P ROBL E M S. Section 30.1 The Biot Savart Law

Transcription

1 roblems Eplin wh it is desirble to use hrd ferromgnetic mterils to mke permnent mgnets. 28. Would ou epect the tpe from tpe recorder to be ttrcted to mgnet? (Tr it, but not with recording ou wish to sve.) 29. Given onl strong mgnet nd screwdriver, how would ou first mgnetie nd then demgnetie the screwdriver? 30. Which w would compss point if ou were t the north mgnetic pole of the Erth? 31. Figure Q30.31 shows two permnent mgnets, ech hving hole through its center. Note tht the upper mgnet is levitted bove the lower one. () How does this occur? (b) Wht purpose does the pencil serve? (c) Wht cn ou s bout the poles of the mgnets from this observtion? (d) f the upper mgnet were inverted, wht do ou suppose would hppen? Figure Q30.31 Courtes of Centrl Scientific Compn OBL E M S 1, 2, 3 = strightforwrd, intermedite, chllenging = full solution vilble in the Student Solutions Mnul nd Stud Guide = coched solution with hints vilble t = computer useful in solving problem = pired numericl nd smbolic problems Section 30.1 The Biot Svrt Lw 1. n Niels Bohr s 1913 model of the hdrogen tom, n electron circles the proton t distnce of m with speed of m/s. Compute the mgnitude of the mgnetic field tht this motion produces t the loction of the proton. 2. A lightning bolt m crr current of A for short period of time. Wht is the resulting mgnetic field 100 m from the bolt? Suppose tht the bolt etends fr bove nd below the point of observtion. 3. () A conductor in the shpe of squre loop of edge length m crries current 10.0 A s in Fig Clculte the mgnitude nd direction of the mgnetic field t the center of the squre. (b) Wht f? f this conductor is formed into single circulr turn nd crries the sme current, wht is the vlue of the mgnetic field t the center? 4. Clculte the mgnitude of the mgnetic field t point 100 cm from long, thin conductor crring current of 1.00 A. 5. Determine the mgnetic field t point locted distnce from the corner of n infinitel long wire bent t right ngle, s shown in Figure The wire crries sted current. Figure A conductor consists of circulr loop of rdius nd two stright, long sections, s shown in Figure The wire = 7.00 A Figure 30.3 Figure 30.6

2 958 CHATE 3 0 Sources of the Mgnetic Field lies in the plne of the pper nd crries current. Find n epression for the vector mgnetic field t the center of the loop. 7. The segment of wire in Figure 30.7 crries current of 5.00 A, where the rdius of the circulr rc is 3.00 cm. Determine the mgnitude nd direction of the mgnetic field t the origin. Figure 30.7 n rc of circle of rdius r, s shown in Figure Determine the mgnetic field t the center of the rc. 11. One ver long wire crries current 30.0 A to the left long the is. A second ver long wire crries current 50.0 A to the right long the line ( m, 0). () Where in the plne of the two wires is the totl mgnetic field equl to ero? (b) A prticle with chrge of 2.00 C is moving with velocit of 150î Mm/s long the line ( m, 0). Clculte the vector mgnetic force cting on the prticle. (c) Wht f? A uniform electric field is pplied to llow this prticle to pss through this region undeflected. Clculte the required vector electric field. 12. Consider the current-crring loop shown in Figure 30.12, formed of rdil lines nd segments of circles whose centers re t point. Find the mgnitude nd direction of B t. 8. Consider flt circulr current loop of rdius crring current. Choose the is to be long the is of the loop, with the origin t the center of the loop. lot grph of the rtio of the mgnitude of the mgnetic field t coordinte to tht t the origin, for 0 to 5. t m be useful to use progrmmble clcultor or computer to solve this problem. 9. Two ver long, stright, prllel wires crr currents tht re directed perpendiculr to the pge, s in Figure Wire 1 crries current 1 into the pge (in the direction) nd psses through the is t. Wire 2 psses through the is t 2 nd crries n unknown current 2. The totl mgnetic field t the origin due to the current-crring wires hs the mgnitude /(2). The current 2 cn hve either of two possible vlues. () Find the vlue of 2 with the smller mgnitude, stting it in terms of 1 nd giving its direction. (b) Find the other possible vlue of 2. Figure A wire crring current is bent into the shpe of n equilterl tringle of side L. () Find the mgnitude of the mgnetic field t the center of the tringle. (b) At point hlfw between the center nd n verte, is the field stronger or weker thn t the center? 14. Determine the mgnetic field (in terms of,, nd d) t the origin due to the current loop in Figure b Figure A ver long stright wire crries current. n the middle of the wire right-ngle bend is mde. The bend forms r d + O Figure Figure Two long, prllel conductors crr currents A nd A, both directed into the pge in Figure Determine the mgnitude nd direction of the resultnt mgnetic field t.

3 roblems 959 Section cm 1 2 Figure cm The Mgnetic Force Between Two rllel Conductors 16. Two long, prllel conductors, seprted b 10.0 cm, crr currents in the sme direction. The first wire crries current A nd the second crries A. () Wht is the mgnitude of the mgnetic field creted b 1 t the loction of 2? (b) Wht is the force per unit length eerted b 1 on 2? (c) Wht is the mgnitude of the mgnetic field creted b 2 t the loction of 1? (d) Wht is the force per length eerted b 2 on 1? 17. n Figure 30.17, the current in the long, stright wire is A nd the wire lies in the plne of the rectngulr loop, which crries the current A. The dimensions re c m, m, nd m. Find the mgnitude nd direction of the net force eerted on the loop b the mgnetic field creted b the wire cm Wire 3 is locted such tht when it crries certin current, ech wire eperiences no net force. Find () the position of wire 3, nd (b) the mgnitude nd direction of the current in wire The unit of mgnetic flu is nmed for Wilhelm Weber. The prcticl-sie unit of mgnetic field is nmed for Johnn Krl Friedrich Guss. Both were scientists t Göttingen, Germn. Along with their individul ccomplishments, together the built telegrph in t consisted of btter nd switch, t one end of trnsmission line 3 km long, operting n electromgnet t the other end. (André Ampère suggested electricl signling in 1821; Smuel Morse built telegrph line between Bltimore nd Wshington in 1844.) Suppose tht Weber nd Guss s trnsmission line ws s digrmmed in Figure Two long, prllel wires, ech hving mss per unit length of 40.0 g/m, re supported in horiontl plne b strings 6.00 cm long. When both wires crr the sme current, the wires repel ech other so tht the ngle between the supporting strings is () Are the currents in the sme direction or in opposite directions? (b) Find the mgnitude of the current. Figure cm θ 16.0 Section 30.3 Ampère s Lw Four long, prllel conductors crr equl currents of 5.00 A. Figure is n end view of the conductors. The current direction is into the pge t points A nd B (indicted b the crosses) nd out of the pge t C nd D (indicted b the dots). Clculte the mgnitude nd direction of the mgnetic field t point, locted t the center of the squre of edge length m. A C c Figure m 18. Two long, prllel wires re ttrcted to ech other b force per unit length of 320 N/m when the re seprted b verticl distnce of m. The current in the upper wire is 20.0 A to the right. Determine the loction of the line in the plne of the two wires long which the totl mgnetic field is ero. 19. Three long wires (wire 1, wire 2, nd wire 3) hng verticll. The distnce between wire 1 nd wire 2 is 20.0 cm. On the left, wire 1 crries n upwrd current of 1.50 A. To the right, wire 2 crries downwrd current of 4.00 A. B m Figure A long stright wire lies on horiontl tble nd crries current of 1.20 A. n vcuum, proton moves prllel to the wire (opposite the current) with constnt speed of m/s t distnce d bove the wire. Determine the vlue of d. You m ignore the mgnetic field due to the Erth. D

4 960 CHATE 3 0 Sources of the Mgnetic Field 23. Figure is cross-sectionl view of coil cble. The center conductor is surrounded b rubber ler, which is surrounded b n outer conductor, which is surrounded b nother rubber ler. n prticulr ppliction, the current in the inner conductor is 1.00 A out of the pge nd the current in the outer conductor is 3.00 A into the pge. Determine the mgnitude nd direction of the mgnetic field t points nd b A 3.00 A. b. 1 mm 1 mm 1 mm Figure The mgnetic field 40.0 cm w from long stright wire crring current 2.00 A is 1.00 T. () At wht distnce is it T? (b) Wht f? At one instnt, the two conductors in long household etension cord crr equl 2.00-A currents in opposite directions. The two wires re 3.00 mm prt. Find the mgnetic field 40.0 cm w from the middle of the stright cord, in the plne of the two wires. (c) At wht distnce is it one tenth s lrge? (d) The center wire in coil cble crries current 2.00 A in one direction nd the sheth round it crries current 2.00 A in the opposite direction. Wht mgnetic field does the cble crete t points outside? 25. A pcked bundle of 100 long, stright, insulted wires forms clinder of rdius cm. () f ech wire crries 2.00 A, wht re the mgnitude nd direction of the mgnetic force per unit length cting on wire locted cm from the center of the bundle? (b) Wht f? Would wire on the outer edge of the bundle eperience force greter or smller thn the vlue clculted in prt ()? 26. The mgnetic coils of tokmk fusion rector re in the shpe of toroid hving n inner rdius of m nd n outer rdius of 1.30 m. The toroid hs 900 turns of lrgedimeter wire, ech of which crries current of 14.0 ka. Find the mgnitude of the mgnetic field inside the toroid long () the inner rdius nd (b) the outer rdius. 27. Consider column of electric current pssing through plsm (ionied gs). Filments of current within the column re mgneticll ttrcted to one nother. The cn crowd together to ield ver gret current densit nd ver strong mgnetic field in smll region. Sometimes the current cn be cut off momentril b this pinch effect. (n metllic wire pinch effect is not importnt, becuse the current-crring electrons repel one nother with electric forces.) The pinch effect cn be demonstrted b mking n empt luminum cn crr lrge current prllel to its is. Let represent the rdius of the cn nd the upwrd current, uniforml distributed over its curved wll. Determine the mgnetic field () just inside the wll nd (b) just outside. (c) Determine the pressure on the wll. 28. Niobium metl becomes superconductor when cooled below 9 K. ts superconductivit is destroed when the surfce mgnetic field eceeds T. Determine the mimum current 2.00-mm-dimeter niobium wire cn crr nd remin superconducting, in the bsence of n eternl mgnetic field. 29. A long clindricl conductor of rdius crries current s shown in Figure The current densit J, however, is not uniform over the cross section of the conductor but is function of the rdius ccording to J br, where b is constnt. Find n epression for the mgnetic field B () t distnce r 1 nd (b) t distnce r 2, mesured from the is. 30. n Figure 30.30, both currents in the infinitel long wires re in the negtive direction. () Sketch the mgnetic field pttern in the plne. (b) At wht distnce d long the is is the mgnetic field mimum? Section 30.4 r 2 r 1 Figure Figure The Mgnetic Field of Solenoid 31. Wht current is required in the windings of long solenoid tht hs turns uniforml distributed over length of m, to produce t the center of the solenoid mgnetic field of mgnitude T? 32. Consider solenoid of length nd rdius, contining N closel spced turns nd crring sted current. () n terms of these prmeters, find the mgnetic field t point long the is s function of distnce from the end of the solenoid. (b) Show tht s becomes ver long, B pproches 0 N/2 t ech end of the solenoid.

5 roblems A single-turn squre loop of wire, 2.00 cm on ech edge, crries clockwise current of A. The loop is inside solenoid, with the plne of the loop perpendiculr to the mgnetic field of the solenoid. The solenoid hs 30 turns/cm nd crries clockwise current of 15.0 A. Find the force on ech side of the loop nd the torque cting on the loop. the solenoid, s shown in Figure (b) Figure 30.36b shows n enlrged end view of the sme solenoid. Clculte the flu through the blue re, which is defined b n nnulus tht hs n inner rdius of cm nd outer rdius of cm. Section 30.5 Mgnetic Flu 34. Consider the hemisphericl closed surfce in Figure The hemisphere is in uniform mgnetic field tht mkes n ngle with the verticl. Clculte the mgnetic flu through () the flt surfce S 1 nd (b) the hemisphericl surfce S 2. () B 1.25 cm θ S 1 S 2 (b) Figure Figure A cube of edge length 2.50 cm is positioned s shown in Figure A uniform mgnetic field given b B (5î 4 ĵ 3ˆk)T eists throughout the region. () Clculte the flu through the shded fce. (b) Wht is the totl flu through the si fces? B Section 30.7 Displcement Current nd the Generl Form of Ampère s Lw 37. A A current is chrging cpcitor tht hs squre pltes 5.00 cm on ech side. The plte seprtion is 4.00 mm. Find () the time rte of chnge of electric flu between the pltes nd (b) the displcement current between the pltes. 38. A A current is chrging cpcitor tht hs circulr pltes 10.0 cm in rdius. f the plte seprtion is 4.00 mm, () wht is the time rte of increse of electric field between the pltes? (b) Wht is the mgnetic field between the pltes 5.00 cm from the center? Figure A solenoid 2.50 cm in dimeter nd 30.0 cm long hs 300 turns nd crries 12.0 A. () Clculte the flu through the surfce of disk of rdius 5.00 cm tht is positioned perpendiculr to nd centered on the is of Section 30.8 Mgnetism in Mtter 39. n Bohr s 1913 model of the hdrogen tom, the electron is in circulr orbit of rdius m nd its speed is m/s. () Wht is the mgnitude of the mgnetic moment due to the electron s motion? (b) f the electron moves in horiontl circle, counterclockwise s seen from bove, wht is the direction of this mgnetic moment vector? 40. A mgnetic field of 1.30 T is to be set up in n iron-core toroid. The toroid hs men rdius of 10.0 cm, nd mgnetic permebilit of Wht current is required if the winding hs 470 turns of wire? The thickness of the iron ring is smll compred to 10 cm, so the field in the mteril is nerl uniform. 41. A toroid with men rdius of 20.0 cm nd 630 turns (see Fig ) is filled with powdered steel whose mgnetic

6 962 CHATE 3 0 Sources of the Mgnetic Field susceptibilit is 100. The current in the windings is 3.00 A. Find B (ssumed uniform) inside the toroid. 42. A prticulr prmgnetic substnce chieves 10.0% of its sturtion mgnetition when plced in mgnetic field of 5.00 T t temperture of 4.00 K. The densit of mgnetic toms in the smple is toms/m 3, nd the mgnetic moment per tom is 5.00 Bohr mgnetons. Clculte the Curie constnt for this substnce. 43. Clculte the mgnetic field strength H of mgnetied substnce in which the mgnetition is A/m nd the mgnetic field hs mgnitude 4.40 T. 44. At sturtion, when nerl ll of the toms hve their mgnetic moments ligned, the mgnetic field in smple of iron cn be 2.00 T. f ech electron contributes mgnetic moment of A m 2 (one Bohr mgneton), how mn electrons per tom contribute to the sturted field of iron? ron contins pproimtel toms/m () Show tht Curie s lw cn be stted in the following w: The mgnetic susceptibilit of prmgnetic substnce is inversel proportionl to the bsolute temperture, ccording to C 0 /T, where C is Curie s constnt. (b) Evlute Curie s constnt for chromium. Section 30.9 The Mgnetic Field of the Erth 46. A circulr coil of 5 turns nd dimeter of 30.0 cm is oriented in verticl plne with its is perpendiculr to the horiontl component of the Erth s mgnetic field. A horiontl compss plced t the center of the coil is mde to deflect 45.0 from mgnetic north b current of A in the coil. () Wht is the horiontl component of the Erth s mgnetic field? (b) The current in the coil is switched off. A dip needle is mgnetic compss mounted so tht it cn rotte in verticl north south plne. At this loction dip needle mkes n ngle of 13.0 from the verticl. Wht is the totl mgnitude of the Erth s mgnetic field t this loction? 47. The mgnetic moment of the Erth is pproimtel A m 2. () f this were cused b the complete mgnetition of huge iron deposit, how mn unpired electrons would this correspond to? (b) At two unpired electrons per iron tom, how mn kilogrms of iron would this correspond to? (ron hs densit of kg/m 3, nd pproimtel iron toms/m 3.) Additionl roblems 48. The mgnitude of the Erth s mgnetic field t either pole is pproimtel T. Suppose tht the field fdes w, before its net reversl. Scouts, silors, nd conservtive politicins round the world join together in progrm to replce the field. One pln is to use current loop round the equtor, without reling on mgnetition of n mterils inside the Erth. Determine the current tht would generte such field if this pln were crried out. (Tke the rdius of the Erth s E m.) 49. A ver long, thin strip of metl of width w crries current long its length s shown in Figure Find the mgnetic field t the point in the digrm. The point is in the plne of the strip t distnce b w from it. 0 Figure Suppose ou instll compss on the center of the dshbord of cr. Compute n order-of-mgnitude estimte for the mgnetic field t this loction produced b the current when ou switch on the hedlights. How does it compre with the Erth s mgnetic field? You m suppose the dshbord is mde mostl of plstic. 51. For reserch project, student needs solenoid tht produces n interior mgnetic field of T. She decides to use current of 1.00 A nd wire mm in dimeter. She winds the solenoid in lers on n insulting form 1.00 cm in dimeter nd 10.0 cm long. Determine the number of lers of wire needed nd the totl length of the wire. 52. A thin copper br of length 10.0 cm is supported horiontll b two (nonmgnetic) contcts. The br crries current A in the direction, s shown in Figure At distnce h cm below one end of the br, long stright wire crries current A in the direction. Determine the mgnetic force eerted on the br. 1 h Figure A nonconducting ring of rdius 10.0 cm is uniforml chrged with totl positive chrge 10.0 C. The ring rottes t constnt ngulr speed 20.0 rd/s bout n 2 b w

7 roblems is through its center, perpendiculr to the plne of the ring. Wht is the mgnitude of the mgnetic field on the is of the ring 5.00 cm from its center? A nonconducting ring of rdius is uniforml chrged with totl positive chrge q. The ring rottes t constnt ngulr speed bout n is through its center, perpendiculr to the plne of the ring. Wht is the mgnitude of the mgnetic field on the is of the ring distnce /2 from its center? 55. Two circulr coils of rdius, ech with N turns, re perpendiculr to common is. The coil centers re distnce prt. Ech coil crries sted current in the sme direction, s shown in Figure () Show tht the mgnetic field on the is t distnce from the center of one coil is B N ( 2 2 ) 3/2 1 3/2 ( ) (b) Show tht db/d nd d 2 B/d 2 re both ero t the point midw between the coils. This mens the mgnetic field in the region midw between the coils is uniform. Coils in this configurtion re clled Helmholt coils. Figure roblems 55 nd Two identicl, flt, circulr coils of wire ech hve 100 turns nd rdius of m. The coils re rrnged s set of Helmholt coils (see Fig ), prllel nd with seprtion m. Ech coil crries current of 10.0 A. Determine the mgnitude of the mgnetic field t point on the common is of the coils nd hlfw between them. 57. We hve seen tht long solenoid produces uniform mgnetic field directed long the is of clindricl region. However, to produce uniform mgnetic field directed prllel to dimeter of clindricl region, one cn use the sddle coils illustrted in Figure The loops re wrpped over somewht flttened tube. Assume the stright sections of wire re ver long. The end view of the tube shows how the windings re pplied. The overll current distribution is the superposition of two overlpping circulr clinders of uniforml distributed current, one towrd ou nd one w from ou. The current densit J is the sme for ech clinder. The position of the is of one clinder is described b position vector reltive to the other clinder. rove tht the mgnetic field inside the hollow tube is 0 J/2 downwrd. Suggestion: The use of vector methods simplifies the clcultion. () Figure () Generl view of one turn of ech sddle coil. (b) End view of the coils crring current into the pper on the left nd out of the pper on the right. 58. A ver lrge prllel-plte cpcitor crries chrge with uniform chrge per unit re on the upper plte nd on the lower plte. The pltes re horiontl nd both move horiontll with speed v to the right. () Wht is the mgnetic field between the pltes? (b) Wht is the mgnetic field close to the pltes but outside of the cpcitor? (c) Wht is the mgnitude nd direction of the mgnetic force per unit re on the upper plte? (d) At wht etrpolted speed v will the mgnetic force on plte blnce the electric force on the plte? Clculte this speed numericll. 59. Two circulr loops re prllel, coil, nd lmost in contct, 1.00 mm prt (Fig ). Ech loop is 10.0 cm in rdius. The top loop crries clockwise current of 140 A. The bottom loop crries counterclockwise current of 140 A. () Clculte the mgnetic force eerted b the bottom loop on the top loop. (b) The upper loop hs mss of kg. Clculte its ccelertion, ssuming tht the onl forces cting on it re the force in prt () nd the grvittionl force. Suggestion: Think bout how one loop looks to bug perched on the other loop. 140 A 140 A Figure Wht objects eperience force in n electric field? Chpter 23 gives the nswer: n electric chrge, sttionr or moving, other thn the chrge tht creted the field. Wht cretes n electric field? An electric chrge, sttionr or moving, s ou studied in Chpter 23. Wht objects eperience force in mgnetic field? An electric current or moving electric chrge, other thn the current or chrge tht creted the field, s discussed in Chpter 29. Wht cretes mgnetic field? An electric current, s ou studied in Section 30.1, or moving electric chrge, s shown in this problem. () To displ how moving chrge cretes mgnetic field, consider chrge q moving with velocit v. Define the vector r r ˆr (b)

8 964 CHATE 3 0 Sources of the Mgnetic Field to led from the chrge to some loction. Show tht the mgnetic field t tht loction is A B 0 4 C q v ˆr r 2 (b) Find the mgnitude of the mgnetic field 1.00 mm to the side of proton moving t m/s. (c) Find the mgnetic force on second proton t this point, moving with the sme speed in the opposite direction. (d) Find the electric force on the second proton. 61. il guns hve been suggested for lunching projectiles into spce without chemicl rockets, nd for ground-to-ir ntimissile wepons of wr. A tbletop model ril gun (Fig ) consists of two long prllel horiontl rils 3.50 cm prt, bridged b br BD of mss 3.00 g. The br is originll t rest t the midpoint of the rils nd is free to slide without friction. When the switch is closed, electric current is quickl estblished in the circuit ABCDEA. The rils nd br hve low electric resistnce, nd the current is limited to constnt 24.0 A b the power suppl. () Find the mgnitude of the mgnetic field 1.75 cm from single ver long stright wire crring current 24.0 A. (b) Find the mgnitude nd direction of the mgnetic field t point C in the digrm, the midpoint of the br, immeditel fter the switch is closed. Suggestion: Consider wht conclusions ou cn drw from the Biot Svrt lw. (c) At other points long the br BD, the field is in the sme direction s t point C, but lrger in mgnitude. Assume tht the verge effective mgnetic field long BD is five times lrger thn the field t C. With this ssumption, find the mgnitude nd direction of the force on the br. (d) Find the ccelertion of the br when it is in motion. (e) Does the br move with constnt ccelertion? (f) Find the velocit of the br fter it hs trveled 130 cm to the end of the rils. B is g/cm, wht vlue of current B will result in equilibrium when the distnce between the two conductors is 2.50 cm? B A Figure Chrge is spred onto lrge nonconducting belt bove the left-hnd roller in Figure The belt crries the chrge with uniform surfce chrge densit s it moves with speed v between the rollers s shown. The chrge is removed b wiper t the right-hnd roller. Consider point just bove the surfce of the moving belt. () Find n epression for the mgnitude of the mgnetic field B t this point. (b) f the belt is positivel chrged, wht is the direction of B? (Note tht the belt m be considered s n infinite sheet.) Figure A B v E D Figure An infinitel long stright wire crring current 1 is prtill surrounded b loop s shown in Figure Fift turns of insulted wire cm in dimeter re tightl wound to form flt spirl. The spirl fills disk surrounding circle of rdius 5.00 cm nd etending to rdius cm t the outer edge. Assume the wire crries current t the center of its cross section. Approimte ech turn of wire s circle. Then loop of current eists t rdius 5.05 cm, nother t 5.15 cm, nd so on. Numericll clculte the mgnetic field t the center of the coil. 63. Two long, prllel conductors crr currents in the sme direction s shown in Figure Conductor A crries current of 150 A nd is held firml in position. Conductor B crries current B nd is llowed to slide freel up nd down (prllel to A) between set of nonconducting guides. f the mss per unit length of conductor B L 2 1 Figure 30.65

9 roblems 965 The loop hs length L, rdius, nd crries current 2. The is of the loop coincides with the wire. Clculte the force eerted on the loop. r = e θ 66. Mesurements of the mgnetic field of lrge torndo were mde t the Geophsicl Observtor in Tuls, Oklhom, in The torndo s field ws mesured to be B T pointing north when the torndo ws 9.00 km est of the observtor. Wht current ws crried up or down the funnel of the torndo, modeled s long stright wire? 67. A wire is formed into the shpe of squre of edge length L (Fig ). Show tht when the current in the loop is, the mgnetic field t point, distnce from the center of the squre long its is is θ r rˆ Figure dr ds β = π/4 B L 0L 2 2( 2 L 2 /4) 2 L 2 /2 70. Tble contins dt tken for ferromgnetic mteril. () Construct mgnetition curve from the dt. emember tht B B 0 0 M. (b) Determine the rtio B/B 0 for ech pir of vlues of B nd B 0, nd construct grph of B/B 0 versus B 0. (The frction B/B 0 is clled the reltive permebilit, nd it is mesure of the induced mgnetic field.) L Figure The force on mgnetic dipole ligned with nonuniform mgnetic field in the direction is given b F db/d. Suppose tht two flt loops of wire ech hve rdius nd crr current. () The loops re rrnged coill nd seprted b vrible distnce, lrge compred to. Show tht the mgnetic force between them vries s 1/ 4. (b) Evlute the mgnitude of this force if 10.0 A, cm, nd 5.00 cm. 69. A wire crring current is bent into the shpe of n eponentil spirl, r e, from 0 to 2 s suggested in Figure To complete loop, the ends of the spirl re connected b stright wire long the is. Find the mgnitude nd direction of B t the origin. Suggestions: Use the Biot Svrt lw. The ngle between rdil line nd its tngent line t n point on the curve r f () is relted to the function in the following w: tn r dr/d Tble B (T) B 0 (T) A sphere of rdius hs uniform volume chrge densit. Determine the mgnetic field t the center of the sphere when it rottes s rigid object with ngulr speed bout n is through its center (Fig ). Thus in this cse r e, tn 1 nd /4. Therefore, the ngle between ds nd ˆr is 3/4. Also ds dr sin(/4) 2 dr Figure roblems 71 nd 72.

10 966 CHATE 3 0 Sources of the Mgnetic Field 72. A sphere of rdius hs uniform volume chrge densit. Determine the mgnetic dipole moment of the sphere when it rottes s rigid bod with ngulr speed bout n is through its center (Fig ). 73. A long clindricl conductor of rdius hs two clindricl cvities of dimeter through its entire length, s shown in Figure A current is directed out of the pge nd is uniform through cross section of the conductor. Find the mgnitude nd direction of the mgnetic field in terms of 0,, r, nd t () point 1 nd (b) point 2. Answers to Quick Quies r 1 Figure B, C, A. oint B is closest to the current element. oint C is frther w nd the field is further reduced b the sin fctor in the cross product ds ˆr. The field t A is ero becuse (c). F 1 F 2 s required b Newton s third lw. Another w to rrive t this nswer is to relie tht Eqution r gives the sme result whether the multipliction of currents is (2 A)(6 A) or (6 A)(2 A) (). The coils ct like wires crring prllel currents in the sme direction nd hence ttrct one nother b, d,, c. Eqution indictes tht the vlue of the line integrl depends onl on the net current through ech closed pth. th b encloses 1 A, pth d encloses 3 A, pth encloses 4 A, nd pth c encloses 6 A b, then c d. ths, c, nd d ll give the sme nonero vlue 0 becuse the sie nd shpe of the pths do not mtter. th b does not enclose the current, nd hence its line integrl is ero (c). The mgnetic field in ver long solenoid is independent of its length or rdius. Overwrpping with n dditionl ler of wire increses the number of turns per unit length (b). There cn be no conduction current becuse there is no conductor between the pltes. There is timevring electric field becuse of the decresing chrge on the pltes, nd the time-vring electric flu represents displcement current (c). There is time-vring electric field becuse of the decresing chrge on the pltes. This time-vring electric field produces mgnetic field (). The loop tht looks like Figure is better becuse the remnent mgnetition t the point corresponding to point b in Figure is greter (b). The lines of the Erth s mgnetic field enter the plnet in Hudson B nd emerge from Antrctic; thus, the field lines resulting from the current would hve to go in the opposite direction. Compre Figure 30.7 with Figure Clvin nd Hobbes Wtterson. eprinted with permission of Universl ress Sndicte. All rights reserved.