Physics for Scientist and Engineers: A Strategic Approach

Randall Knight

Chapter 31

Current and Resistance - all with Video Answers

Educators

Chapter Questions

Problem 1

$1.0 \times 10^{20}$ electrons flow through a cross section of a 2.0 -mm diameter iron wire in 5.0 s. What is the electron drift speed?

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

The electron drift speed in a 1.0 -mm-diameter gold wire is $5.0 \times 10^{-5} \mathrm{m} / \mathrm{s} .$ How long does it take 1 mole of electrons to flow through a cross section of the wire?

Keshav Singh

Numerade Educator

Problem 3

$1.0 \times 10^{16}$ electrons flow through a cross section of silver wire in $320 \mu$ s with a drift speed of $8.0 \times 10^{-4} \mathrm{m} / \mathrm{s}$. What is the diameter of the wire?

Keshav Singh

Numerade Educator

Problem 4

Electrons flow through a 1.6 -mm-diameter aluminum wire at $2.0 \times 10^{-4} \mathrm{m} / \mathrm{s} .$ How many electrons move through a cross section of the wire each day?

Keshav Singh

Numerade Educator

Problem 5

$1.44 \times 10^{14}$ electrons flow through a cross section of a $2.00 \mathrm{mm} \times 2.00 \mathrm{mm}$ square wire in $3.0 \mu \mathrm{s}$. The electron drift speed is $2.00 \times 10^{-4} \mathrm{m} / \mathrm{s} .$ What metal is the wire made of?

Keshav Singh

Numerade Educator

Problem 6

The electron drift speed is $2.0 \times 10^{-4} \mathrm{m} / \mathrm{s}$ in a metal with a mean free time between collisions of $5.0 \times 10^{-14} \mathrm{s}$. What is the electric field strength?

Keshav Singh

Numerade Educator

Problem 7

A 1.0 -mm-diameter wire has 1000 excess electrons per centimeter of length. What is the surface charge density?

Mayukh Banik

Numerade Educator

Problem 8

a. How many conduction electrons are there in a 1.0 -mm diameter gold wire that is $10 \mathrm{cm}$ long?
b. How far must the sea of electrons in the wire move to deliver $-32 \mathrm{ nC }$ of charge to an electrode?

Keshav Singh

Numerade Educator

Problem 9

The mean free time between collisions in iron is $4.2 \times 10^{-15} \mathrm{s}$ What electric field strength causes a $5.0 \times 10^{19} \mathrm{s}^{-1}$ electron current in a 1.8 -mm-diameter iron wire?

Keshav Singh

Numerade Educator

Problem 10

A $2.0 \times 10^{-3}$ V/m electric field creates a $3.5 \times 10^{17}$ electrons/s current in a 1.0 -mm-diameter aluminum wire. What are (a) the drift speed and (b) the mean time between collisions for electrons in this wire?

Keshav Singh

Numerade Educator

Problem 11

The wires leading to and from a 0.12 -mm-diameter light-bulb filament are 1.5 mm in diameter. The wire to the filament carries a current with a current density of $4.5 \times 10^{5} \mathrm{A} / \mathrm{m}^{2} .$ What are (a) the current and (b) the current density in the filament?

Keshav Singh

Numerade Educator

Problem 12

The current in a 100 watt light-bulb is 0.85 A. The filament inside the bulb is 0.25 mm in diameter.
a. What is the current density in the filament?
b. What is the electron current in the filament?

Keshav Singh

Numerade Educator

Problem 13

The electron drift speed in a gold wire is $3.0 \times 10^{-4} \mathrm{m} / \mathrm{s}$.
a. What is the current density in the wire?
b. What is the current if the wire diameter is $0.50 \mathrm{mm} ?$

Mayukh Banik

Numerade Educator

Problem 14

In an integrated circuit, the current density in a $2.5-\mu \mathrm{m}$ -thick $\times$ $75-\mu \mathrm{m}-$ wide gold film is $7.5 \times 10^{5} \mathrm{A} / \mathrm{m}^{2} .$ What is the current in the film?

Keshav Singh

Numerade Educator

Problem 15

The current in an electric hair dryer is $10.0 \mathrm{A}$. How much charge and how many electrons flow through the hair dryer in 5.0 min?

Keshav Singh

Numerade Educator

Problem 16

$2.0 \times 10^{13}$ electrons flow through a transistor in $1.0 \mathrm{ms}$. What is the current through the transistor?

Keshav Singh

Numerade Educator

Problem 17

In an ionic solution, $5.0 \times 10^{15}$ positive ions with charge $+2 e$ pass to the right each second while $6.0 \times 10^{15}$ negative ions with charge $-e$ pass to the left. What is the current in the solution?

Keshav Singh

Numerade Educator

Problem 18

The current in a $2.0 \mathrm{mm} \times 2.0 \mathrm{mm}$ square aluminum wire is 2.5 A. What are (a) the current density and (b) the electron drift speed?

Keshav Singh

Numerade Educator

Problem 19

A hollow copper wire with an inner diameter of $1.0 \mathrm{mm}$ and an outer diameter of 2.0 mm carries a current of 10 A. What is the current density in the wire?

Keshav Singh

Numerade Educator

Problem 20

What is the mean free time between collisions for electrons in an aluminum wire and in an iron wire?

Keshav Singh

Numerade Educator

Problem 21

What is the mean free time between collisions for electrons in silver and in gold?

Keshav Singh

Numerade Educator

Problem 22

The electric field in a $2.0 \mathrm{mm} \times 2.0 \mathrm{mm}$ square aluminum wire is $0.012 \mathrm{V} / \mathrm{m}$. What is the current in the wire?

Keshav Singh

Numerade Educator

Problem 23

What electric field strength is needed to create a 5.0 A current in a 2.0 -mm-diameter iron wire?

Keshav Singh

Numerade Educator

Problem 24

A 3.0 -mm-diameter wire carries a 12 A current when the electric field is $0.085 \mathrm{V} / \mathrm{m}$. What is the wire's resistivity?

Keshav Singh

Numerade Educator

Problem 25

A 0.0075 V/m electric field creates a 3.9 mA current in a 1.0-mm-diameter wire. What material is the wire made of?

Keshav Singh

Numerade Educator

Problem 26

A 0.50 -mm-diameter silver wire carries a 20 mA current. What are (a) the electric field and (b) the electron drift speed in the wire?

Keshav Singh

Numerade Educator

Problem 27

The two segments of the wire in FIGURE EX 31.27 have equal diameters but different conductivities $\sigma_{1}$ and $\sigma_{2} .$ Current $I$ passes through this wire. If the conductivities have the ratio $\sigma_{2} / \sigma_{1}=2,$ what is the ratio $E_{2} / E_{1}$ of the electric field strengths in the two segments of the wire? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 28

A metal cube $1.0 \mathrm{cm}$ on each side is sandwiched between two electrodes. The electrodes create a $0.0050 \mathrm{V} / \mathrm{m}$ electric field in the metal. A current of 9.0 A passes through the cube, from the positive electrode to the negative electrode. Identify the metal.

Keshav Singh

Numerade Educator

Problem 29

A $1.5 \mathrm{V}$ battery provides 0.50 A of current.
a. At what rate (C/s) is charge lifted by the charge escalator?
b. How much work does the charge escalator do to lift $1.0 \mathrm{C}$ of charge?
c. What is the power output of the charge escalator?

Keshav Singh

Numerade Educator

Problem 30

Wires 1 and 2 are made of the same metal. Wire 2 has twice the length and twice the diameter of wire $1 .$ What are the ratios (a) $\rho_{2} / \rho_{1}$ of the resistivities and (b) $R_{2} / R_{1}$ of the resistances of the two wires?

Keshav Singh

Numerade Educator

Problem 31

What is the resistance of
a. $A 1.0-m-$ long copper wire that is 0.50 mm in diameter?
b. A 10 -cm-long piece of carbon with a $1.0 \mathrm{mm} \times 1.0 \mathrm{mm}$ square cross section?

Keshav Singh

Numerade Educator

Problem 32

A 10 -m-long wire with a diameter of 0.80 mm has a resistance of $1.1 \Omega .$ Of what material is the wire made?

Keshav Singh

Numerade Educator

Problem 33

The electric field inside a 30 -cm-long copper wire is $5.0 \mathrm{mV} / \mathrm{m}$ What is the potential difference between the ends of the wire?

Keshav Singh

Numerade Educator

Problem 34

a. How long must a 0.60 -mm-diameter aluminum wire be to have a 0.50 A current when connected to the terminals of a $1.5 \mathrm{V}$ flashlight battery?
b. What is the current if the wire is half this length?

Keshav Singh

Numerade Educator

Problem 35

The terminals of a $0.70 \mathrm{V}$ watch battery are connected by a $100-\mathrm{m}$ -long gold wire with a diameter of $0.10 \mathrm{mm}$. What is the current in the wire?

Keshav Singh

Numerade Educator

Problem 36

Pencil "lead" is actually carbon. What is the resistance of the $0.70-\mathrm{mm}$ -diameter, $6.0-\mathrm{cm}$ -long lead from a mechanical pencil?

Keshav Singh

Numerade Educator

Problem 37

The resistance of a very fine aluminum wire with a $10 \mu \mathrm{m} \times$ $10 \mu \mathrm{m}$ square cross section is $1000 \Omega$.
a. How long is the wire?
b. A $1000 \Omega$ resistor is made by wrapping this wire in a spiral around a $3.0-$ mm-diameter glass core. How many turns of wire are needed?

Keshav Singh

Numerade Educator

Problem 38

FIGURE EX 31.38 is a current-versus-potential-difference graph for a material. What is the material's resistance? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 39

A circuit calls for a $0.50-$ mm-diameter copper wire to be stretched between two points. You don't have any copper wire, but you do have aluminum wire in a wide variety of diameters. What diameter aluminum wire will provide the same resistance?

Keshav Singh

Numerade Educator

Problem 40

For what electric field strength would the current in a 2.0 -mm-diameter nichrome wire be the same as the current in a 1.0 -mm-diameter aluminum wire in which the electric field strength is $0.0080 \mathrm{V} / \mathrm{m} ?$

Keshav Singh

Numerade Educator

Problem 41

The density of aluminum is $2700 \mathrm{kg} / \mathrm{m}^{3} .$ Verify the conduction-electron density for aluminum given in Table $31.1 .$ The mass of an aluminum atom is 27$\mathrm{u}$.

Mayukh Banik

Numerade Educator

Problem 42

The electron beam inside a television picture tube is $0.40 \mathrm{mm}$ in diameter and carries a current of $50 \mu$ A. This electron beam impinges on the inside of the picture tube screen.
a. How many electrons strike the screen each second?
b. What is the current density in the electron beam?
c. The electrons move with a velocity of $4.0 \times 10^{7} \mathrm{m} / \mathrm{s}$. What electric field strength is needed to accelerate electrons from rest to this velocity in a distance of $5.0 \mathrm{mm} ?$
d. Each electron transfers its kinetic energy to the picture tube screen upon impact. What is the power delivered to the screen by the electron beam?

Mayukh Banik

Numerade Educator

Problem 43

FIGURE P31.43 shows a $4.0-\mathrm{cm}-$wide plastic film being wrapped onto a $2.0-\mathrm{cm}$ - diameter roller that turns at 90 rpm. The plastic has a uniform surface charge density $-2.0 \mathrm{nC} / \mathrm{cm}^{2}$.
a. What is the current of the moving film?
b. How long does it take the roller to accumulate a charge of $-10 \mu C ?$
(FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 44

In a classical model of the hydrogen atom, the electron moves around the proton in a circular orbit of radius $0.053 \mathrm{nm}$.
a. What is the electron's orbital frequency?
b. What is the effective current of the electron?

Keshav Singh

Numerade Educator

Problem 45

A sculptor has asked you to help electroplate gold onto a brass statue. You know that the charge carriers in the ionic solution are gold ions, and you've calculated that you must deposit $0.50 \mathrm{g}$ of gold to reach the necessary thickness. How much current do you need, in $\mathrm{mA}$, to plate the statue in 3.0 hours?

Keshav Singh

Numerade Educator

Problem 46

The biochemistry that takes place inside cells depends on various elements, such as sodium, potassium, and calcium, that are dissolved in water as ions. These ions enter cells through narrow pores in the cell membrane known as ion channels. Bach ion channel, which is formed from a specialized protein molecule, is selective for one type of ion. Measurements with microelectrodes have shown that a 0.30 -nm-diameter potassium ion $\left(\mathrm{K}^{+}\right)$ channel carries a current of $1.8 \mathrm{pA}$.
a. How many potassium ions pass through if the ion channel opens for $1.0 \mathrm{ms} ?$
b. What is the current density in the ion channel?

Keshav Singh

Numerade Educator

Problem 47

The starter motor of a car engine draws a current of $150 \mathrm{A}$ from the battery. The copper wire to the motor is $5.0 \mathrm{mm}$ in diameter and $1.2 \mathrm{m}$ long. The starter motor runs for 0.80 s until the car engine starts.
a. How much charge passes through the starter motor?
b. How far does an electron travel along the wire while the starter motor is on?

Keshav Singh

Numerade Educator

Problem 48

A car battery is rated at 90 A hr, meaning that it can supply a 90 A current for 1 hr before being completely discharged. If you leave your headlights on until the battery is completely dead, how much charge leaves the battery?

Keshav Singh

Numerade Educator

Problem 49

What fraction of the current in a wire of radius $R$ flows in the part of the wire with radius $r \leq \frac{1}{2} R ?$

Keshav Singh

Numerade Educator

Problem 50

You need to design a 1.0 A fuse that "blows" if the current exceeds $1.0 \mathrm{A}$. The fuse material in your stockroom melts at a current density of $500 \mathrm{A} / \mathrm{cm}^{2} .$ What diameter wire of this material will do the job?

Keshav Singh

Numerade Educator

Problem 51

A hollow metal cylinder has inner radius $a$, outer radius $b$, length $L$, and conductivity $\sigma .$ The current $I$ is radially outward from the inner surface to the outer surface.
a. Find an expression for the electric field strength inside the metal as a function of the radius $r$ from the cylinder's axis.
b. Evaluate the electric field strength at the inner and outer surfaces of an iron cylinder if $a=1.0 \mathrm{cm}, b=2.5 \mathrm{cm}$ $L=10 \mathrm{cm},$ and $I=25 \mathrm{A}$.

Mayukh Banik

Numerade Educator

Problem 52

A hollow metal sphere has inner radius $a$, outer radius $b$, and conductivity $\sigma .$ The current $I$ is radially outward from the inner surface to the outer surface.
a. Find an expression for the electric field strength inside the metal as a function of the radius $r$ from the center.
b. Evaluate the clectric ficld strength at the inner and outer surfaces of a copper sphere if $a=1.0 \mathrm{cm}, b=2.5 \mathrm{cm},$ and $I=25 \mathrm{A}$.

Keshav Singh

Numerade Educator

Problem 53

The total amount of charge in coulombs that has entered a wire at time $t$ is given by the expression $Q=4 t-t^{2},$ where $t$ is in seconds and $t \geq 0$.
a. Graph $Q$ versus $t$ for the interval $0 \leq t \leq 4$ s.
b. Find an expression for the current in the wire at time $t$.
c. Graph $I$ versus $t$ for the interval $0 \leq t \leq 4$ s.
d. Explain why $I$ has the value at $t=2.0 \mathrm{s}$ that you observe.

Mayukh Banik

Numerade Educator

Problem 54

The total amount of charge that has entered a wire at time $t$ is given by the expression $Q=(20 \mathrm{C})\left(1-e^{-\mu(20 \mathrm{v})}\right),$ where $t$ is in seconds and $t \geq 0$.
a. Graph $Q$ versus $t$ for the interval $0 \leq t \leq 10$ s.
b. Find an expression for the current in the wire at time $t$.
c. What is the maximum value of the current?
d. Graph $I$ versus $t$ for the interval $0 \leq t \leq 10$ s.

Mayukh Banik

Numerade Educator

Problem 55

The current in a wire at time $t$ is given by the expression $I=(2.0 \mathrm{A}) e^{-t /(20 \mu \mathrm{a})},$ where $t$ is in microseconds and $t \geq 0$.
a. Graph $I$ versus $t$ for the interval $0 \leq t \leq 10 \mu \mathrm{s}$.
b. Find an expression for the total amount of charge (in coulombs) that has entered the wire at time $t .$ The initial conditions are $Q=0 \mathrm{Cat} t=0 \mu \mathrm{s}$.
c. Graph $Q$ versus $t$ for the interval $0 \leq t \leq 10 \mu \mathrm{s}$.

Mayukh Banik

Numerade Educator

Problem 56

The electric field in a current-carrying wire can be modeled as the electric field at the midpoint between two charged rings. Model a 3.0 -mm-diameter aluminum wire as two 3.0 mm-diameter rings $2.0 \mathrm{mm}$ apart. What is the current in the wire after 20 electrons are transferred from one ring to the other?

Mayukh Banik

Numerade Educator

Problem 57

The two wires in FIGURE P31.57 are made of the same material. What are the current and the electron drift speed in the 2.0 -mm-diameter segment of the wire? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 58

What is the electron drift speed at the 3.0 -mm-diameter end (the left end) of the wire in FIGURE P31.58? (FIGURE CAN'T COPY)

Mayukh Banik

Numerade Educator

Problem 59

What diameter should the nichrome wire in FIGURE P31.59 be in order for the electric field strength to be the same in both wires? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 60

An aluminum wire consists of the three segments shown in FIGURE P31.60. The current in the top segment is $10 \mathrm{A}$ For each of these three segments, find the
a. Current $I$.
b. Current density $J$.
c. Electric field $E$.
d. Drift velocity $v_{d}$.
e. Mean time between collisions $\tau$.
f. Electron current $i$.
Place your results in a table for easy viewing. (FIGURE CAN'T COPY)

Mayukh Banik

Numerade Educator

Problem 61

A 15 -cm-long nichrome wire is connected across the terminals of a 1.5 V battery.
a. What is the electric field inside the wire?
b. What is the current density inside the wire?
c. If the current in the wire is $2.0 \mathrm{A},$ what is the wire's diameter?

Vishal Gupta

Numerade Educator

Problem 62

A 20 -cm-long hollow nichrome tube of inner diameter $2.8 \mathrm{mm},$ outer diameter $3.0 \mathrm{mm}$ is connected to a $3.0 \mathrm{V}$ battery. What is the current in the tube?

Keshav Singh

Numerade Educator

Problem 63

A $1.5 \mathrm{V}$ flashlight battery is connected to a wire with a resistance of $3.0 \Omega .$ FIGURE P31.63 shows the battery's potential difference as a function of time. What is the total charge lifted by the charge escalator? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 64

Two 10 -cm-diameter metal plates are $1.0 \mathrm{cm}$ apart. They are charged to $\pm 12.5 \mathrm{nC} .$ They are suddenly connected together by a 0.224 -mm-diameter copper wire stretched taut from the center of one plate to the center of the other.
a. What is the maximum current in the wire?
b. What is the largest electric field in the wire?
c. Does the current increase with time, decrease with time, or remain steady? Explain.
d. What is the total amount of energy dissipated in the wire?

Mayukh Banik

Numerade Educator

Problem 65

A long, round wire has resistance $R .$ What will the wire's resistance be if you stretch it to twice its initial length?

Keshav Singh

Numerade Educator

Problem 66

FIGURE P31.66 shows the potential along a tungsten wire. What is the current density in the wire? (FIGURE CAN'T COPY)

Keshav Singh

Numerade Educator

Problem 67

Household wiring often uses 2.0 -mm-diameter copper wires. The wires can get rather long as they snake through the walls from the fuse box to the farthest corners of your house. What is the potential difference across a $20-\mathrm{m}$ -long, 2.0 -mm-diameter copper wire carrying an 8.0 A current?

Keshav Singh

Numerade Educator

Problem 68

You've decided to protect your house by placing a 5.0 -m-tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you've learned that lightning bolts can carry up to 50 kA of current and last up to $50 \mu$ s.
a. How much charge is delivered by a lightning bolt with these parameters?
b. You don't want the potential difference between the top and bottom of the lightning rod to exceed $100 \mathrm{V}$. What is the minimum diameter, in cm, the rod can be?

Keshav Singh

Numerade Educator

Problem 69

A 62 g hollow copper cylinder is $10 \mathrm{cm}$ long and has an inner diameter of $1.0 \mathrm{cm} .$ The current density along the length of the cylinder is $150,000 \mathrm{A} / \mathrm{m}^{2} .$ What is the current in the cylinder?

Mayukh Banik

Numerade Educator

Problem 70

The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is $I=(0.75 \mathrm{~A}) e^{-i /(6 \mathrm{br})}$ . What is the total number of electrons transported from the positive electrode to the negative electrode by the charge escalator from the time the battery is first used until it is completely dead?

Keshav Singh

Numerade Educator

Problem 71

Assume the conduction electrons in a metal can be treated as classical particles in an ideal gas.
a. What is the rms velocity of electrons in room-temperature copper?
b. How far, on average, does an electron move between collisions?

Mayukh Banik

Numerade Educator

Problem 72

A 5.0 -mm-diameter proton beam carries a total current of 1.5 mA. The current density in the proton beam, which increases with distance from the center, is given by $J=J_{edge}(r / R),$ where $R$ is the radius of the beam and $J_{edge}$ is the current density at the edge.
a. How many protons per second are delivered by this proton beam?
b. Determine the value of $J_{edge}$.

Keshav Singh

Numerade Educator

Problem 73

A metal wire connecting the terminals of a battery with potential difference $\Delta V_{\mathrm{bat}}$ gets warm as it draws a current $I$.
a. What is $\Delta U,$ the change in potential energy of charge $Q$ as it passes through the wire?
b. Where does this energy go?
c. Power is the rate of transfer of energy. Based on your answer to part $a,$ find an expression for the power supplied by the battery to warm the wire.
d. What power does a $1.5 \mathrm{V}$ battery supply to a wire drawing a 1.2 A current?

Keshav Singh

Numerade Educator

Problem 74

FIGURE CP31.74 shows a wire that is made of two equal-diameter segments with conductivities $\sigma_{1}$ and $\sigma_{2} .$ When current $I$ passes through the wire, a thin layer of charge appears at the boundary between the segments.
a. Find an expression for the surface charge density $\eta$ on the boundary. Give your result in terms of $I, \sigma_{1}, \sigma_{2},$ and the wire's cross-section area $A$.
b. $A 1.0$ -mm-diameter wire made of copper and iron segments carries a 5 A current. How much charge accumulates at the boundary between the segments?
(FIGURE CAN'T COPY)

Mayukh Banik

Numerade Educator