Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 26

Current and Resistance - all with Video Answers

Educators

Chapter Questions

Problem 1

During the $4.0 \mathrm{~min}$ a $5.0 \mathrm{~A}$ current is set up in a wire, how many (a) coulombs and (b) electrons pass through any cross section across the wire's width?

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

An isolated conducting sphere has a $10 \mathrm{~cm}$ radius. One wire carries a current of $1.0000020 \mathrm{~A}$ into it. Another wire carries a current of $1.0000000 \mathrm{~A}$ out of it. How long would it take for the sphere to increase in potential by $1000 \mathrm{~V}$ ?

Zachary Warner

Numerade Educator

Problem 3

A charged belt, $50 \mathrm{~cm}$ wide, travels at $30 \mathrm{~m} / \mathrm{s}$ between a source of charge and a sphere. The belt carries charge into the sphere at a rate corresponding to $100 \mu \mathrm{A}$. Compute the surface charge density on the belt.

Vishal Gupta

Numerade Educator

Problem 4

The (United States) National Electric Code, which sets maximum safe currents for insulated copper wires of various diameters, is given (in part) in the table. Plot the safe current density as a function of diameter. Which wire gauge has the maximum safe current density? ("Gauge" is a way of identifying wire diameters, and $1 \mathrm{mil}=10^{-3} \mathrm{in}$.)
$$
\begin{array}{lrrrrrrrr}
\hline \text { Gauge } & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\
\text { Diameter, mils } & 204 & 162 & 129 & 102 & 81 & 64 & 51 & 40 \\
\text { Safe current, A } & 70 & 50 & 35 & 25 & 20 & 15 & 6 & 3 \\
\hline
\end{array}
$$

Zachary Warner

Numerade Educator

Problem 5

A beam contains $2.0 \times 10^8$ doubly charged positive ions per cubic centimeter, all of which are moving north with a speed of $1.0 \times 10^5 \mathrm{~m} / \mathrm{s}$. What are the (a) magnitude and (b) direction of the current density $\vec{J}$ ? (c) What additional quantity do you need to calculate the total current $i$ in this ion beam?

Salamat Ali

Numerade Educator

Problem 6

A certain cylindrical wire carries current. We draw a circle of radius $r$ around its central axis in Fig. $26.10 a$ to determine the current $i$ within the circle. Figure $26.10 b$ shows current $i$ as a function of $r^2$. The vertical scale is set by $i_s=4.0 \mathrm{~mA}$, and the horizontal scale is set by $r_s^2=4.0 \mathrm{~mm}^2$. (a) Is the current density uniform? (b) If so, what is its magnitude?
FIGURE CANT COPY
Fig. 26.10

Vishal Gupta

Numerade Educator

Problem 7

A fuse in an electric circuit is a wire that is designed to melt, and thereby open the circuit, if the current exceeds a predetermined value. Suppose that the material to be used in a fuse melts when the current density rises to $440 \mathrm{~A} / \mathrm{cm}^2$. What diameter of cylindrical wire should be used to make a fuse that will limit the current to $0.50 \mathrm{~A}$ ?

Vishal Gupta

Numerade Educator

Problem 8

A small but measurable current of $1.2 \times 10^{-10} \mathrm{~A}$ exists in a copper wire whose diameter is $2.5 \mathrm{~mm}$. The number of charge carriers per unit volume is $8.49 \times 10^{28} \mathrm{~m}^{-3}$. Assuming the current is uniform, calculate the (a) current density and (b) electron drift speed.

Zachary Warner

Numerade Educator

Problem 9

The magnitude $J(r)$ of the current density in a certain cylindrical wire is given as a function of radial distance from the center of the wire's cross section as $J(r)=B r$, where $r$ is in meters, $J$ is in amperes per square meter, and $B=2.00 \times 10^5 \mathrm{~A} / \mathrm{m}^3$. This function applies out to the wire's radius of $2.00 \mathrm{~mm}$. How much current is contained within the width of a thin ring concentric with the wire if the ring has a radial width of $10.0 \mu \mathrm{m}$ and is at a radial distance of $1.20 \mathrm{~mm}$ ?

Salamat Ali

Numerade Educator

Problem 10

The magnitude $J$ of the current density in a certain lab wire with a circular cross section of radius $R=2.00 \mathrm{~mm}$ is given by $J=\left(3.00 \times 10^8\right) r^2$, with $J$ in amperes per square meter and radial distance $r$ in meters. What is the current through the outer section bounded by $r=0.900 R$ and $r=R$ ?

Zachary Warner

Numerade Educator

Problem 11

What is the current in a wire of radius $R=3.40 \mathrm{~mm}$ if the magnitude of the current density is given by (a) $J_a=J_0 r / R$ and (b) $J_b=J_0(1-r / R)$, in which $r$ is the radial distance and $J_0=5.50 \times 10^4 \mathrm{~A} / \mathrm{m}^2$ ? (c) Which function maximizes the current density near the wire's surface?

Salamat Ali

Numerade Educator

Problem 13

Near Earth, the density of protons in the solar wind (a stream of particles from the Sun) is $8.70 \mathrm{~cm}^{-3}$, and their speed is $470 \mathrm{~km} / \mathrm{s}$. (a) Find the current density of these protons. (b) If Earth's magnetic field did not deflect the protons, what total current would Earth receive?
$13 \mathbf{M}$ How long does it take electrons to get from a car battery to the starting motor? Assume the current is $300 \mathrm{~A}$ and the electrons travel through a copper wire with cross-sectional area $0.21 \mathrm{~cm}^2$ and length $0.85 \mathrm{~m}$. The number of charge carriers per unit volume is $8.49 \times 10^{28} \mathrm{~m}^{-3}$.

Vishal Gupta

Numerade Educator

Problem 14

A human being can be electrocuted if a current as small as $50 \mathrm{~mA}$ passes near the heart. An electrician working with sweaty hands makes good contact with the two conductors he is holding, one in each hand. If his resistance is $2000 \Omega$, what might the fatal voltage be?

Zachary Warner

Numerade Educator

Problem 15

A coil is formed by winding 250 turns of insulated 16-gauge copper wire $($ diameter $=1.3 \mathrm{~mm})$ in a single layer on a cylindrical form of radius $12 \mathrm{~cm}$. What is the resistance of the coil? Neglect the thickness of the insulation. (Use Table 26.3.1.)

Stanley Enemuo

Numerade Educator

Problem 16

Copper and aluminum are being considered for a high-voltage transmission line that must carry a current of $60.0 \mathrm{~A}$. The resistance per unit length is to be $0.150 \Omega / \mathrm{km}$. The densities of copper and aluminum are 8960 and $2600 \mathrm{~kg} / \mathrm{m}^3$, respectively. Compute (a) the magnitude $J$ of the current density and (b) the mass per unit length $\lambda$ for a copper cable and (c) $J$ and (d) $\lambda$ for an aluminum cable.

Zachary Warner

Numerade Educator

Problem 17

A wire of Nichrome (a nickel-chromium-iron alloy commonly used in heating elements) is $1.0 \mathrm{~m}$ long and $1.0 \mathrm{~mm}^2$ in cross-sectional area. It carries a current of $4.0 \mathrm{~A}$ when a $2.0 \mathrm{~V}$ potential difference is applied between its ends. Calculate the conductivity $\sigma$ of Nichrome.

Salamat Ali

Numerade Educator

Problem 18

A wire $4.00 \mathrm{~m}$ long and $6.00 \mathrm{~mm}$ in diameter has a resistance of $15.0 \mathrm{~m} \Omega$. A potential difference of $23.0 \mathrm{~V}$ is applied between the ends. (a) What is the current in the wire? (b) What is the magnitude of the current density? (c) Calculate the resistivity of the wire material. (d) Using Table 26.3.1, identify the material.

Zachary Warner

Numerade Educator

Problem 19

What is the resistivity of a wire of $1.0 \mathrm{~mm}$ diameter, $2.0 \mathrm{~m}$ length, and $50 \mathrm{~m} \Omega$ resistance?

Salamat Ali

Numerade Educator

Problem 20

A certain wire has a resistance $R$. What is the resistance of a second wire, made of the same material, that is half as long and has half the diameter?

Vishal Gupta

Numerade Educator

Problem 21

A common flashlight bulb is rated at $0.30 \mathrm{~A}$ and $2.9 \mathrm{~V}$ (the values of the current and voltage under operating conditions). If the resistance of the tungsten bulb filament at room temperature $\left(20^{\circ} \mathrm{C}\right)$ is $1.1 \Omega$, what is the temperature of the filament when the bulb is on?

Vishal Gupta

Numerade Educator

Problem 22

The legend that Benjamin Franklin flew a kite as a storm approached is only a legend-he was neither stupid nor suicidal. Suppose a kite string of radius $2.00 \mathrm{~mm}$ extends directly upward by $0.800 \mathrm{~km}$ and is coated with a $0.500 \mathrm{~mm}$ layer of water having resistivity $150 \Omega \cdot \mathrm{m}$. If the potential difference between the two ends of the string is 160 MV, what is the current through the water layer? The danger is not this current but the chance that the string draws a lightning strike, which can have a current as large as $500000 \mathrm{~A}$ (way beyond just being lethal).

Alex Garger

Numerade Educator

Problem 23

When $115 \mathrm{~V}$ is applied across a wire that is $10 \mathrm{~m}$ long and has a $0.30 \mathrm{~mm}$ radius, the magnitude of the current density is $1.4 \times 10^8 \mathrm{~A} / \mathrm{m}^2$. Find the resistivity of the wire.

Vishal Gupta

Numerade Educator

Problem 24

Figure 26.11a gives the magnitude $E(x)$ of the electric fields that have been set up by a battery along a resistive rod of length $9.00 \mathrm{~mm}$ (Fig. 26.11b). The vertical scale is set by $E_s=4.00 \times 10^3 \mathrm{~V} / \mathrm{m}$. The rod consists of three sections of the same material but with different radii. (The schematic diagram of Fig. $26.11 b$ does not indicate the different radii.) The radius of section 3 is $2.00 \mathrm{~mm}$. What is the radius of (a) section 1 and (b) section 2?
FIGURE CANT COPY
Fig. 26.11

Vishal Gupta

Numerade Educator

Problem 25

A wire with a resistance of $6.0 \Omega$ is drawn out through a die so that its new length is three times its original length. Find the resistance of the longer wire, assuming that the resistivity and density of the material are unchanged.

Salamat Ali

Numerade Educator

Problem 26

In Fig. $26.12 a$, a $9.00 \mathrm{~V}$ battery is connected to a resistive strip that consists of three sections with the same cross-sectional areas but different conductivities. Figure $26.12 b$ gives the electric potential $V(x)$ versus position $x$ along the strip. The horizontal scale is set by $x_s=8.00 \mathrm{~mm}$. Section 3 has conductivity $3.00 \times$ $10^7(\Omega \cdot \mathrm{m})^{-1}$. What is the conductivity of section (a) 1 and (b) 2 ?
FIGURE CANT COPY
Fig. 26.12

Zachary Warner

Numerade Educator

Problem 27

Two conductors are made of the same material and have the same length. Conductor $A$ is a solid wire of diameter $1.0 \mathrm{~mm}$. Conductor $B$ is a hollow tube of outside diameter $2.0 \mathrm{~mm}$ and inside diameter $1.0 \mathrm{~mm}$. What is the resistance ratio $R_A / R_B$, measured between their ends?

Salamat Ali

Numerade Educator

Problem 28

Figure 26.13 gives the electric potential $V(x)$ along a copper wire carrying uniform current, from a point of higher potential $V_s=12.0 \mu \mathrm{V}$ at $x=0$ to a point of zero potential at $x_s=3.00 \mathrm{~m}$. The wire has a radius of $2.00 \mathrm{~mm}$. What is the current in the wire?
FIGURE CANT COPY
Fig. 26.13

Sunita Kumari

Numerade Educator

Problem 29

A potential difference of $3.00 \mathrm{nV}$ is set up across a $2.00 \mathrm{~cm}$ length of copper wire that has a radius of $2.00 \mathrm{~mm}$. How much charge drifts through a cross section in $3.00 \mathrm{~ms}$ ?

Salamat Ali

Numerade Educator

Problem 30

If the gauge number of a wire is increased by 6 , the diameter is halved; if a gauge number is increased by 1 , the diameter decreases by the factor $2^{1 / 6}$ (see the table in Problem 4). Knowing this, and knowing that $1000 \mathrm{ft}$ of 10 -gauge copper wire has a resistance of approximately $1.00 \Omega$, estimate the resistance of $25 \mathrm{ft}$ of 22-gauge copper wire.

Vishal Gupta

Numerade Educator

Problem 31

An electrical cable consists of 125 strands of fine wire, each having $2.65 \mu \Omega$ resistance. The same potential difference is applied between the ends of all the strands and results in a total current of $0.750 \mathrm{~A}$. (a) What is the current in each strand? (b) What is the applied potential difference? (c) What is the resistance of the cable?

Salamat Ali

Numerade Educator

Problem 32

Earth's lower atmosphere contains negative and positive ions that are produced by radioactive elements in the soil and cosmic rays from space. In a certain region, the atmospheric electric field strength is $120 \mathrm{~V} / \mathrm{m}$ and the field is directed vertically down. This field causes singly charged positive ions, at a density of $620 \mathrm{~cm}^{-3}$, to drift downward and singly charged negative ions, at a density of $550 \mathrm{~cm}^{-3}$, to drift upward (Fig. 26.14). The measured conductivity of the air in that region is $2.70 \times 10^{-14}(\Omega \cdot \mathrm{m})^{-1}$. Calculate (a) the magnitude of the current density and (b) the ion drift speed, assumed to be the same for positive and negative ions.
FIGURE CANT COPY
Fig. 26.14

Zachary Warner

Numerade Educator

Problem 33

A block in the shape of a rectangular solid has a crosssectional area of $3.50 \mathrm{~cm}^2$ across its width, a front-to-rear length of $15.8 \mathrm{~cm}$, and a resistance of $935 \Omega$. The block's material contains $5.33 \times 10^{22}$ conduction electrons $/ \mathrm{m}^3$. A potential difference of $35.8 \mathrm{~V}$ is maintained between its front and rear faces. (a) What is the current in the block? (b) If the current density is uniform, what is its magnitude? What are (c) the drift velocity of the conduction electrons and (d) the magnitude of the electric field in the block?

Vishal Gupta

Numerade Educator

Problem 34

Figure 26.15 shows wire section 1 of diameter $D_1=4.00 R$ and wire section 2 of diameter $D_2=$ $2.00 R$, connected by a tapered section. The wire is copper and carries a current. Assume that the current is uniformly distributed across any cross-sectional area through the wire's width. The electric potential change $V$ along the length $L=2.00 \mathrm{~m}$ shown in section 2 is $10.0 \mu \mathrm{V}$. The number of charge carriers per unit volume is $8.49 \times 10^{28} \mathrm{~m}^{-3}$. What is the drift speed of the conduction electrons in section 1 ?
FIGURE CANT COPY
Fig. 26.15

Vishal Gupta

Numerade Educator

Problem 35

In Fig. 26.16, current is set up through a truncated right circular cone of resistivity $731 \Omega \cdot \mathrm{m}$, left radius $a=2.00 \mathrm{~mm}$, right radius $b=2.30 \mathrm{~mm}$, and length $L=1.94 \mathrm{~cm}$. Assume that the current density is uniform across any cross section taken perpendicular to the length. What is the resistance of the cone?
FIGURE CANT COPY
Fig. 26.16

GA

Gabriel A

Numerade Educator

Problem 36

Figure 26.17 shows a swimmer at distance $D=35.0 \mathrm{~m}$ from a lightning strike to the water, with current $I=78 \mathrm{kA}$. The water has resistivity $30 \Omega \cdot \mathrm{m}$, the width of the swimmer along a radial line from the strike is $0.70 \mathrm{~m}$, and his resistance across that width is $4.00 \mathrm{k} \Omega$. Assume that the current spreads through the water over a hemisphere centered on the strike point. What is the current through the swimmer?
FIGURE CANT COPY
Fig. 26.17

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 37

Show that, according to the free-electron model of electrical conduction in metals and classical physics, the resistivity of metals should be proportional to $\sqrt{T}$, where $T$ is the temperature in kelvins. (See Eq. 19.6.5.)

Salamat Ali

Numerade Educator

Problem 38

In Fig. 26.18a, a $20 \Omega$ resistor is connected to a battery. Figure $26.18 \mathrm{~b}$ shows the increase of thermal energy $E_{\text {th }}$ in the resistor as a function of time $t$. The vertical scale is set by $E_{\text {th }, s}=$ $2.50 \mathrm{~mJ}$, and the horizontal scale is set by $t_s=4.0 \mathrm{~s}$. What is the electric potential across the battery?
FIGURE CANT COPY
Fig. 26.18

Sunita Kumari

Numerade Educator

Problem 39

A certain brand of hot-dog cooker works by applying a potential difference of $120 \mathrm{~V}$ across opposite ends of a hot dog and allowing it to cook by means of the thermal energy produced. The current is $10.0 \mathrm{~A}$, and the energy required to cook one hot dog is $60.0 \mathrm{~kJ}$. If the rate at which energy is supplied is unchanged, how long will it take to cook three hot dogs simultaneously?

Salamat Ali

Numerade Educator

Problem 40

Thermal energy is produced in a resistor at a rate of $100 \mathrm{~W}$ when the current is $3.00 \mathrm{~A}$. What is the resistance?

Zachary Warner

Numerade Educator

Problem 41

A $120 \mathrm{~V}$ potential difference is applied to a space heater whose resistance is $14 \Omega$ when hot. (a) At what rate is electrical energy transferred to thermal energy? (b) What is the cost for $5.0 \mathrm{~h}$ at US $$\$ 0.05 / \mathrm{kW} \cdot \mathrm{h}$$ ?

Vishal Gupta

Numerade Educator

Problem 42

In Fig. 26.19, a battery of potential difference $V=12 \mathrm{~V}$ is connected to a resistive strip of resistance $R=6.0 \Omega$. When an electron moves through the strip from one end to the other, (a) in which direction in the figure does the electron move, (b) how much work is done on the electron by the electric field in the strip, and (c) how much energy is transferred to the thermal energy of the strip by the electron?
FIGURE CANT COPY
Fig. 26.19

Sunita Kumari

Numerade Educator

Problem 43

An unknown resistor is connected between the terminals of a $3.00 \mathrm{~V}$ battery. Energy is dissipated in the resistor at the rate of $0.540 \mathrm{~W}$. The same resistor is then connected between the terminals of a $1.50 \mathrm{~V}$ battery. At what rate is energy now dissipated?

Salamat Ali

Numerade Educator

Problem 44

A student kept his $9.0 \mathrm{~V}, 7.0 \mathrm{~W}$ radio turned on at full volume from 9:00 p.M. until 2:00 A.M. How much charge went through it?

Zachary Warner

Numerade Educator

Problem 45

A $1250 \mathrm{~W}$ radiant heater is constructed to operate at $115 \mathrm{~V}$. (a) What is the current in the heater when the unit is operating? (b) What is the resistance of the heating coil? (c) How much thermal energy is produced in $1.0 \mathrm{~h}$ ?

Salamat Ali

Numerade Educator

Problem 46

A copper wire of cross-sectional area $2.00 \times 10^{-6} \mathrm{~m}^2$ and length $4.00 \mathrm{~m}$ has a current of $2.00 \mathrm{~A}$ uniformly distributed across that area. (a) What is the magnitude of the electric field along the wire? (b) How much electrical energy is transferred to thermal energy in $30 \mathrm{~min}$ ?

Zachary Warner

Numerade Educator

Problem 47

A heating element is made by maintaining a potential difference of $75.0 \mathrm{~V}$ across the length of a Nichrome wire that has a $2.60 \times 10^{-6} \mathrm{~m}^2$ cross section. Nichrome has a resistivity of $5.00 \times 10^{-7} \Omega \cdot \mathrm{m}$. (a) If the element dissipates $5000 \mathrm{~W}$, what is its length? (b) If $100 \mathrm{~V}$ is used to obtain the same dissipation rate, what should the length be?

Salamat Ali

Numerade Educator

Problem 48

The rain-soaked shoes of a person may explode if ground current from nearby lightning vaporizes the water. The sudden conversion of water to water vapor causes a dramatic expansion that can rip apart shoes. Water has density $1000 \mathrm{~kg} / \mathrm{m}^3$ and requires $2256 \mathrm{~kJ} / \mathrm{kg}$ to be vaporized. If horizontal current lasts $2.00 \mathrm{~ms}$ and encounters water with resistivity $150 \Omega \cdot \mathrm{m}$, length $12.0 \mathrm{~cm}$, and vertical cross-sectional area $15 \times 10^{-5} \mathrm{~m}^2$, what average current is required to vaporize the water?

Sunita Kumari

Numerade Educator

Problem 49

A $100 \mathrm{~W}$ lightbulb is plugged into a standard $120 \mathrm{~V}$ outlet. (a) How much does it cost per 31-day month to leave the light turned on continuously? Assume electrical energy costs US $$\$ 0.06 / \mathrm{kW} \cdot \mathrm{h}$$. (b) What is the resistance of the bulb? (c) What is the current in the bulb?

Vishal Gupta

Numerade Educator

Problem 50

The current through the battery and resistors 1 and 2 in Fig. $26.20 a$ is $2.00 \mathrm{~A}$. Energy is transferred from the current to thermal energy $E_{\mathrm{th}}$ in both resistors. Curves 1 and 2 in Fig. $26.20 b$ give that thermal energy $E_{\text {th }}$ for resistors 1 and 2, respectively, as a function of time $t$. The vertical scale is set by $E_{\mathrm{th}, s}=40.0 \mathrm{~mJ}$, and the horizontal scale is set by $t_s=5.00 \mathrm{~s}$. What is the power of the battery?
FIGURE CANT COPY
Fig. 26.20

Zachary Warner

Numerade Educator

Problem 51

Wire $C$ and wire $D$ are made from different materials and have length $L_C=L_D=$ $1.0 \mathrm{~m}$. The resistivity and diameter of wire $C$ are $2.0 \times 10^{-6} \Omega \cdot \mathrm{m}$ and $1.00 \mathrm{~mm}$, and those of wire $D$ are $1.0 \times 10^{-6} \Omega \cdot \mathrm{m}$ and $0.50 \mathrm{~mm}$. The wires are joined as shown in Fig. 26.21, and a current of $2.0 \mathrm{~A}$ is set up in them. What is the electric potential difference between (a) points 1 and 2 and (b) points 2 and 3? What is the rate at which energy is dissipated between (c) points 1 and 2 and (d) points 2 and 3 ?
FIGURE CANT COPY
Fig. 26.21

Vishal Gupta

Numerade Educator

Problem 52

The current-density magnitude in a certain circular wire is $J=\left(2.75 \times 10^{10} \mathrm{~A} / \mathrm{m}^4\right) r^2$, where $r$ is the radial distance out to the wire's radius of $3.00 \mathrm{~mm}$. The potential applied to the wire (end to end) is $60.0 \mathrm{~V}$. How much energy is converted to thermal energy in $1.00 \mathrm{~h}$ ?

Zachary Warner

Numerade Educator

Problem 53

A $120 \mathrm{~V}$ potential difference is applied to a space heater that dissipates $500 \mathrm{~W}$ during operation. (a) What is its resistance during operation? (b) At what rate do electrons flow through any cross section of the heater element?

Salamat Ali

Numerade Educator

Problem 54

Figure $26.22 a$ shows a rod of resistive material. The resistance per unit length of the rod increases in the positive direction of the $x$ axis. At any position $x$ along the rod, the resistance $d R$ of a narrow (differential) section of width $d x$ is given by $d R=5.00 x d x$, where $d R$ is in ohms and $x$ is in meters. Figure $26.22 b$ shows such a narrow section. You are to slice off a length of the rod between $x=0$ and some position $x=L$ and then connect that length to a battery with potential difference $V=5.0 \mathrm{~V}$ (Fig. 26.22c). You want the current in the length to transfer energy to thermal energy at the rate of $200 \mathrm{~W}$. At what position $x=L$ should you cut the rod?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 55

A Nichrome heater dissipates $500 \mathrm{~W}$ when the applied potential difference is $110 \mathrm{~V}$ and the wire temperature is $800^{\circ} \mathrm{C}$. What would be the dissipation rate if the wire temperature were held at $200^{\circ} \mathrm{C}$ by immersing the wire in a bath of cooling oil? The applied potential difference remains the same, and $\alpha$ for Nichrome at $800^{\circ} \mathrm{C}$ is $4.0 \times 10^{-4} \mathrm{~K}^{-1}$.

Salamat Ali

Numerade Educator

Problem 56

A potential difference of $1.20 \mathrm{~V}$ will be applied to a $33.0 \mathrm{~m}$ length of 18 -gauge copper wire (diameter $=0.0400 \mathrm{in}$.). Calculate (a) the current, (b) the magnitude of the current density, (c) the magnitude of the electric field within the wire, and (d) the rate at which thermal energy will appear in the wire.

Zachary Warner

Numerade Educator

Problem 57

An $18.0 \mathrm{~W}$ device has $9.00 \mathrm{~V}$ across it. How much charge goes through the device in $4.00 \mathrm{~h}$ ?

Vishal Gupta

Numerade Educator

Problem 58

An aluminum rod with a square cross section is $1.3 \mathrm{~m}$ long and $5.2 \mathrm{~mm}$ on edge. (a) What is the resistance between its ends? (b) What must be the diameter of a cylindrical copper rod of length $1.3 \mathrm{~m}$ if its resistance is to be the same as that of the aluminum rod?

Zachary Warner

Numerade Educator

Problem 59

A cylindrical metal rod is $1.60 \mathrm{~m}$ long and $5.50 \mathrm{~mm}$ in diameter. The resistance between its two ends (at $20^{\circ} \mathrm{C}$ ) is $1.09 \times 10^{-3} \Omega$. (a) What is the material? (b) A round disk, $2.00 \mathrm{~cm}$ in diameter and $1.00 \mathrm{~mm}$ thick, is formed of the same material. What is the resistance between the round faces, assuming that each face is an equipotential surface?

Salamat Ali

Numerade Educator

Problem 60

The chocolate crumb mystery. This story begins with Problem 60 in Chapter 23 and continues through Chapters 24 and 25 . The chocolate crumb powder moved to the silo through a pipe of radius $R$ with uniform speed $v$ and uniform charge density $\rho$. (a) Find an expression for the current $i$ (the rate at which charge on the powder moved) through a perpendicular cross section of the pipe. (b) Evaluate $i$ for the conditions at the factory: pipe radius $R=5.0 \mathrm{~cm}$, speed $v=2.0 \mathrm{~m} / \mathrm{s}$, and charge density $\rho=1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^3$.
If the powder were to flow through a change $V$ in electric potential, its energy could be transferred to a spark at the rate $P=i V$. (c) Could there be such a transfer within the pipe due to the radial potential difference discussed in Problem 70 of Chapter 24 ?
As the powder flowed from the pipe into the silo, the electric potential of the powder changed. The magnitude of that change was at least equal to the radial potential difference within the pipe (as evaluated in Problem 70 of Chapter 24). (d) Assuming that value for the potential difference and using the current found in (b) above, find the rate at which energy could have been transferred from the powder to a spark as the powder exited the pipe. (e) If a spark did occur at the exit and lasted for $0.20 \mathrm{~s}$ (a reasonable expectation), how much energy would have been transferred to the spark? Recall from Problem 60 in Chapter 23 that a minimum energy transfer of $150 \mathrm{~mJ}$ is needed to cause an explosion. (f) Where did the powder explosion most likely occur: in the powder cloud at the unloading bin (Problem 60 of Chapter 25), within the pipe, or at the exit of the pipe into the silo?

Zachary Warner

Numerade Educator

Problem 61

A steady beam of alpha particles $(q=+2 e)$ traveling with constant kinetic energy $20 \mathrm{MeV}$ carries a current of $0.25 \mu \mathrm{A}$. (a) If the beam is directed perpendicular to a flat surface, how many alpha particles strike the surface in $3.0 \mathrm{~s}$ ? (b) At any instant, how many alpha particles are there in a given $20 \mathrm{~cm}$ length of the beam? (c) Through what potential difference is it necessary to accelerate each alpha particle from rest to bring it to an energy of $20 \mathrm{MeV}$ ?

Salamat Ali

Numerade Educator

Problem 62

A resistor with a potential difference of $200 \mathrm{~V}$ across it transfers electrical energy to thermal energy at the rate of $3000 \mathrm{~W}$. What is the resistance of the resistor?

Zachary Warner

Numerade Educator

Problem 63

A $2.0 \mathrm{~kW}$ heater element from a dryer has a length of $80 \mathrm{~cm}$. If a $10 \mathrm{~cm}$ section is removed, what power is used by the now shortened element at $120 \mathrm{~V}$ ?

Prabhat Tyagi

Numerade Educator

Problem 64

A cylindrical resistor of radius $5.0 \mathrm{~mm}$ and length $2.0 \mathrm{~cm}$ is made of material that has a resistivity of $3.5 \times 10^{-5} \Omega \cdot \mathrm{m}$. What are (a) the magnitude of the current density and (b) the potential difference when the energy dissipation rate in the resistor is $1.0 \mathrm{~W}$ ?

Vishal Gupta

Numerade Educator

Problem 65

A potential difference $V$ is applied to a wire of crosssectional area $A$, length $L$, and resistivity $\rho$. You want to change the applied potential difference and stretch the wire so that the energy dissipation rate is multiplied by 30.0 and the current is multiplied by 4.00. Assuming the wire's density does not change, what are (a) the ratio of the new length to $L$ and (b) the ratio of the new cross-sectional area to $A$ ?

Salamat Ali

Numerade Educator

Problem 66

The headlights of a moving car require about $10 \mathrm{~A}$ from the $12 \mathrm{~V}$ alternator, which is driven by the engine. Assume the alternator is $80 \%$ efficient (its output electrical power is $80 \%$ of its input mechanical power), and calculate the horsepower the engine must supply to run the lights.

Zachary Warner

Numerade Educator

Problem 67

A $500 \mathrm{~W}$ heating unit is designed to operate with an applied potential difference of $115 \mathrm{~V}$. (a) By what percentage will its heat output drop if the applied potential difference drops to $110 \mathrm{~V}$ ? Assume no change in resistance. (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than that calculated in (a)?

Salamat Ali

Numerade Educator

Problem 68

The copper windings of a motor have a resistance of $50 \Omega$ at $20^{\circ} \mathrm{C}$ when the motor is idle. After the motor has run for several hours, the resistance rises to $58 \Omega$. What is the temperature of the windings now? Ignore changes in the dimensions of the windings. (Use Table 26.3.1.)

Vishal Gupta

Numerade Educator

Problem 69

How much electrical energy is transferred to thermal energy in $2.00 \mathrm{~h}$ by an electrical resistance of $400 \Omega$ when the potential applied across it is $90.0 \mathrm{~V}$ ?

Sunita Kumari

Numerade Educator

Problem 70

A caterpillar of length $4.0 \mathrm{~cm}$ crawls in the direction of electron drift along a $5.2-\mathrm{mm}$-diameter bare copper wire that carries a uniform current of $12 \mathrm{~A}$. (a) What is the potential difference between the two ends of the caterpillar? (b) Is its tail positive or negative relative to its head? (c) How much time does the caterpillar take to crawl $1.0 \mathrm{~cm}$ if it crawls at the drift speed of the electrons in the wire? (The number of charge carriers per unit volume is $8.49 \times 10^{28} \mathrm{~m}^{-3}$.)

Zachary Warner

Numerade Educator

Problem 71

(a) At what temperature would the resistance of a copper conductor be double its resistance at $20.0^{\circ} \mathrm{C}$ ? (Use $20.0^{\circ} \mathrm{C}$ as the reference point in Eq. 26.3.10; compare your answer with Fig. 26.3.4.) (b) Does this same "doubling temperature" hold for all copper conductors, regardless of shape or size?

Salamat Ali

Numerade Educator

Problem 72

A steel trolley-car rail has a cross-sectional area of $56.0 \mathrm{~cm}^2$. What is the resistance of $10.0 \mathrm{~km}$ of rail? The resistivity of the steel is $3.00 \times 10^{-7} \Omega \cdot \mathrm{m}$.

Zachary Warner

Numerade Educator

Problem 73

A coil of current-carrying Nichrome wire is immersed in a liquid. (Nichrome is a nickel-chromium-iron alloy commonly used in heating elements.) When the potential difference across the coil is $12 \mathrm{~V}$ and the current through the coil is $5.2 \mathrm{~A}$, the liquid evaporates at the steady rate of $21 \mathrm{mg} / \mathrm{s}$. Calculate the heat of vaporization of the liquid (see Module 18.4).

Salamat Ali

Numerade Educator

Problem 74

(c) The current density in a wire is uniform and has magnitude $2.0 \times 10^6 \mathrm{~A} / \mathrm{m}^2$, the wire's length is $5.0 \mathrm{~m}$, and the density of conduction electrons is $8.49 \times 10^{28} \mathrm{~m}^{-3}$. How long does an electron take (on the average) to travel the length of the wire?

Vishal Gupta

Numerade Educator

Problem 75

A certain $x$-ray tube operates at a current of $7.00 \mathrm{~mA}$ and a potential difference of $80.0 \mathrm{kV}$. What is its power in watts?

Salamat Ali

Numerade Educator

Problem 76

A current is established in a gas discharge tube when a sufficiently high potential difference is applied across the two electrodes in the tube. The gas ionizes; electrons move toward the positive terminal and singly charged positive ions toward the negative terminal. (a) What is the current in a hydrogen discharge tube in which $3.1 \times 10^{18}$ electrons and $1.1 \times 10^{18}$ protons move past a cross-sectional area of the tube each second? (b) Is the direction of the current density $\vec{J}$ toward or away from the negative terminal?

Zachary Warner

Numerade Educator

Problem 77

Two drift speeds. One end of an aluminum wire with diameter $2.5 \mathrm{~mm}$ is welded to one end of a copper wire with diameter $1.8 \mathrm{~mm}$. The composite carries a steady current $i$ of $1.3 \mathrm{~A}$. For points that are not next to the junction, what is the current density in (a) the aluminum wire and (b) the copper wire? (c) What is the magnitude of the electric field in the copper?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 78

A strip of silicon has width $w=3.2 \mathrm{~mm}$ and thickness $t=250 \mu \mathrm{m}$ and carries current $i=5.2 \mathrm{~mA}$. The silicon is said to be an n-type semiconductor because it has been "doped" with a controlled phosphorus impurity. The process greatly increased $n$, the number of charge carriers per unit volume, to a value of $1.5 \times 10^{23} \mathrm{~m}^{-3}$. What are (a) the current density, (b) the drift speed, and (c) the electric field magnitude in the strip?

Chai Santi

Numerade Educator