ when hot. Assume the resistivity of tungsten varies linearly with temperature even over the large temperature range involved here. Find the temperature of the hot filament.

Kayla Gephart

Numerade Educator

Problem 23

What is the fractional change in the resistance of an iron filament when its temperature changes from $25.0^{\circ} \mathrm{C}$ to $50,0^{\circ} \mathrm{C}$?

Kayla Gephart

Numerade Educator

Problem 24

Plethysmographs are devices used for measuring changes in the volume of internal organs or limbs. In one form of this device, a rubber capillary tube with an inside diameter of 1.00 $\mathrm{mm}$ is filled with mercury at $20.0^{\circ} \mathrm{C}$ . The resistance of the mercury is measured with the aid of electrodes sealed into the ends of the tube. If 100 $\mathrm{cm}$ of the tube is wound in a helix around a patient's upper arm, the blood flow during a heartbeat causes the arm to expand, stretching the length of the tube by $0.0400 \mathrm{cm} .$ From this observation and assuming cylindrical symmetry, you can find the change in volume of the arm, which gives an indication of blood flow. Taking the resistivity of mercury to be $9.58 \times 10^{-7} \Omega \cdot \mathrm{m},$ calculate (a) the resistance of the mercury and (b) the fractional change in resistance during the heartbeat. Hint: The fraction by which the cross-sectional area of the mercury column decreases is the fraction by which the length increases because the volume of mercury is constant.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 25

An aluminum wire with a diameter of 0.100 mm has a uniform electric field of 0.200 V/m imposed along its entire length. The temperature of the wire is $50.0^{\circ} \mathrm{C}$ . Assume one free electron per atom. (a) Use the information in Table 27.2 to determine the resistivity of aluminum at this temperature. (b) What is the current density in the wire? (c) What is the total current in the wire? (d) What is the drift speed of the conduction electrons? (e) What potential difference must exist between the ends of a $2.00-\mathrm{m}$ length of the wire to produce the stated electric field?

Kayla Gephart

Numerade Educator

Problem 26

An engineer needs a resistor with a zero overall temperature coefficient of resistance at $20.0^{\circ} \mathrm{C}$ . She designs a pair of circular cylinders, one of carbon and one of Nichrome as shown in Figure $\mathrm{P} 27.26 .$ The device must have an overall resistance of $R_{1}+R_{2}=10.0 \Omega$ independent of temperature and a uniform radius of $r=1.50 \mathrm{mm} .$ Ignore thermal expansion of the cylinders and assume both are always at the same temperature. (a) Can she meet the design goal with this method? (b) If so, state what you can determine about the lengths $\ell_{1}$ and $\ell_{2}$ of each segment. If not, explain.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 27

At what temperature will aluminum have a resistivity that is three times the resistivity copper has at room temperature?

Kayla Gephart

Numerade Educator

Problem 28

Review. An aluminum rod has a resistance of 1.23$\Omega$ at $20.0^{\circ} \mathrm{C}$ . Calculate the resistance of the rod at $120^{\circ} \mathrm{C}$ by accounting for the changes in both the resistivity and the dimensions of the rod. The coefficient of linear expansion for aluminum is $2.40 \times 10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$ .

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 29

A certain waffle iron is rated at 1.00 $\mathrm{kW}$ when connected to a $120-\mathrm{V}$ source. (a) What current does the waffle iron carry? (b) What is its resistance?

Kayla Gephart

Numerade Educator

Problem 30

A Van de Graaff generator (see Fig. 25.24) is operating so that the potential difference between the high-potential electrode and the charging needles at is 15.0 kV. Calculate the power required to drive the belt against electrical forces at an instant when the effective current delivered to the high-potential electrode is 500 $\mu \mathrm{A}$.

Kayla Gephart

Numerade Educator

Problem 31

Suppose your portable DVD player draws a current of 350 $\mathrm{mA}$ at 6.00 $\mathrm{V}$ . How much power does the player require?

Kayla Gephart

Numerade Educator

Problem 32

The potential difference across a resting neuron in the human body is about 75.0 $\mathrm{mV}$ and carries a current of about 0.200 $\mathrm{mA}$ . How much power does the neuron release?

Nishant Kumar

Numerade Educator

Problem 33

A $100-\mathrm{W}$ lightbulb connected to a $120-\mathrm{V}$ source experiences a voltage surge that produces 140 $\mathrm{V}$ for a moment. By what percentage does its power output increase? Assume its resistance does not change.

Kayla Gephart

Numerade Educator

Problem 34

Review. A well-insulated electric water heater warms 109 $\mathrm{kg}$ of water from $20.0^{\circ} \mathrm{C}$ to $49.0^{\circ} \mathrm{C}$ in 25.0 $\mathrm{min}$ . Find the resistance of its heating element, which is connected across a $240-\mathrm{V}$ potential difference.

Kayla Gephart

Numerade Educator

Problem 35

Batteries are rated in terms of ampere-hours $(\mathrm{A} \cdot \mathrm{h}) .$ For example, a battery that can produce a current of 2.00 A for 3.00 $\mathrm{h}$ is rated at $6.00 \mathrm{A} \cdot \mathrm{h} .$ (a) What is the total energy, in kilowatt-hours, stored in a $12.0-\mathrm{V}$ battery rated at 55.0 $\mathrm{A} \cdot \mathrm{h}$ ? (b) At $\$ 0.110$ per kilowatt-hour, what is the value of the electricity produced by this battery?

Kayla Gephart

Numerade Educator

Problem 36

Residential building codes typically require the use of 12-gauge copper wire (diameter 0.205 cm) for wiring receptacles. Such circuits carry currents as large as 20.0 A. If a wire of smaller diameter (with a higher gauge number) carried that much current, the wire could rise to a high temperature and cause a fire. (a) Calculate the rate at which internal energy is produced in 1.00 $\mathrm{m}$ of 12 -gauge copper wire carrying 20.0 $\mathrm{A}$ . (b) What If? Repeat the calculation for a 12 -gauge aluminum wire. (c) Explain whether a 12 -gauge aluminum wire would be as safe as a copper wire.

Kayla Gephart

Numerade Educator

Problem 37

An 11.0 -W energy-efficient fluorescent lightbulb is designed to produce the same illumination as a conventional $40.0-\mathrm{W}$ incandescent lightbulb. Assuming a cost of $\$ 0.110 / \mathrm{kWh}$ for energy from the electric company, how much money does the user of the energy-efficient bulb save during 100 $\mathrm{h}$ of use?

Kayla Gephart

Numerade Educator

Problem 38

The cost of energy delivered to residences by electrical transmission varies from $\$ 0.070 / \mathrm{kWh}$ to $\$ 0.258 / \mathrm{kWh}$ throughout the United States; $\$ 0.110 / \mathrm{kWh}$ is the average value. At this average price, calculate the cost of (a) leaving a $40.0-\mathrm{W}$ porch light on for two weeks while you are on vacation, (b) making a piece of dark toast in 3.00 $\mathrm{min}$ with a $970-\mathrm{W}$ toaster, and $(\mathrm{c})$ drying a load of clothes in 40.0 $\mathrm{min}$ in a $5.20 \times 10^{3}-\mathrm{W}$ dryer.

Kayla Gephart

Numerade Educator

Problem 39

Assuming the cost of energy from the electric company is $\$ 0.110 / \mathrm{kWh}$ , compute the cost per day of operating a lamp that draws a current of 1.70 $\mathrm{A}$ from a $110-\mathrm{V}$ line.

Kayla Gephart

Numerade Educator

Problem 40

Review. A rechargeable battery of mass 15.0 g delivers an average current of 18.0 mA to a portable DVD player at 1.60 V for 2.40 h before the battery must be recharged. The recharger maintains a potential difference of 2.30 V across the battery and delivers a charging current of 13.5 mA for 4.20 h. (a) What is the efficiency of the battery as an energy storage device? (b) How much internal energy is produced in the battery during one charge–discharge cycle? (c) If the battery is surrounded by ideal thermal insulation and has an effective specific heat of $975 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C},$ by how much will its temperature increase during the cycle?

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 41

A coil of Nichrome wire is 25.0 $\mathrm{m}$ long. The wire has a diameter of 0.400 $\mathrm{mm}$ and is at $20.0^{\circ} \mathrm{C}$ . If it carries a current of 0.500 $\mathrm{A}$ , what are (a) the magnitude of the electric field in the wire and (b) the power delivered to it? (c) What If? If the temperature is increased to $340^{\circ} \mathrm{C}$ and the potential difference across the wire remains constant, what is the power delivered?

Kayla Gephart

Numerade Educator

Problem 42

Why is the following situation impossible? A politician is decrying wasteful uses of energy and decides to focus on energy used to operate plug-in electric clocks in the United States. He estimates there are 270 million of these clocks, approximately one clock for each person in the population. The clocks transform energy taken in by electrical transmission at the average rate 2.50 $\mathrm{W}$ . The politician gives a speech in which he complains that, at today's electrical rates, the nation is losing $\$ 100$ million every year to operate these clocks.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 43

A 500 -W heating coil designed to operate from 110 $\mathrm{V}$ is made of Nichrome wire 0.500 $\mathrm{mm}$ in diameter. (a) Assuming the resistivity of the Nichrome remains constant at its $20.0^{\circ} \mathrm{C}$ value, find the length of wire used. (b) What If? Now consider the variation of resistivity with temperature. What power is delivered to the coil of part (a) when it is warmed to $1200^{\circ} \mathrm{C} ?$

Supratim Pal

Numerade Educator

Problem 44

A $120-\mathrm{V}$ motor has mechanical power output of 2.50 $\mathrm{hp}$ . It is 90.0$\%$ efficient in converting power that it takes in by electrical transmission into mechanical power. (a) Find the current in the motor. (b) Find the energy delivered to the motor by electrical transmission in 3.00 $\mathrm{h}$ of operation. (c) If the electric company charges $\$ 0.110 / \mathrm{kWh},$ what does it cost to run the motor for 3.00 $\mathrm{h}$ ?

Kayla Gephart

Numerade Educator

Problem 45

A certain toaster has a heating element made of Nichrome wire. When the toaster is first connected to a 120-V source (and the wire is at a temperature of 20.0°C), the initial current is 1.80 A. The current decreases as the heating element warms up. When the toaster reaches its final operating temperature, the current is 1.53 A. (a) Find the power delivered to the toaster when it is at its operating temperature. (b) What is the final temperature of the heating element?

Kayla Gephart

Numerade Educator

Problem 46

Make an order-of-magnitude estimate of the cost of one person's routine use of a handheld hair dryer for 1 year. If you do not use a hair dryer yourself, observe or interview someone who does. State the quantities you estimate and their values.

Kayla Gephart

Numerade Educator

Problem 47

Review. The heating element of an electric coffee maker operates at 120 $\mathrm{V}$ and carries a current of 2.00 $\mathrm{A}$ . Assuming the water absorbs all the energy delivered to the resistor, calculate the time interval during which the temperature of 0.500 $\mathrm{kg}$ of water rises from room temperature $\left(23.0^{\circ} \mathrm{C}\right)$ to the boiling point.

Kayla Gephart

Numerade Educator

Problem 48

Determine the temperature at which the resistance of an aluminum wire will be twice its value at $20.0^{\circ} \mathrm{C}$ . Assume its coefficient of resistivity remains constant.

Kayla Gephart

Numerade Educator

Problem 49

A car owner forgets to turn off the headlights of his car while it is parked in his garage. If the $12.0-\mathrm{V}$ battery in his car is rated at 90.0 $\mathrm{A} \cdot \mathrm{h}$ and each headlight requires 36.0 $\mathrm{W}$ of power, how long will it take the battery to completely discharge?

Kayla Gephart

Numerade Educator

Problem 50

Lightbulb A is marked “25 W 120 V,” and lightbulb B is marked " 100 $\mathrm{W} 120 \mathrm{V}$ " These labels mean that each lightbulb has its respective power delivered to it when it is connected to a constant $120-\mathrm{V}$ source. (a) Find the resistance of each lightbulb. (b) During what time interval does 1.00 $\mathrm{C}$ pass into lightbulb A? (c) Is this charge different upon its exit versus its entry into the lightbulb? Explain. (d) In what time interval does 1.00 $\mathrm{J}$ pass into lightbulb A? (e) By what mechanisms does this energy enter and exit the lightbulb? Explain. (f) Find the cost of running lightbulb A continuously for 30.0 days, assuming the electric company sells its product at $\$ 0.110$ per $\mathrm{kWh}$ .

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 51

One wire in a high-voltage transmission line carries 1000 $\mathrm{A}$ starting at 700 $\mathrm{kV}$ for a distance of 100 $\mathrm{mi}$ . If the resistance in the wire is 0.500$\Omega / \mathrm{mi}$ , what is the power loss due to the resistance of the wire?

Kayla Gephart

Numerade Educator

Problem 52

Review. An office worker uses an immersion heater to warm 250 g of water in a light, covered, insulated cup from $20.0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ in 4.00 $\mathrm{min}$ . The heater is a Nichrome resistance wire connected to a $120-\mathrm{V}$ power supply. Assume the wire is at $100^{\circ} \mathrm{C}$ throughout the 4.00 -min time interval. (a) Specify a relationship between a diameter and a length that the wire can have. (b) Can it be made from less than 0.500 $\mathrm{cm}^{3}$ of Nichrome?

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 53

A charge $Q$ is placed on a capacitor of capacitance $C .$ The capacitor is connected into the circuit shown in Figure $\mathrm{P} 27.53$ , with an open switch, a resistor, and an initially uncharged capacitor of capacitance 3$C$ . The switch is then closed, and the circuit comes to equilibrium. In terms of $Q$ and $C,$ find (a) the final potential difference between the plates of each capacitor, (b) the charge on each capacitor, and (c) the final energy stored in each capacitor. (d) Find the internal energy appearing in the resistor.

Rashmi Sinha

Numerade Educator

Problem 54

An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30-gauge wire, which has a cross-sectional area of $7.30 \times 10^{-8} \mathrm{m}^{2} .$ The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. (a) For each set of measurements given in the table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding values of the resistivity. (b) What is the average value of the resistivity? (c) Explain how this value compares with the value given in Table $27.2 .$
$$\begin{array}{lll} \hline {{L(\mathbf{m})}} & {\Delta V(\mathbf{V})} & {\mathbf{I}(\mathbf{A})} & {\mathbf{R}(\mathbf{\Omega})} & \rho(\Omega \cdot \mathrm{m})\\ \hline {0.540} & {5.22} & {0.72} \\ {1.028} & {5.82} & {0.414} \\ {1.543} & {5.94} & {0.281} \\ \hline\end{array}$$

Kayla Gephart

Numerade Educator

Problem 55

A high-voltage copper transmission line with a diameter of $2.00 \mathrm{~cm}$ and a length of $200 \mathrm{~km}$ carries a steady current of $1.00 \times 10^{3}$ A. If copper has a free charge density of $8.46 \times 10^{28}$ electrons $/ \mathrm{m}^{3},$ over what time interval does one electron travel the full length of the line?

Kayla Gephart

Numerade Educator

Problem 56

An all-electric car (not a hybrid) is designed to run from a bank of $12.0-\mathrm{V}$ batteries with total energy storage of $2.00 \times 10^{7} \mathrm{J}$ . If the electric motor draws 8.00 $\mathrm{kW}$ as the car moves at a steady speed of $20.0 \mathrm{m} / \mathrm{s},$ (a) what is the current delivered to the motor? (b) How far can the car travel before it is "out of juice"?

Kayla Gephart

Numerade Educator

Problem 57

A straight, cylindrical wire lying along the $x$ axis has a length of 0.500 $\mathrm{m}$ and a diameter of $0.200 \mathrm{mm} .$ It is made of a material described by Ohm's law with a resistivity of $\rho=4.00 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Assume a potential of 4.00 $\mathrm{V}$ is maintained at the left end of the wire at $x=0 .$ Also assume $V=0$ at $x=0.500 \mathrm{m}$ . Find $(a)$ the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire. (e) Show that $E=\rho J$.

Kayla Gephart

Numerade Educator

Problem 58

A straight, cylindrical wire lying along the $x$ axis has a length $L$ and a diameter $d .$ It is made of a material described by Ohm's law with a resistivity $\rho$ . Assume potential $V$ is maintained at the left end of the wire at $x=0$ . Also assume the potential is zero at $x=L .$ In terms of $L, d$ , $V, \rho,$ and physical constants, derive expressions for (a) the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire. (e) Show that $E=\rho J$.

Kayla Gephart

Numerade Educator

Problem 59

An electric utility company supplies a customer's house from the main power lines $(120 \mathrm{V})$ with two copper wires, each of which is 50.0 $\mathrm{m}$ long and has a resistance of 0.108$\Omega$ per 300 $\mathrm{m}$ . (a) Find the potential difference at the customer's house for a load current of 110 $\mathrm{A}$ . For this load current, find (b) the power delivered to the customer and (c) the rate at which internal energy is produced in the copper wires.

Kayla Gephart

Numerade Educator

Problem 60

The strain in a wire can be monitored and computed by measuring the resistance of the wire. Let $L_{i}$ represent the original length of the wire, $A_{i}$ its original cross-sectional area, $R_{i}=\rho L_{i} / A_{i}$ the original resistance between its ends, and $\delta=\Delta L / L_{i}=\left(L-L_{i}\right) / L_{i}$ the strain resulting from the application of tension. Assume the resistivity and the volume of the wire do not change as the wire stretches. (a) Show that the resistance between the ends of the wire under strain is given by $R=R_{i}\left(1+2 \delta+\delta^{2}\right) .$ (b) If the assumptions are precisely true, is this result exact or approximate? Explain your answer.

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 61

The temperature coefficients of resistivity $\alpha$ in Table 27.2 are based on a reference temperature $T_{0}$ of $20.0^{\circ} \mathrm{C}$ . Suppose the coefficients were given the symbol $\alpha^{\prime}$ and were based on a $T_{0}$ of $0^{\circ} \mathrm{C}$ . What would the coefficient $\alpha^{\prime}$ for silver be? Note: The coefficient $\alpha$ satisfies $\rho=$ $\rho_{0}\left[1+\alpha\left(T-T_{0}\right)\right],$ where $\rho_{0}$ is the resistivity of the material
at $T_{0}=20.0^{\circ} \mathrm{C} .$ The coefficient $\alpha^{\prime}$ must satisfy the expression $\rho=\rho_{0}^{\prime}\left[1+\alpha^{\prime} T\right],$ where $\rho_{0}^{\prime}$ is the resistivity of the material at $0^{\circ} \mathrm{C} .$

Kayla Gephart

Numerade Educator

Problem 62

A close analogy exists between the flow of energy by heat because of a temperature difference (see Section 20.7$)$ and the flow of electric charge because of a potential difference. In a metal, energy $d Q$ and electrical charge $d q$ are both transported by free electrons. Consequently, a good electrical conductor is usually a good thermal conductor as well. Consider a thin conducting slab of thickness $d x,$ area $A,$ and electrical conductivity $\sigma,$ with a potential difference $d V$ between opposite faces. (a) Show that the current $I=d q / d t$ is given by the equation on the left:
$$\begin{array}{ll}{\text { Charge conduction }} & {\text { Thermal conduction }} \\ {\frac{d q}{d t}=\sigma A\left|\frac{d V}{d x}\right|} & {\frac{d Q}{d t}=k A\left|\frac{d T}{d x}\right|}\end{array}$$
In the analogous thermal conduction equation on the right (Eq. 20.15 ), the rate $d Q / d t$ of energy flow by heat (in SI units of joules per second) is due to a temperature gradient $d T / d x$ in a material of thermal conductivity $k .$ (b) State analogous rules relating the direction of the electric current to the change in potential and relating the direction of energy flow to the change in temperature.

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 63

An oceanographer is studying how the ion concentration in seawater depends on depth. She makes a measurement by lowering into the water a pair of concentric metallic cylinders (Fig.
P27.63) at the end of a cable and taking data to determine the resistance between these electrodes as a function of depth. The water between the two cylinders forms a cylindrical shell of inner radius $r_{a},$ outer radius $r_{b},$ and length $L$ much larger than $r_{b} .$ The scientist applies a potential difference $\Delta V$ between the inner and outer surfaces, producing an outward radial current $I$ . Let $\rho$ represent the resistivity of the water. (a) Find the resistance of the water between the cylinders in terms of $L, \rho, r_{a},$ and $r_{b} .$ (b) Express the resistivity of the water in terms of the measured quantities $L, r_{a}$ $r_{b}, \Delta V,$ and $I .$

Vidhi Bhatt

Numerade Educator

Problem 64

Why is the following situation impossible? An inquisitive physics student takes a $100-W$ lightbulb out of its socket and measures its resistance with an ohmmeter. He measures a value of 10.5$\Omega$ . He is able to connect an ammeter to the lightbulb socket to correctly measure the current drawn by the bulb while operating. Inserting the bulb back into the socket and operating the bulb from a $120-V$ source, he measures the current to be 11.4 $\mathrm{A}$ .

Kayla Gephart

Numerade Educator

Problem 65

Review. When a straight wire is warmed, its resistance is given by $R=R_{0}\left[1+\alpha\left(T-T_{0}\right)\right]$ according to Equation $27.19,$ where $\alpha$ is the temperature coefficient of resistivity. This expression needs to be modified if we include the change in dimensions of the wire due to thermal expansion. For a copper wire of radius 0.1000 $\mathrm{mm}$ and length 2.000 $\mathrm{m}$ , find its resistance at $100.0^{\circ} \mathrm{C}$ , including the effects of both thermal expansion and temperature variation of resistivity. Assume the coefficients are known to four significant figures.

Vidhi Bhatt

Numerade Educator

Problem 66

Review. When a straight wire is warmed, its resistance is given by $R=R_{0}\left[1+\alpha\left(T-T_{0}\right)\right]$ according to Equation $27.19,$ where $\alpha$ is the temperature coefficient of resistivity. This expression needs to be modified if we include the change in dimensions of the wire due to thermal expansion. Find a more precise expression for the resistance, one that includes the effects of changes in the dimensions of the wire when it is warmed. Your final expression should be in terms of $R_{0}, T, T_{0},$ the temperature coefficient of resistivity $\alpha,$ and the coefficient of linear expansion $\alpha^{\prime} .$

Vidhi Bhatt

Numerade Educator

Problem 67

Review. A parallel-plate capacitor consists of square plates of edge length $\ell$ that are separated by a distance $d,$ where $d<\ell$ . A potential difference $\Delta V$ is maintained between the plates. A material of dielectric constant $\kappa$ fills half the space between the plates. The dielectric slab is withdrawn from the capacitor as shown in Figure $\mathrm{P} 27.67$. (a) Find the capacitance when the left edge of the dielectric is at a distance $x$ from the center of the capacitor. (b) If the dielectric is removed at a constant speed $v$ , what is the current in the circuit as the dielectric is being withdrawn?

Jonathan Ibarra

Jonathan Ibarra

Numerade Educator

Problem 68

The dielectric material between the plates of a parallelplate capacitor always has some nonzero conductivity $\sigma$ . Let $A$ represent the area of each plate and $d$ the distance between them. Let $\kappa$ represent the dielectric constant of the material. (a) Show that the resistance $R$ and the capacitance $C$ of the capacitor are related by
$$R C=\frac{\kappa \epsilon_{0}}{\sigma}$$
(b) Find the resistance between the plates of a $14.0-\mathrm{nF}$ capacitor with a fused quartz dielectric.

Ravindra Yadav

Numerade Educator

Problem 69

Gold is the most ductile of all metals. For example, one gram of gold can be drawn into a wire 2.40 $\mathrm{km}$ long. The density of gold is $19.3 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ , and its resistivity is $2.44 \times 10^{-8} \Omega \cdot \mathrm{m} .$ What is the resistance of such a wire at $20.0^{\circ} \mathrm{C}$ ?

Kayla Gephart

Numerade Educator

Problem 70

The current-voltage characteristic curve for a semiconductor diode as a function of temperature $T$ is given by $$I=I_{0}\left(e^{e \Delta V / k_{\mathrm{B}} T}-1\right).$$ Here the first symbol $e$ represents Euler's number, the base of natural logarithms. The second $e$ is the magnitude of the electron charge, the $k_{\mathrm{B}}$ stands for Boltzmann's constant, and $T$ is the absolute temperature. (a) Set up a spreadsheet to calculate $I$ and $R=\Delta V / I$ for $\Delta V=0.400 \mathrm{~V}$ to $0.600 \mathrm{~V}$ in increments of $0.005 \mathrm{~V}$. Assume $I_{0}=1.00 \mathrm{nA}$. (b) Plot $R$ versus $\Delta V$ for $T=280 \mathrm{~K}, 300 \mathrm{~K},$ and $320 \mathrm{~K}$.

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 71

The potential difference across the filament of a lightbulb is maintained at a constant value while equilibrium temperature is being reached. The steady-state current in the bulb is only one-tenth of the current drawn by the bulb when it is first turned on. If the temperature coefficient of resistivity for the bulb at $20.0^{\circ} \mathrm{C}$ is 0.00450$\left(^{\circ} \mathrm{C}\right)^{-1}$ and the resistance increases linearly with increasing temperature, what is the final operating temperature of the filament?

Kayla Gephart

Numerade Educator

Problem 72

Material with uniform resistivity $p$ is formed into a wedge as shown in Figure P27.72. Show that the resistance between face A and face B of this wedge is
$$R=\rho \frac{L}{w\left(y_{2}-y_{1}\right)} \ln \frac{y_{2}}{y_{1}}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

A material of resistivity $p$ is formed into the shape of a truncated cone of height $h$ as shown in Figure P27.73. The bottom end has radius $b,$ and the top end has radius $a$ . Assume the current is distributed uniformly over any circular cross section of the cone so that the current density does not depend on radial position. (The current density does vary with position along the axis of the cone.) Show that the resistance between the two ends is
$$R=\frac{\rho}{\pi}\left(\frac{h}{a b}\right)$$

Vidhi Bhatt

Numerade Educator

Problem 74

A more general definition of the temperature coefficient of resistivity is
$$\alpha=\frac{1}{\rho} \frac{d \rho}{d T}$$
where $\rho$ is the resistivity at temperature $T .$ (a) Assuming $\alpha$ is constant, show that
$$\rho=\rho_{0} e^{\alpha\left(T-T_{0}\right)}$$
where $\rho_{0}$ is the resistivity at temperature $T_{0} .$ (b) Using the series expansion $e^{x} \approx 1+x$ for $x<<1$ , show that the resistivity is given approximately by the expression
$$\rho=\rho_{0}\left[1+\alpha\left(T-T_{0}\right)\right] \text { for } \alpha\left(T-T_{0}\right)<<1$$

Sheh Lit Chang

University of Washington

Problem 75

If A spherical shell with inner radius $r_{a}$ and outer radius $r_{b}$ is formed from a material of resistivity $\rho$ . It carries current radially, with uniform density in all directions. Show that its resistance is
$$R=\frac{\rho}{4 \pi}\left(\frac{1}{r_{a}}-\frac{1}{r_{b}}\right)$$

Vidhi Bhatt

Numerade Educator