Chapter 26

During the \(4.0 \mathrm{~min}\) a 5.0 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?

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 A out of it. How long would it take for the sphere to increase in potential by \(1000 \mathrm{~V} ?\)

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 \(100 \mu\) A. Compute the surface charge density on the belt.

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 \(1 \mathrm{mil}=10^{-3}\) in. \()\) $$ \begin{array}{lrrrrrrrr} \hline \text { Gauge } & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ \text { Diameter, mils } & 204 & 162 & 129 & 102 & 81 & 64 & 51 & 40 \\ \text { Safe current, A } & 70 & 50 & 35 & 25 & 20 & 15 & 6 & 3 \\ \hline \end{array} $$

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?

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 \(0.50 \mathrm{~A} ?\)

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.

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 \(1.20 \mathrm{~mm} ?\)

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 \(r=R ?\)

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?

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?

A human being can be electrocuted if a current as small as 50 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?

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 A when a \(2.0 \mathrm{~V}\) potential difference is applied between its ends. Calculate the conductivity \(\sigma\) of Nichrome.

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

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?

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?

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 \(160 \mathrm{MV},\) 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 A (way beyond just being lethal).

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.

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.

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?

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 \(3.00 \mathrm{~ms} ?\)

An electrical cable cons?sts of 125 strands of tine 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 A. (a) What is the current in each strand? (b) What is the applied potential difference? (c) What is the resistance of the cable?

A block in the shape of a rectangular solid has a cross-sectional 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?

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?

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

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 \(\mathrm{US} \$ 0.05 / \mathrm{k} \mathrm{W} \cdot \mathrm{h} ?\)

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?

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?

A 1250 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 \(1.0 \mathrm{~h} ?\)

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 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 \(30 \mathrm{~min} ?\)

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?

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?

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?

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 \(1.00 \mathrm{~h} ?\)

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?

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 \(4.0 \times 10^{-4} \mathrm{~K}^{-1}\)

A potential difference of \(1.20 \mathrm{~V}\) will be applied to a \(33.0 \mathrm{~m}\) length of 18 -gauge copper wire (diameter \(=0.0400\) 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.

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

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 \(20 \mathrm{MeV} ?\)

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?

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 \(120 \mathrm{~V} ?\)

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 \(1.0 \mathrm{~W} ?\)

A potential difference \(V\) is applied to a wire of cross-sectional 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 \(A ?\)

The headlights of a moving car require about 10 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.

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)?

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 \(90.0 \mathrm{~V} ?\)

A caterpillar of length \(4.0 \mathrm{~cm}\) crawls in the direction of electron drift along a 5.2-mm-diameter bare copper wire that carries a uniform current of 12 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 \(8.49 \times 10^{28} \mathrm{~m}^{-3}\).

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 \(3.00 \times 10^{-7} \Omega \cdot \mathrm{m}\)

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 \(18-4\) ).

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?