Review & Summary. Current An electric current i in a conductor is defined by. Drift Speed of the Charge Carriers When an electric field

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1 evew & Summary Current An electrc current n a conductor s defned by dq (26-1) dt. Here dq s the amount of (postve) charge that passes n tme dt through a hypothetcal surface that cuts across the conductor. By conventon, the drecton of electrc current s taken as the drecton n whch postve charge carrers would move. The SI unt of electrc current s the ampere (A) 1 A 1 C/s. Current Densty Current (a scalar) s related to current densty J (a vector) by J da, (26-4) where da s a vector perpendcular to a surface element of area da and the ntegral s taken over any surface cuttng across the conductor. J has the same drecton as the velocty of the movng charges f they are postve and the opposte drecton f they are negatve. Drft Speed of the Charge Carrers When an electrc feld E s establshed n a conductor, the charge carrers (assumed postve) acqure a drft speed v d n the drecton of E ; the velocty v d s related to the current densty by J (ne)v d, (26-7) where ne s the carrer charge densty. esstance of a Conductor s defned as The resstance of a conductor V (defnton of ), (26-8) where V s the potental dfference across the conductor and s the current.the SI unt of resstance s the ohm ( ) 1 1V/A. Smlar equatons defne the resstvty r and conductvty s of a materal r 1 s E J (defntons of r and s), (26-12, 26-1)

2 764 CHAPTE 26 CUENT AND ESISTANCE where E s the magntude of the appled electrc feld. The SI unt of resstvty s the ohm-meter ( m). Equaton 26-1 corresponds to the vector equaton E r J. (26-11) The resstance of a conductng wre of length L and unform cross secton s r L (26-16) A, where A s the cross-sectonal area. Change of r wth Temperature The resstvty r for most materals changes wth temperature. For many materals, ncludng metals, the relaton between r and temperature T s approxmated by the equaton r r r a(t T ). (26-17) Here T s a reference temperature, r s the resstvty at T, and a s the temperature coeffcent of resstvty for the materal. Ohm s Law A gven devce (conductor, resstor, or any other electrcal devce) obeys Ohm s law f ts resstance, defned by Eq as V/, s ndependent of the appled potental dfference V. A gven materal obeys Ohm s law f ts resstvty, defned by Eq. 26-1, s ndependent of the magntude and drecton of the appled electrc feld E. esstvty of a Metal By assumng that the conducton electrons n a metal are free to move lke the molecules of a gas, t s possble to derve an expresson for the resstvty of a metal r m (26-22) e 2 nt. Here n s the number of free electrons per unt volume and t s the mean tme between the collsons of an electron wth the atoms of the metal. We can explan why metals obey Ohm s law by pontng out that t s essentally ndependent of the magntude E of any electrc feld appled to a metal. Power The power P, or rate of energy transfer, n an electrcal devce across whch a potental dfference V s mantaned s P V (rate of electrcal energy transfer). (26-26) esstve Dsspaton If the devce s a resstor, we can wrte Eq as P 2 V 2 (resstve dsspaton). (26-27, 26-28) In a resstor, electrc potental energy s converted to nternal thermal energy va collsons between charge carrers and atoms. Semconductors Semconductors are materals that have few conducton electrons but can become conductors when they are doped wth other atoms that contrbute charge carrers. Superconductors Superconductors are materals that lose all electrcal resstance at low temperatures. Some materals are superconductng at surprsngly hgh temperatures. Questons 1 Fgure shows cross sectons through three long conductors of the same length and materal, wth square cross sectons of edge lengths as shown. Conductor B fts snugly wthn conductor A, and conductor C fts snugly wthn conductor B. ank the followng accordng to ther end-to-end resstances, greatest frst the ndvdual conductors and the combnatons of A B (B nsde A), B C (C nsde B), and A B C (B nsde A nsde C). A l B 2 l Fgure Queston 1. 2 Fgure shows cross sectons through three wres of dentcal length and materal; the sdes are gven n mllmeters. ank the wres accordng to ther resstance (measured end to end along each wre s length), greatest frst. 2 5 Fgure Queston 2. 3 (c) C l 6 3 Fgure shows a rectangular 3L sold conductor of edge lengths L,2L, 2L and 3L. A potental dfference V s to be appled unformly between pars of opposte faces of the conductor as n L Fg. 26-8b. (The potental dfference s appled between the entre face on Fgure Queston 3. one sde and the entre face on the other sde.) Frst V s appled between the left rght faces, then between the top bottom faces, and then between the front back faces. ank those pars, greatest frst, accordng to the followng (wthn the conductor) the magntude of the electrc feld, the current densty, (c) the current, and (d) the drft speed of the electrons. 4 Fgure shows plots of the current through a certan cross secton of a wre over four dfferent tme perods. ank the perods accordng to the net charge that passes through the cross secton durng the perod, greatest frst. a b Fgure Queston 4. c d t

3 POBLEMS Fgure shows four stuatons n whch postve and negatve charges move horzontally and gves the rate at whch each charge moves. ank the stuatons accordng to the effectve current through the regons, greatest frst. 7 C/s 3 C/s 2 C/s 6 C/s 4 C/s 5 C/s 1 C/s (c) (d) Fgure Queston 5. 6 In Fg. 26-2, a wre that carres a current conssts of three sectons wth dfferent rad. ank the sectons accordng to the followng quanttes, greatest frst current, magntude of current densty, and (c) magntude of electrc feld. 7 Fgure gves the electrc potental V(x) versus poston x along a copper wre carryng current. The wre conssts of three sectons that dffer n radus. ank the three sectons accordng to the magntude of the electrc feld and current densty, greatest frst A B C Fgure 26-2 Queston 6. 8 The followng table gves the Fgure Queston 7. lengths of three copper rods, ther dameters, and the potental dfferences between ther ends. ank the rods accordng to the magntude of the electrc feld wthn them, the current densty wthn them, and (c) the drft speed of electrons through them, greatest frst. V A B C x od Length Dameter Potental Dfference 1 L 3d V 2 2L d 2V 3 3L 2d 2V 9 Fgure gves the drft speed v d of conducton electrons n a copper wre versus poston x along the wre. The wre conssts of three sectons that dffer n radus. ank the three A B C x quanttes, greatest frst radus, sectons accordng to the followng Fgure Queston number of conducton electrons per cubc meter, (c) magntude of electrc feld, (d) conductvty. 1 Three wres, of the same dameter, are connected n turn between two ponts mantaned at a constant potental dfference. Ther resstvtes and lengths are r and L (wre A), 1.2r and 1.2L (wre B), and.9r and L (wre C). ank the wres accordng to the rate at whch energy s transferred to thermal energy wthn them, greatest frst. 11 Fgure gves, for three a wres of radus, the current densty J(r) versus radus r, as meas- c b J ured from the center of a crcular cross secton through the wre. The wres are all made from the same r materal. ank the wres accordng Fgure Queston 11. to the magntude of the electrc feld at the center, halfway to the surface, and (c) at the surface, greatest frst. v d Problems SSM Tutorng problem avalable (at nstructor s dscreton) n WleyPLUS and WebAssgn Worked-out soluton avalable n Student Solutons Manual WWW Worked-out soluton s at Number of dots ndcates level of problem dffculty ILW Interactve soluton s at Addtonal nformaton avalable n The Flyng Crcus of Physcs and at flyngcrcusofphyscs.com http// Module 26-1 Electrc Current 1 Durng the 4. mn a 5. A current s set up n a wre, how many coulombs and electrons pass through any cross secton across the wre s wdth? 2 An solated conductng sphere has a 1 cm radus. One wre carres a current of 1. 2 A nto t. Another wre carres a current of 1. A out of t. How long would t take for the sphere to ncrease n potental by 1 V? 3 A charged belt, 5 cm wde, travels at 3 m/s between a source of charge and a sphere.the belt carres charge nto the sphere at a rate correspondng to 1 ma. Compute the surface charge densty on the belt. Module 26-2 Current Densty 4 The (Unted States) Natonal Electrc Code, whch sets maxmum safe currents for nsulated copper wres of varous dameters, s gven (n part) n the table. Plot the safe current densty as a functon of dameter. Whch wre gauge has the maxmum safe current densty? ( Gauge s a way of dentfyng wre dameters, and 1 ml 1 3 n.) Gauge Dameter, mls Safe current, A SSM WWW A beam contans doubly charged postve ons per cubc centmeter, all of whch are movng north wth a speed of m/s.what are the magntude and drecton of the current densty J? (c) What addtonal quantty do you need to calculate the total current n ths on beam? 6 A certan cylndrcal wre carres current. We draw s a crcle of radus r around ts central axs n Fg a to r determne the current wthn the crcle. Fgure 26- r 24b shows current as a functon (mm 2 ) r 2 s of r 2. The vertcal scale s set by s 4. ma, and the Fgure Problem 6. (ma)

4 766 CHAPTE 26 CUENT AND ESISTANCE horzontal scale s set by r 2 4. mm 2. Is the current densty unform? 17 A wre of Nchrome (a nckel chromum ron alloy com- s If so, what s ts magntude? monly used n heatng elements) s 1. m long and 1. mm 2 n cross-sectonal area. It carres a current of 4. A when a 2. V 7 A fuse n an electrc crcut s a wre that s desgned to melt, potental dfference s appled between ts ends. Calculate the and thereby open the crcut, f the current exceeds a predetermned value. Suppose that the materal to be used n a fuse melts conductvty s of Nchrome. when the current densty rses to 44 A/cm 2. What dameter of 18 A wre 4. m long and 6. mm n dameter has a resstance cylndrcal wre should be used to make a fuse that wll lmt the current to.5 A? of 15. m.a potental dfference of 23. V s appled between the ends. What s the current n the wre? What s the magntude of the current densty? (c) Calculate the resstvty of the wre materal. (d) Usng Table 26-1, dentfy the materal. 8 A small but measurable current of A exsts n a copper wre whose dameter s 2.5 mm. The number of charge carrers per unt volume s m 3. Assumng the current s unform, 19 SSM What s the resstvty of a wre of 1. mm dameter, 2. m calculate the current densty and electron drft speed. length, and 5 m resstance? 9 The magntude J(r) of the current densty n a certan cylndrcal wre s gven as a functon of radal dstance from the center second wre, made of the same materal, that s half as long and has 2 A certan wre has a resstance. What s the resstance of a of the wre s cross secton as J(r) Br, where r s n meters, J s n half the dameter? amperes per square meter, and B A/m 3.Ths functon 21 ILW A common flashlght bulb s rated at.3 A and 2.9 V apples out to the wre s radus of 2. mm. How much current s contaned wthn the wdth of a thn rng concentrc wth the wre f the rng has a radal wdth of 1. mm and s at a radal dstance of 1.2 mm? (the values of the current and voltage under operatng condtons). If the resstance of the tungsten bulb flament at room temperature (2 C) s 1.1, what s the temperature of the flament when the bulb s on? 1 The magntude J of the current densty n a certan lab 22 Ktng durng a storm. The legend that Benjamn wre wth a crcular cross secton of radus 2. mm s gven by J ( )r 2, wth J n amperes per square meter and radal dstance r n meters. What s the current through the outer secton bounded by r.9 and r? 11 What s the current n a wre of radus 3.4 mm f the magntude of the current densty s gven by J a J r/ and J b J (1 r/), n whch r s the radal dstance and J A/m 2? (c) Whch functon maxmzes the current Frankln flew a kte as a storm approached s only a legend he was nether stupd nor sucdal. Suppose a kte strng of radus 2. mm extends drectly upward by.8 km and s coated wth a.5 mm layer of water havng resstvty 15 m. If the potental dfference between the two ends of the strng s 16 MV, what s the current through the water layer? The danger s not ths current but the chance that the strng draws a lghtnng strke, whch can have a current as large as 5 A (way beyond just beng lethal). densty near the wre s surface? 23 When 115 V s appled across a wre that s 1 m long and 12 Near Earth, the densty of protons n the solar wnd (a stream of partcles from the Sun) s 8.7 cm 3, and ther speed has a.3 mm radus, the magntude of the current densty s A/m 2. Fnd the resstvty of the wre. s 47 km/s. Fnd the current densty of these protons. If 24 Fgure 26-25a gves the magntude E(x) of the electrc Earth s magnetc feld dd not deflect the protons, what total current would Earth receve? length 9. mm (Fg b).The vertcal scale s set by E s 4. felds that have been set up by a battery along a resstve rod of 13 ILW How long does t take electrons to get from a car battery to the startng motor? Assume the current s 3 A and the electrons travel through a copper wre wth cross-sectonal area.21 cm 2 and length.85 m. The number of charge carrers per unt 1 3 V/m. The rod conssts of three sectons of the same materal but wth dfferent rad. (The schematc dagram of Fg b does not ndcate the dfferent rad.) The radus of secton 3 s 2. mm. What s the radus of secton 1 and secton 2? volume s m 3. Module 26-3 esstance and esstvty E s 14 A human beng can be electrocuted f a current as x = small as 5 ma passes near the heart. An electrcan workng wth sweaty hands makes good contact wth the two conductors V he s holdng, one n each hand. If hs resstance s 2, what x = 9 mm mght the fatal voltage be? SSM A col s formed by wndng 25 turns of nsulated x (mm) 16-gauge copper wre (dameter 1.3 mm) n a sngle layer on a cylndrcal form of radus 12 cm. What s the resstance Fgure Problem 24. of the col? Neglect the thckness of the nsulaton. (Use Table 26-1.) 16 Copper and alumnum are beng consdered for a 25 SSM ILW A wre wth a resstance of 6. s drawn out hgh-voltage transmsson lne that must carry a current of 6. A. through a de so that ts new length s three tmes ts orgnal The resstance per unt length s to be.15 /km. The denstes of length. Fnd the resstance of the longer wre, assumng that the resstvty and densty of the materal are unchanged. copper and alumnum are 896 and 26 kg/m 3, respectvely. Compute the magntude J of the current densty and the 26 In Fg a, a 9. V battery s connected to a resstve mass per unt length l for a copper cable and (c) J and (d) l for an alumnum cable. strp that conssts of three sectons wth the same cross-sectonal areas but dfferent conductvtes. Fgure 26-26b gves the electrc E (1 3 V/m)

5 POBLEMS 767 potental V(x) versus poston x along the strp. The horzontal scale s set by x s 8. mm. Secton 3 has conductvty ( m) 1.What s the conductvty of secton 1 and 2? V x = x = x s V (V) Fgure Problem 26. E Fgure Problem x (mm) 27 SSM WWW Two conductors are made of the same materal and have the same length. Conductor A s a sold wre of dameter 1. mm. Conductor B s a hollow tube of outsde dameter 2. mm and nsde dameter 1. mm. What s the resstance rato A / B, measured between ther ends? 28 Fgure gves the V s electrc potental V(x) along a copper wre carryng unform current, from a pont of hgher potental V s 12. mv at x to a pont of zero potental at x s 3. m. The wre has a radus of 2. mm. What s the current n the wre? x s x (m) 29 A potental dfference of Fgure Problem nv s set up across a 2. cm length of copper wre that has a radus of 2. mm. How much charge drfts through a cross secton n 3. ms? 3 If the gauge number of a wre s ncreased by 6, the dameter s halved; f a gauge number s ncreased by 1, the dameter decreases by the factor 2 1/6 (see the table n Problem 4). Knowng ths, and knowng that 1 ft of 1-gauge copper wre has a resstance of approxmately 1., estmate the resstance of 25 ft of 22-gauge copper wre. 31 An electrcal cable conssts of 125 strands of fne wre, each havng 2.65 m resstance. The same potental dfference s appled between the ends of all the strands and results n a total current of.75 A. What s the current n each strand? What s the appled potental dfference? (c) What s the resstance of the cable? 32 Earth s lower atmosphere contans negatve and postve ons that are produced by radoactve elements n the sol and cosmc rays from space. In a certan regon, the atmospherc V ( μ V) 3 x s electrc feld strength s 12 V/m and the feld s drected vertcally down.ths feld causes sngly charged postve ons, at a densty of 62 cm 3, to drft downward and sngly charged negatve ons, at a densty of 55 cm 3, to drft upward (Fg ). The measured conductvty of the ar n that regon s ( m) 1. Calculate the magntude of the current densty and the on drft speed, assumed to be the same for postve and negatve ons. 33 A block n the shape of a rectangular sold has a crosssectonal area of 3.5 cm 2 across ts wdth, a front-to-rear length of 15.8 cm, and a resstance of 935. The block s materal contans conducton electrons/m 3. A potental dfference of 35.8 V s mantaned between ts front and rear faces. What s the current n the block? If the current densty s unform, what s ts magntude? What are (c) the drft velocty of the conducton electrons and (d) the magntude of the electrc feld n the L block? D 1 D 2 34 Fgure shows wre (2) secton 1 of dameter D 1 4. (1) Fgure Problem 34. and wre secton 2 of dameter D 2 2., connected by a tapered secton. The wre s copper and carres a current. Assume that the current s unformly dstrbuted across any cross-sectonal area through the wre s wdth. The electrc potental change V along the length L 2. m shown n secton 2 s 1. mv. The number of charge carrers per unt volume s m 3. What s the drft speed of the conducton electrons n secton 1? 35 In Fg. 26-3, current s set up through a truncated rght crcular cone of resstvty 731 m, left radus a 2. mm, rght radus b 2.3 mm, and length L 1.94 cm. Assume that the current densty s unform across any cross secton taken perpendcular to the length.what s the resstance of the cone? a L Fgure 26-3 Problem Swmmng durng a storm. Fgure shows a swmmer at dstance D 35. m from a lghtnng strke to the water, wth current I 78 ka. The water has Δr resstvty 3 m, the wdth of the swmmer along a radal lne from the strke s.7 m, and hs resstance across that wdth s 4. k. D Assume that the current spreads Fgure Problem 36. through the water over a hemsphere centered on the strke pont. What s the current through the swmmer? Module 26-4 Ohm s Law 37 Show that, accordng to the free-electron model of electrcal conducton n metals and classcal physcs, the resstvty of metals should be proportonal to 1T, where T s the temperature n kelvns. (See Eq ) b

6 768 CHAPTE 26 CUENT AND ESISTANCE Module 26-5 Power, Semconductors, Superconductors 38 In Fg a, a 2 resstor s connected to a battery. Fgure 26-32b shows the ncrease of thermal energy E th n the resstor as a functon of tme t. The vertcal scale s set by E th,s 2.5 mj, and the horzontal scale s set by t s 4. s. What s the electrc potental across the battery? Fgure Problem A certan brand of hot-dog cooker works by applyng a potental dfference of 12 V across opposte ends of a hot dog and allowng t to cook by means of the thermal energy produced. The current s 1. A, and the energy requred to cook one hot dog s 6. kj. If the rate at whch energy s suppled s unchanged, how long wll t take to cook three hot dogs smultaneously? 4 Thermal energy s produced n a resstor at a rate of 1 W when the current s 3. A. What s the resstance? 41 SSM A 12 V potental dfference s appled to a space heater whose resstance s 14 when hot. At what rate s electrcal energy transferred to thermal energy? What s the cost for 5. h at US$.5/kW h? 42 In Fg , a battery of potental dfference V 12 V s connected to a resstve strp of resstance 6.. When an electron moves through the strp from one end to the other, n whch drecton n the fgure does the electron move, how much work s done on the electron by the electrc E th (mj) E th,s t s t (s) feld n the strp, and (c) how much energy s transferred to the thermal energy of the strp by the electron? 43 ILW An unknown resstor s connected between the termnals of a 3. V battery. Energy s dsspated n the resstor at the rate of.54 W. The same resstor s then connected between the termnals of a 1.5 V battery. At what rate s energy now dsspated? 44 A student kept hs 9. V, 7. W rado turned on at full volume from 9 P.M. untl 2 A.M. How much charge went through t? 45 SSM ILW A 125 W radant heater s constructed to operate at 115 V. What s the current n the heater when the unt s operatng? What s the resstance of the heatng col? (c) How much thermal energy s produced n 1. h? 46 A copper wre of cross-sectonal area m 2 and length 4. m has a current of 2. A unformly dstrbuted across that area. What s the magntude of the electrc feld along the wre? How much electrcal energy s transferred to thermal energy n 3 mn? 47 A heatng element s made by mantanng a potental dfference of 75. V across the length of a Nchrome wre that V Fgure Problem 42. has a m 2 cross secton. Nchrome has a resstvty of m. If the element dsspates 5 W, what s ts length? If 1 V s used to obtan the same dsspaton rate, what should the length be? 48 Explodng shoes. The ran-soaked shoes of a person may explode f ground current from nearby lghtnng vaporzes the water. The sudden converson of water to water vapor causes a dramatc expanson that can rp apart shoes. Water has densty 1 kg/m 3 and requres 2256 kj/kg to be vaporzed. If horzontal current lasts 2. ms and encounters water wth resstvty 15 m, length 12. cm, and vertcal cross-sectonal area m 2, what average current s requred to vaporze the water? 49 A 1 W lghtbulb s plugged nto a standard 12 V outlet. How much does t cost per 31-day month to leave the lght turned on contnuously? Assume electrcal energy costs US$.6/kW h. What s the resstance of the bulb? (c) What s the current n the bulb? 5 The current through the battery and resstors 1 and 2 n Fg a s 2. A. Energy s transferred from the current to thermal energy E th n both resstors. Curves 1 and 2 n Fg b gve that thermal energy E th for resstors 1 and 2, respectvely, as a functon of tme t. The vertcal scale s set by E th,s 4. mj, and the horzontal scale s set by t s 5. s. What s the power of the battery? 1 2 E th (mj) E th,s Fgure Problem 5. t (s) 51 SSM WWW Wre C and C D wre D are made from dfferent materals and have length L C L D 1. m. The resstvty and dameter of wre C are C L D L m and 1. mm, and those of wre D are Fgure Problem 51. m and.5 mm. The wres are joned as shown n Fg , and a current of 2. A s set up n them. What s the electrc potental dfference between ponts 1 and 2 and ponts 2 and 3? What s the rate at whch energy s dsspated between (c) ponts 1 and 2 and (d) ponts 2 and 3? 52 The current-densty magntude n a certan crcular wre s J ( A/m 4 )r 2, where r s the radal dstance out to the wre s radus of 3. mm. The potental appled to the wre (end to end) s 6. V. How much energy s converted to thermal energy n 1. h? 53 A 12 V potental dfference s appled to a space heater that dsspates 5 W durng operaton. What s ts resstance durng operaton? At what rate do electrons flow through any cross secton of the heater element? 1 2 t s

7 POBLEMS Fgure 26-36a shows a rod of resstve materal. The resstance per unt length of the rod ncreases dx n the postve drecton of the x axs. At any poston x along the rod, the 1. x (m) resstance d of a narrow (dfferental) secton of wdth dx s gven by (c) V d 5.xdx, where d s n ohms and x s n meters. Fgure 26-36b Fgure Problem 54. shows such a narrow secton. You are to slce off a length of the rod between x and some poston x Land then connect that length to a battery wth potental dfference V 5. V (Fg c). You want the current n the length to transfer energy to thermal energy at the rate of 2 W. At what poston x L should you cut the rod? Addtonal Problems 55 SSM A Nchrome heater dsspates 5 W when the appled potental dfference s 11 V and the wre temperature s 88C. What would be the dsspaton rate f the wre temperature were held at 28C by mmersng the wre n a bath of coolng ol? The appled potental dfference remans the same, and a for Nchrome at 88C s K A potental dfference of 1.2 V wll be appled to a 33. m length of 18-gauge copper wre (dameter.4 n.). Calculate the current, the magntude of the current densty, (c) the magntude of the electrc feld wthn the wre, and (d) the rate at whch thermal energy wll appear n the wre. 57 An 18. W devce has 9. V across t. How much charge goes through the devce n 4. h? 58 An alumnum rod wth a square cross secton s 1.3 m long and 5.2 mm on edge. What s the resstance between ts ends? What must be the dameter of a cylndrcal copper rod of length 1.3 m f ts resstance s to be the same as that of the alumnum rod? 59 A cylndrcal metal rod s 1.6 m long and 5.5 mm n dameter. The resstance between ts two ends (at 2 C) s What s the materal? A round dsk, 2. cm n dameter and 1. mm thck, s formed of the same materal. What s the resstance between the round faces, assumng that each face s an equpotental surface? 6 The chocolate crumb mystery. Ths story begns wth Problem 6 n Chapter 23 and contnues through Chapters 24 and 25.The chocolate crumb powder moved to the slo through a ppe of radus wth unform speed v and unform charge densty r. Fnd an expresson for the current (the rate at whch charge on the powder moved) through a perpendcular cross secton of the ppe. Evaluate for the condtons at the factory ppe radus 5. cm, speed v 2. m/s, and charge densty r C/m 3. If the powder were to flow through a change V n electrc potental, ts energy could be transferred to a spark at the rate P V. (c) Could there be such a transfer wthn the ppe due to the radal potental dfference dscussed n Problem 7 of Chapter 24? As the powder flowed from the ppe nto the slo, the electrc potental of the powder changed. The magntude of that change was at least equal to the radal potental dfference wthn the ppe (as evaluated n Problem 7 of Chapter 24). (d) Assumng that value for the potental dfference and usng the current found n above, fnd the rate at whch energy could have been transferred from the powder to a spark as the powder exted the ppe. (e) If a spark dd occur at the ext and lasted for.2 s (a reasonable expectaton), how much energy would have been transferred to the spark? ecall from Problem 6 n Chapter 23 that a mnmum energy transfer of 15 mj s needed to cause an exploson. (f) Where dd the powder exploson most lkely occur n the powder cloud at the unloadng bn (Problem 6 of Chapter 25), wthn the ppe, or at the ext of the ppe nto the slo? 61 SSM A steady beam of alpha partcles (q 2e) travelng wth constant knetc energy 2 MeV carres a current of.25 ma. If the beam s drected perpendcular to a flat surface, how many alpha partcles strke the surface n 3. s? At any nstant, how many alpha partcles are there n a gven 2 cm length of the beam? (c) Through what potental dfference s t necessary to accelerate each alpha partcle from rest to brng t to an energy of 2 MeV? 62 A resstor wth a potental dfference of 2 V across t transfers electrcal energy to thermal energy at the rate of 3 W. What s the resstance of the resstor? 63 A 2. kw heater element from a dryer has a length of 8 cm. If a 1 cm secton s removed, what power s used by the now shortened element at 12 V? 64 A cylndrcal resstor of radus 5. mm and length 2. cm s made of materal that has a resstvty of m.what are the magntude of the current densty and the potental dfference when the energy dsspaton rate n the resstor s 1. W? 65 A potental dfference V s appled to a wre of cross-sectonal area A, length L, and resstvty r. You want to change the appled potental dfference and stretch the wre so that the energy dsspaton rate s multpled by 3. and the current s multpled by 4.. Assumng the wre s densty does not change, what are the rato of the new length to L and the rato of the new cross-sectonal area to A? 66 The headlghts of a movng car requre about 1 A from the 12 V alternator, whch s drven by the engne. Assume the alternator s 8% effcent (ts output electrcal power s 8% of ts nput mechancal power), and calculate the horsepower the engne must supply to run the lghts. 67 A 5 W heatng unt s desgned to operate wth an appled potental dfference of 115 V. By what percentage wll ts heat output drop f the appled potental dfference drops to 11 V? Assume no change n resstance. If you took the varaton of resstance wth temperature nto account, would the actual drop n heat output be larger or smaller than that calculated n? 68 The copper wndngs of a motor have a resstance of 5 at 2 C when the motor s dle. After the motor has run for several hours, the resstance rses to 58. What s the temperature of the wndngs now? Ignore changes n the dmensons of the wndngs. (Use Table 26-1.) 69 How much electrcal energy s transferred to thermal energy n 2. h by an electrcal resstance of 4 when the potental appled across t s 9. V? 7 A caterpllar of length 4. cm crawls n the drecton of electron drft along a 5.2-mm-dameter bare copper wre that carres a unform current of 12 A. What s the potental dfference between the two ends of the caterpllar? Is ts tal postve or negatve relatve to ts head? (c) How much tme does the caterpllar take to crawl 1. cm f t crawls at the drft speed of the electrons n the wre? (The number of charge carrers per unt volume s m 3.) 71 SSM At what temperature would the resstance of a copper conductor be double ts resstance at 2.8C? (Use 2.8C as the reference pont n Eq ; compare your answer wth

8 77 CHAPTE 26 CUENT AND ESISTANCE Fg ) Does ths same doublng temperature hold for all copper conductors, regardless of shape or sze? 72 A steel trolley-car ral has a cross-sectonal area of 56. cm 2. What s the resstance of 1. km of ral? The resstvty of the steel s m. 73 A col of current-carryng Nchrome wre s mmersed n a lqud. (Nchrome s a nckel chromum ron alloy commonly used n heatng elements.) When the potental dfference across the col s 12 V and the current through the col s 5.2 A, the lqud evaporates at the steady rate of 21 mg/s. Calculate the heat of vaporzaton of the lqud (see Module 18-4). 74 The current densty n a wre s unform and has magntude A/m 2, the wre s length s 5. m, and the densty of conducton electrons s m 3. How long does an electron take (on the average) to travel the length of the wre? 75 A certan x-ray tube operates at a current of 7. ma and a potental dfference of 8. kv. What s ts power n watts? 76 A current s establshed n a gas dscharge tube when a suffcently hgh potental dfference s appled across the two electrodes n the tube. The gas onzes; electrons move toward the postve termnal and sngly charged postve ons toward the negatve termnal. What s the current n a hydrogen dscharge tube n whch electrons and protons move past a crosssectonal area of the tube each second? Is the drecton of the current densty J toward or away from the negatve termnal? 77 In Fg , a resstance col, wred to an external battery, s placed nsde a v thermally nsulated cylnder ftted wth a m frctonless pston and contanng an deal gas. A current 24 ma flows through the col, whch has a resstance 55. At what speed v must the pston, of mass m 12 kg, move upward n order that the temperature of the gas remans unchanged? 78 An nsulatng belt moves at speed 3 m/s and has a wdth of 5 cm. It carres Fgure Problem 77. charge nto an expermental devce at a rate correspondng to 1 ma. What s the surface charge densty on the belt? 79 In a hypothetcal fuson research lab, hgh temperature helum gas s completely onzed and each helum atom s separated nto two free electrons and the remanng postvely charged nucleus, whch s called an alpha partcle. An appled electrc feld causes the alpha partcles to drft to the east at 25. m/s whle the electrons drft to the west at 88. m/s.the alpha partcle densty s cm 3. What are the net current densty and the current drecton? 8 When a metal rod s heated, not only ts resstance but also ts length and cross-sectonal area change. The relaton rl/a suggests that all three factors should be taken nto account n measurng r at varous temperatures. If the temperature changes by 1. C, what percentage changes n L, A, and (c) occur for a copper conductor? (d) What concluson do you draw? The coeffcent of lnear expanson s K A beam of 16 MeV deuterons from a cyclotron strkes a copper block. The beam s equvalent to current of 15 ma. At what rate do deuterons strke the block? At what rate s thermal energy produced n the block? 82 A lnear accelerator produces a pulsed beam of electrons. The pulse current s.5 A, and the pulse duraton s.1 ms. How many electrons are accelerated per pulse? What s the average current for a machne operatng at 5 pulses/s? If the electrons are accelerated to an energy of 5 MeV, what are the (c) average power and (d) peak power of the accelerator? 83 An electrc mmerson heater normally takes 1 mn to brng cold water n a well-nsulated contaner to a certan temperature, after whch a thermostat swtches the heater off. One day the lne voltage s reduced by 6.% because of a laboratory overload. How long does heatng the water now take? Assume that the resstance of the heatng element does not change. 84 A 4 W mmerson heater s placed n a pot contanng 2. L of water at 2 C. How long wll the water take to rse to the bolng temperature, assumng that 8% of the avalable energy s absorbed by the water? How much longer s requred to evaporate half of the water? 85 A 3 mf capactor s connected across a programmed power supply. Durng the nterval from t to t 3. s the output voltage of the supply s gven by V(t) 6. 4.t 2.t 2 volts. At t.5 s fnd the charge on the capactor, the current nto the capactor, and (c) the power output from the power supply.