Solved: A steady beam of alpha particles (q = +2e)

Figure 26-15 shows cross sections through three long conductors of the same length and material, with square cross sections of edge lengths as shown. Conductor B fits snugly within conductor A, and conductor C fits snugly within conductor B. Rank the following according to their end-to-end resistances, greatest first: the individual conductors and the combinations of A + B (B inside A), B + C (C inside B), and A + B + C (B inside A inside C).

Figure 26-16 shows cross sections through three wires of identical length and material; the sides are given in millimeters. Rank the wires according to their resistance (measured end to end along each wire's length), greatest first.

Figure 26-17 shows a rectangular solid conductor of edge lengths L, 2L, and 3L. A potential difference V is to be applied uniformly between pairs of opposite faces of L the conductor as in Fig. 26-8b. First 2L 3L V is applied between the left-right Fig.26-17 Question 3. faces, then between the top-bottom faces, and then between the front - back faces. Rank those pairs, greatest first, according to the following (within the conductor): (a) the magnitude of the electric field, (b) the current density, (c) the current, and (d) the drift speed of the electrons.

Figure 26-18 shows plots of the current i through a certain cross section of a wire over four different time periods. Rank the periods according to the net charge that passes through the cross section during the period, greatest first.

Figure 26-19 shows four situations in which positive and negative charges move horizontally and gives the rate at which each charge moves. Rank the situations according to the effective current through the regions, greatest first.

In Fig. 26-20, a wire that carries a current consists of three sections with different radii. Rank the sections according to the following quantities, greatest first: (a) current, (b) magnitude of current density, and (c) magnitude of electric field.

Figure 26-21 gives the electric potential Vex) versus position x along a copper wire carrying current. The wire consists of three sections that differ in radius. Rank the three sections according to the magnitude of the (a) electric field and (b) current density, greatest first.

The following table gives the lengths of three copper rods, their diameters, and the potential differI A I B Fig. 26-21 c Question 7. ences between their ends. Rank the rods according to (a) the magnitude of the electric field within them, (b) the current density within them, and (c) the drift speed of electrons through them, grea test first.

Figure 26-22 gives the drift speed Vd V d of conduction electrons in a copI . per wire versus position x along the I wire. The wire consists of three sec- I I tions that differ in radius. Rank the A B C I three sections according to the following quantities, greatest first: (a) Fig.26-22 Question 9. radius, (b) number of conduction x electrons per cubic meter, (c) magnitude of electric field, (d) conductivity.

Three wires, of the same diameter, are connected in turn between two points maintained at a constant potential difference. Their resistivities and lengths are p and L (wire A), 1.2p and 1.2L (wire B), and 0.9p and L (wire C). Rank the wires according to the rate at which energy is transferred to thermal energy, greatest first.

What is the current in a wire of radius R = 3.40 mm if the magnitude of the current density is given by (a) la = 10rIR and (b) Ib = 10(1 - r/R), in which r is the radial distance and 10 = 5.50 X 10 4 A/m2 ? (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 cm-3, and their speed is 470 km/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?

How long does it take electrons to get from a car battery to the starting motor? Assume the current is 300 A and the electrons travel through a copper wire with cross-sectional area 0.21 cm2 and length 0.85 m. The number of charge carriers per unit volume is 8.49 X 1028 m-3.

A human being can be electrocuted if a current as small as 50 rnA 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 n, what might the fatal voltage be?

A coil is formed by winding 250 turns of insulated 16- gauge copper wire (diameter = 1.3 mm) in a single layer on a cylindrical form of radius 12 cm. What is the resistance of the coil? Neglect the thickness ofthe insulation. (Use Table 26-1.)

Copper and aluminum are being considered for a high-voltage transmission line that must carry a current of 60.0 A. The resistance per unit length is to be 0.150 Wkm. The densities of copper and aluminum are 8960 and 2600 kg/m3, respectively. Compute (a) the magnitude I of the current density and (b) the mass per unit length ..\ for a copper cable and (c) I and (d) ..\ for an aluminum cable.

A wire of Nichrome (a nickel-chromium-iron alloy commonly used in heating elements) is 1.0 m long and 1.0 mm2 in cross-sectional area. It carries a current of 4.0 A when a 2.0 V potential difference is applied between its ends. Calculate the conductivity (Tof Nichrome.

A wire 4.00 m long and 6.00 mm in diameter has a resistance of 15.0 mD. A potential difference of 23.0 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 26-1, identify the material.

What is the resistivity of a wire of 1.0 mm diameter, 2.0 m length, and 50 mD 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 A and 2.9 V (the values of the current and voltage under operating conditions). If the resistance of the tungsten bulb filament at room temperature (20C) is 1.1 D, what is the temperature of the filament when the bulb is on?

Kiting during a storm. 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 mm extends directly upward by 0.800 km and is coated with a 0.500 mm layer of water having resistivity 150 D . 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 500 000 A (way beyond just being lethal).

When 115 V is applied across a wire that is 10 m long and has a 0.30 mm radius, the magnitude of the current density is 1.4 X 104 A/m2 Find the resistivity of the wire.

Figure 26-24a 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 mm (Fig. 26-24b). The vertical scale is set by Es = 4.00 X 103 Vim. The rod consists of three sections of the same material but with different radii. (The schematic diagram of Fig. 26-24b does not indicate the different radii.) The radius of section 3 is 2.00 mm. What is the radius of (a) section 1 and (b) section 2?

A wire with a resistance of 6.0 D 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.

In Fig. 26-25a, a 9.00 V battery is connected to a resistive 8 x=o 6 11 : 4 ::- 2 x= .\'s 0 Xs (a) (b) ,,(mm) Fig. 26-25 Problem 26. PROBLEMS 701 strip that consists of three sections with the same cross-sectional areas but different conductivities. Figure 26-25b gives the electric potential Vex) versus position x along the strip. The horizontal scale is set by Xs = 8.00 mm. Section 3 has conductivity 3.00 X 107 (D mt I. What is the conductivity of section (a) 1 and (b) 2?

Two conductors are made of the same material and have the same length. Conductor A is a solid wire of diameter 1.0 mm. Conductor B is a hollow tube of outside diameter 2.0 mm and inside diameter 1.0 mm. What is the resistance ratio RAIRS' measured between their ends?

Figure 26-26 gives the electric potential Vex) along a copper wire carrying uniform current, from a point of higher potential ~ v,. = 12.0 /-tV at x = 0 to a point of ::- zero potential at Xs = 3.00 m. The wire has a radius of 2.00 mm. What is the current in the wire?

A potential difference of 3.00 n V is set up across a 2.00 cm Fig. 26-26 Problem 28. length of copper wire that has a radius of 2.00 mm. How much charge drifts through a cross section in 3.00 ms?

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 2116 (see the table in Problem 4). Knowing this, and knowing that 1000 ft of lO-gauge copper wire has a resistance of approximately 1.00 D, estimate the resistance of 25 ft of 22- gauge copper wire.

An electrical cable consists of 125 strands of fine wire, each having 2.65 /-tD 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?

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 Vim and the field is directed vertically down. This field causes singly charged positive ions, at a density of 620 cm-3, to drift downward and singly charged negative ions, at a density of 550 cm- 3, to drift upward (Fig. 26-27). The measured conductivity of the air in that region is 2.70 X 10-14 (D m)-I. Calculate (a) the magnitude of the current density and (b) the ion drift speed, assumed to be the same for positive and negative ions.

A block in the shape of a rectangular solid has a cross-sectional area of 3.50 cm2 across its width, a front-to-rear length of 15.8 cm, and a resistance of 935 D. The block's material contains 5.33 X 10 22 conduction electrons/m3 A potential difference of 35.8 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?

Figure 26-28 shows wire section 1 of diameter DJ = 4.00R and wire section 2 of diameter D2 = 2.00R, connected by a tapered (1) section. The wire is copper and car- Fig. 26-28 Problem 34. ries 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 m shown in section 2 is 10.0 fL V. The number of charge carriers per unit volume is 8.49 X 1028 m- 3. What is the drift speed of the conduction electrons in section 1 ?

In Fig. 26-29, current is set up through a truncated right circular cone of resistivity 731 D m, left radius a = 2.00 mm, right radius b = 2.30 mm, and length L = 1.94 cm. Assume that the current density is uniform across any cross section taken perpendicular to the length. What is the resistance of the cone?

Swimming during a storm. Figure 26-30 shows a swimmer at distance D = 35.0 m from a lightning strike to the water, with current 1= 78 kA. The water has resistivity 30 D . m, the width of the swimmer along a radial line from the strike is 0.70 m, and his resistance across that width is 4.00 kD. Assume that the current spreads through the water over a hemisphere centered on the strike point. What is the current through the swimmer?

Show that, according to the free-electron model of electrical conduction in metals and classical physics, the resistivity of metals should be proportional to {T, where T is the temperature in kelvins. (See Eq.19-31.)

In Fig. 26-31a, a 20 D resistor is connected to a battery. Figure 26-31b shows the increase of thermal energy Eth in the resistor as a function of time t. The vertical scale is set by Eth,s = 2.50 mJ, and the horizontal scale is set by ts = 4.0 s. What is the electric potential across the battery?

A certain brand of hot-dog cooker works by applying a poten- ~ tial difference of 120 V across op- J posite ends of a hot dog and allowing it to cook by means of the thermal energy produced. The current is 10.0 A, and the energy required to cook one hot dog is 60.0 o kJ. If the rate at which energy is Fig. 26-31 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 W when the current is 3.00 A. What is the resistance?

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

In Fig. 26-32, a battery of potential difference V = 12 V is connected to a resistive strip of resistance R = 6.0 D. 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 Fig. 26-32 Problem 42. transferred to the thermal energy of the strip by the electron?

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

A student kept his 9.0 V, 7.0 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 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 h?

A copper wire of cross-sectional area 2.00 X 10 -6 m2 and length 4.00 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 min?

A heating element is made by maintaining a potential difference of 75.0 V across the length of a Nichrome wire that has a 2.60 X 10-6 m2 cross section. Nichrome has a resistivity of 5.00 X 10-7 D m. (a) If the element dissipates 5000 W, what is its length? (b) If 100 V is used to obtain the same dissipation rate, what should the length be?

Exploding shoes. 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 kg/m3 and requires 2256 kJ/kg to be vaporized. If horizontal current lasts 2.00 ms and encounters water with resistivity 150 D . m, length 12.0 cm, and vertical cross-sectional area 15 X 10-5 m2, what average current is required to vaporize the water?

A 100 W lightbulb is plugged into a standard 120 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/kW h. (b) What is the resistance of the bulb? (c) What is the current in the bulb?

The current through the battery and resistors 1 and 2 in Fig. 26-33a is 2.00 A. Energy is transferred from the current to thermal energy Elh in both resistors. Curves 1 and 2 in Fig. 26-33b give that thermal energy Elh for resistors 1 and 2, respectively, as a function of time t. The vertical scale is set by Elh" = 40.0 mJ, and the horizontal scale is set by ts = 5.00 s. What is the power of the battery?

Wire C and wire D are made from different materials and have length Lc = LD = 1.0 m. The resistivity and diameter of wire Care 2.0 X 10-6 D . m and 1.00 mm, and those of wire Dare 1.0 X 10-6 D m and 123 ~Lc--~'I-'--LD~ Fig. 26-34 Problem 51. 0.50 mm. The wires are joined as shown in Fig. 26-34, and a current of 2.0 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?

The current-density magnitude in a certain circular wire is J = (2.75 X 10 10 A/m4)r2, where r is the radial distance out to the wire's radius of 3.00 mm. The potential applied to the wire (end to end) is 60.0 V. How much energy is converted to thermal energy in 1.00 h?

A 120 V potential difference is applied to a space heater that dissipates 500 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?

Figure 26-35a shows a rod of resistive material. The resistance per unit length of the rod in- (a) -' I--,\' (m) creases in the positive direction of 0 1.0 the x axis. At any position x along --II-- do\' row the rod, (differential) the resistance section dR of of a width nar- (b) ~ n~ dx is given by dR = 5.00x dx, (c) V n where dR is in ohms and x is in U meters. Figure 26-35b shows such a narrow section. You are to slice off Fig.26-35 Problem 54. PROBLEMS 703 a length of the rod between x = 0 and some position x = Land then connect that length to a battery with potential difference V = 5.0 V (Fig. 26-35c). You want the current in the length to transfer energy to thermal energy at the rate of 200 W. At what position x = L should you cut the rod?

A Nichrome heater dissipates 500 W when the applied potential difference is 110 V and the wire temperature is SOODe. What would be the dissipation rate if the wire temperature were held at 200DC by immersing the wire in a bath of cooling oil? The applied potential difference remains the same, and IX for Nichrome at SOODCis 4.0 X 1O-4 K-l.

A potential difference of 1.20 V will be applied to a 33.0 m length of IS-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 lS.0 W device has 9.00 V across it. How much charge goes through the device in 4.00 h?

An aluminum rod with a square cross section is 1.3 m long and 5.2 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 m if its resistance is to be the same as that of the aluminum rod?

A cylindrical metal rod is 1.60 m long and 5.50 mm in diameter. The resistance between its two ends (at 20DC) is 1.09 X 10-3 D. (a) What is the material? (b) A round disk, 2.00 cm in diameter and 1.00 mm thick, is formed of the same material. What is the resistance between the round faces, assuming that each face is an equipotential surface?

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 p. (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 cm, speed v = 2.0 mis, and charge density p = 1.1 X 10-3 C/m3 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 = iV. (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 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 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?

A steady beam of alpha particles (q = +2e) traveling with constant kinetic energy 20 Me V carries a current of 0.25 /-LA. (a) If the beam is directed perpendicular to a fiat surface, how many alpha particles strike the surface in 3.0 s? (b) At any instant, how many alpha particles are there in a given 20 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 Me V?

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

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

A cylindrical resistor of radius 5.0 mm and length 2.0 cm is made of material that has a resistivity of 3.5 X 10 -5 n 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 W?

A potential difference V is applied to a wire of cross-sectional area A, length L, and resistivity p. 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 Land (b) the ratio of the new cross-sectional area toA?

The headlights of a moving car require about 10 A from the 12 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 W heating unit is designed to operate with an applied potential difference of 115 V. (a) By what percentage will its heat output drop if the applied potential difference drops to 110 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)?

The copper windings of a motor have a resistance of 50 n at 20C when the motor is idle. After the motor has run for several hours, the resistance rises to 58 n. What is the temperature of the windings now? Ignore changes in the dimensions of the windings. (Use Table 26-1.)

How much electrical energy is transferred to thermal energy in 2.00 h by an electrical resistance of 400 n when the potential applied across it is 90.0 V?

A caterpillar of length 4.0 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 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 X 1028 m-3.)

(a) At what temperature would the resistance of a copper conductor be double its resistance at 20.0C? (Use 20.0C as the reference point in Eq. 26-17; compare your answer with Fig. 26-10.) (b) Does this same "doubling temperature" hold for all copper conductors, regardless of shape or size?

A steel trolley-car rail has a cross-sectional area of 56.0 cm2. What is the resistance of 10.0 km of rail? The resistivity of the steel is 3.00 X 10-7 n 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 V and the current through the coil is 5.2 A, the liquid evaporates at the steady rate of 21 mg/s. Calculate the heat of vaporization of the liquid (see Section 18-8).

The current density in a wire is uniform and has magnitude 2.0 X 106 A/m2, the wire's length is 5.0 m, and the density of conduction electrons is 8.49 X 1028 m-3. How long does an electron take (on the average) to travel the length of the wire?

A certain x-ray tube operates at a current of 7.00 rnA and a potential difference of 80.0 kV. What is its power in watts?

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 X 10 18 electrons and 1.1 X 10 18 protons move past a cross-sectional area of the tube each second? (b) Is the direction of the current density 7 toward or away from the negative terminal?