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1 HAPTE ELETRIAL SYSTEMS 2.1 PROESSES 1. onsider he experimen described in Fig Do he elecromeers conneced o charged spheres measure quaniies of elecriciy (charge) or he inensiy of elecriciy (volage)? Why? Figure 2.1: Differenly charged bodies in (elecrical) conac. A glow lamp lighs up briefly, and he readings of he elecromeers become equal. 2. Why does he phenomenon shown in Fig. 2.2 demonsrae ha volages raher han quaniies of elecric charge equilibrae in communicaing capaciors? Wha does his ell us abou he driving force of he flow of charge? 5 U 1 U 2 olage / X XXXX X XXXXXX X[ Figure 2.2: Two capaciors are conneced wih a resisor o form a circui (circui diagram is a he cener). The diagram on he righ shows he volages of he capaciors as funcions of ime Time / s 3. When you rub non-conducing maerials (plasic, rubber, glass, amber) on fur or cloh, saic elecriciy develops. Where does he charge come from? Has i been produced? 4. A meal coaed pingpong ball is hanging from a hin hread. I can be charged wih he help of a plasic rod rubbed agains fur. Afer his, when we bring he rod close o he ball (wihou ouching i), he ball is repelled. If we bring a glass rod rubbed wih a silk cloh near he charged ball, i is araced by he rod. How can his be explained? QUESTIONS AND EXERISES 1

2 5. Do generaors, power supplies, and baeries generae elecriciy (i.e., elecric charge)? Wha do hey do wih charge? 6. onsider he circui in Fig Should he charge currens hrough he wo lamps be equal? Wha relaion do hey saisfy in general? Figure 2.3: A branching circui wih wo loops. A charge curren flows oward a juncion and splis ino wo which add up o he original curren. I Q1 Juncion Lamp I Q3 I Q2 Lamp 7. In Fig. 2.4 (righ), he elecric poenial in a single loop of a circui is shown as a funcion of posiion along he circui. How many volages are here? Idenify hem in he diagram on he righ. Wha is he relaion beween he volages? Figure 2.4: olages across he elemens in a single loop circui (he elemens are conneced in series). The volmeers are conneced in parallel o an elemen. olages (poenial differences) are denoed by (blue) arrows parallel o he devices. The recangles in he circui diagram symbolize resisors. B A U 1 U B U 2 olmeer D E F Poenial A B D E F A Posiion 8. onsider he circui in Fig Wha mus be he relaion beween he volages across he wo lamps? Why? 9. Which fluid quaniies can be compared o which elecric quaniies? How are hey similar? Are here differences? Are here elemens of fluid sysems (such as pipes or pumps) ha are analogous o elemens of elecric sysems? 10. Skech an elecric circui ha is analogous o he sysem of wo communicaing fluid anks. Do he same for a sysem of wo communicaing anks where one of hem has an addiional ouflow. 2.2 HARGE, URRENTS, AND BALANES 11. A wide piece of ape is orn off an uncharged meal sphere (he sphere is insulaed elecrically). Aferwards he meal sphere shows an elecric charge of How does his occur? Is he ape also charged? If so, how much charge does i have? 12. A neural oxygen aom loses wo elecrons. Wha is is charge hen? 13. A charge curren of 2.0 A flows hrough a wire. How much charge flows 2 PHYSIS AS A SYSTEMS SIENE

3 hrough a cross secion of he wire in 10 s? 14. The curren of charge hrough a resisor is variable and increases linearly in 100 s from 0.20 A o 1.0 A (as in he diagram on he lef of Fig. 2.5). How much charge has flowed in his ime hrough a cross secion of he resisor? 15. Describe he curren in Fig. 2.5 in he diagram on he righ. How do you use he diagram o calculae he charge ranspored during a period of ime? 16. An alernaing curren flows hrough a wire ha behaves according o I asin 2π f, a 20. A, f 50Hz q ()= ( ) = = f is he frequency. How much charge flows in a cross secion in a half-period? How much flows in a whole period? 17. The rae of change of charge of a capacior is a consan 0.20 /s. Wha is he charge of he capacior a = 10 s if i was equal o 3.0 a = 0 s? 18. There are wo poins in he diagram on he lef of Fig. 2.6 where he rae of change of charge of a sysem equals zero. Wha happens o he charge of he sysem a hose poins? 19. Explain howe he charge of an elecric sysem is deermined from a knowledege of he rae of change of charge. 20. The second capacior in Fig. 2.7 receives a quaniy of charge of 2.0 during a cerain period from he firs capacior. During he same period, i loses 3.0 hrough he second resisor. Wha are he changes of charge of he capaciors (separaely)? 21. Use an explici expression for he ineracion rule for he capaciors in Fig Wha is he form of he laws of balance of charge for he capaciors in his case? 22. One ofen inroduces a hird elecric curren in he diagram of Fig. 2.7 leading from he juncion o he second capacior. all his curren I Q3. Rewrie he laws of balance of charge for boh capaciors. Is he sysem of equaions sill equivalen o wha was formulaed above (Exercise 21)? 23. Someimes he juncion rule is formulaed as follows: The sum of all currens meeing a a juncion is equal o zero. How does his agree wih he example shown in Equ. 2.1 (as i migh apply o he juncion in Fig. 2.7)? I Q Figure 2.5: urrens of elecric charge as a funcions of ime. urrens can change slowly or quickly, hey can be posiive or negaive (his has nohing o do wih posiive or negaive charge). Figure 2.6: Rase of change of charge of an elecric sysem as a funcion of ime. I Q1 Q I Q I Q2 R Figure 2.7: Skech of an elecric circui having wo capaciors and wo resisive elemens. I = I I Q1 Q2 Q3 (2.1) 24. ombine he capaciors in Fig. 2.7 ino a single one (for his o make pracical sense, one would make he resisance of R1 small compared o ha of R2). Rewrie he law(s) of balance for his case. 25. A wire leads from a baery and hen branches off ino wo pars. The curren coming from he baery measures 0.50 A. A curren of 0.70 A flows hrough one branch of he wire from he juncion. Wha is he curren in he second wire and in wha direcion does i flow? QUESTIONS AND EXERISES 3

4 2.3 ELETRI POTENTIAL AND POTENTIAL DIFFERENES (OLTAGES) Poenial A B D E F G H A Posiion Figure 2.8: Elecric poenial as a funcion of posiion in a circui conaining wo capaciors, a baery, and a resisor. The values will change in he course of ime, bu he form of he diagram will basically remain he same. 12 R4 7 Figure 2.9: Skech of an elecric circui having wo branches, a baery (or power supply) and a number of resisors. R1 R2 3 R3 26. Trace he elecric poenial in a circui consising of a real baery and a lamp. Skech a level diagram such as ha in Fig Idenify volages. Wha is he relaion beween all he volages? 27. An elecric moor conains coils of wire, and he wire has resisive properies. Is he volage measured across he moor during operaion equal o he volage ha is said o cause he producion of hea? 28. A galvanic elemen shows a volage of 1.5. How can a 12 baery be made from several such elemens? 29. A baery (12 ) is par of a closed elecrical circui. Aached o i is a resisor, a moor, anoher baery (5.0 ), and finally, a lamp. The volages across he moor and he lamp are 6.0 and 7.0. a. Skech he poenial-posiion diagram. b. Wha is he volage across all he elemens (wih signs)? 30. Deermine he missing volage values in he elecrical circui of Fig There is a volage of 3.0 along a uniform wire of 0.50 m lengh. Wha is he gradien of he elecric poenial along he wire in he direcion of flow of charge? Wha is he sign of U? 2.4 ENERGY IN ELETRI PROESSES 32. onsider an elecric waer pump. Wha is he meaning of driving process? Wha is he driven process? Wha does energy have o do wih one process driving anoher? Wha is he meaning of elecric power? Of hydraulic power? 33. Explain he role of energy in he operaion of a baery. Explain erms such as driving and driven process, power, energy sorage, ec. 34. onsider a baery driving an elecric pump. Wha is he role of energy? 35. Describe releasing and binding (using) of energy in he elecrical processes of (1) an immersion heaer, (2) a baery and an elecric moor. Draw he sysem diagrams. Which processes are coupled in hese devices? Which process drives? Which one is driven? 36. A pump is driven by an elecric engine. Togeher, he moor and he pump have an efficiency of 80%. One wishes o pump a waer curren of m 3 /s 10 m high. Wha mus he elecric curren be for he drive o be a leas 220? Does i need o be greaer han 220? 37. A volage of 210 and a curren of 4.0 A are measured in a running elecric moor. a. Wha is he elecric power of he moor? b. Wha is he released energy used for? 4 PHYSIS AS A SYSTEMS SIENE

5 2.5 FLOW RELATIONS 38. Assume a ype of conducing device for which he elecric curren doubles if he volage across i is doubled. Wha is he characerisic diagram of such a device? How do you explain he meaning of conducance or resisance of he device using he characerisic I Q U R diagram? 39. Meals are said o be ohmic conducors which are said o have linear characerisic curves: double he volage across an ohmic conducor or resisor gives double he curren hrough he device. The filamen of a ligh bulb is a meal. Why is he characerisic of an incandescen lamp no linear? 40. Skech he curren-volage characerisics for he following resisive elemens: a. An elemen wih consan resisance. b. An elemen similar o he one in a, bu wih a greaer resisance value. c. The filamen of a ligh bulb. 41. Derive he expressions for equivalen resisances of resisors in series or in parallel. Imagine wo idenical resisors. Wha is heir equivalen resisance if hey are conneced in parallel? 42. A 100 W incandescen ligh bulb is conneced o 220. Wha is he resisance of he filamen? Is he resisance differen if i is conneced o 150? 43. Wha is he raio of he resisance of copper wires having he same lengh where one has double he diameer of he oher? 44. We have wo equal amouns of a meal. Each is formed ino a wire. The second wire is wice as long as he firs. Wha is is resisance compared o he resisance of he firs one? 45. A 20, plainum has a resisiviy of Ω m. Wha mus he radius of a 0.50 m long plainum wire be so ha i has a resisance of 20 Ω? 46. The emperaure coefficien of he resisiviy of plainum is K 1. Wha is he resisance of he wire in Exercise 45 a 120? 47. The resisances of R1 and R2 in Fig. 2.9 are 20 Ω and 10 Ω. a. Wha is he oal resisance in he branch wih R1 and R2? b. Wha is R3 when he curren hrough his branch is wice as grea as in R1? c. Wha are he equivalen resisances for R1, R2, and R3? 48. Wha is he elecric curren hrough an immersion heaer hooked up o 220 having an elecric power of 300 W? Wha is he hermal power of his device? 49. Two resisors wih resisances of R 1 and are boh conneced o an ideal baery wih a volage of U B. a. Find he elecric power of all hree elemens. b. Why does energy conservaion in his circui lead o Kirchhoff s law? 50. Fig shows he measured I U characerisic of a ligh bulb. a. Deermine he resisance and he elecric power in seps of 5. urren / A [ [ [ [ [ [ [ [ [ [ [ [ [ [ 0.00 [ olage / Figure 2.10: haracerisic diagram of he filamen of a ligh bulb. QUESTIONS AND EXERISES 5

6 b. Explain in words he meaning of he elecric power of a filamen. c. The behavior of he emperaure of he elecrical resisance can be approximaed using he quadraic formula 2 ( ( ) ) R( T )= R 1 α( T T ) β T T α = K, β = K Find he emperaure of he filamen. d. How is he energy ha is released from he filamen ranspored away? 2.6 BATTERIES 51. You find 8.0 Ah wrien on a baery. Wha does his mean? A baery has 1.5 wrien on i. Wha does his mean? 52. Why does he volage measured across he erminals of a baery decrease if he elecric curren hrough he baery is increasing? How are volage and curren relaed? 53. Why is he curren hrough a baery limied, even for load resisors wih very small resisances (such as a simple shor wire connecing he erminals)? Elecric curren / ma olage / Figure 2.11: haracerisic curves of a small array of 21 solar cells in series exposed o he sun and he sky on hree differen winer days. The insolaion was 400, 200, and 60 W/m 2 for he hree curves (from he op). 54. A lamp is conneced o a 4.5 baery. A volage of 4.1 is measured across he lamp. The volages over he wires in he circui are jus abou zero. Wha is going on? 55. The open-circui volage of a baery is 4.5. When i is conneced o a small moor, i shows 4.0. The curren measured in he circui is 0.60 A. a. A wha rae is energy released in he baery? b. Wha happens wih he energy released by he baery? Wha is he energy curren emied by he baery along wih he elecric charge? c. Wha is he elecric power of he moor? 56. Why is here a maximum of he power of a baery in operaion? Wha is is power for eiher maximum volage or maximum curren? 57. Esimae he maximum power of he phoovolaic array in Fig for an insolaion of 400 W. 2.7 STORING HARGE 58. onsider a capacior having consan capaciance. Wha is he form of he capaciive characerisic? 59. Explain he meaning of elecric capaciance. 60. Imagine a capacior being discharged in a simple circui. Wha kind of daa should be aken o derive he capaciive relaion of he capacior? How do you deermine i? 6 PHYSIS AS A SYSTEMS SIENE

7 61. Wha is he charge of a 10 µf capacior which has been charged o 100? 62. A capacior having a capaciance of 100 µf is charged o a volage of 10. How much charge and energy is sored? 63. One wishes o sore a leas 1.0 J of energy in a capacior ha can be charged o 200. How grea mus is capaciance be made for his? 64. Produce a word model ha explains he charging or discharging of a capacior in a simple circui having a resisor (Fig. 2.12). Do he same for discharging of a ank conaining oil hrough a pipe a is boom. ompare he explanaions you use in he elecric and in he hydraulic cases R 1 U olage / [ Time / s olage / [ [[[[ [ [ [[[ [ [[ [ 0.0 [ Time / s Figure 2.12: Discharging and charging of a simple capacior. Lef: Diagram of a circui ha allows for charging and subsequen discharging (R 1 = 9.8 kohm, = 108 kohm). ener: olage across he capacior as a funcion of ime, as i discharges. Righ: olage across capacior during charging. 65. Does i make more sense o compare a parallel plae capacior wih a waer ank or a U-pipe? Which quaniies do you compare? Wha is he capaciance of a U-pipe compared o normal pipe of equal hickness? Wha does his mean for an elecric parallel plae capacior? 66. A capacior wih a capaciance is charged by an ideal baery wih a volage of U B over a resisor having resisance R. a. Wha is he volage across he capacior a he momen he curren has half of is iniial value? b. Wha is he energy curren emied by he baery a he beginning? Where does he energy go? c. Wha is he energy curren emied by he baery when he curren is only half wha i was a he beginning? Where does he energy go? 2.8 DYNAMIAL MODELS 67. In wha sense can we say ha a model such as he one in Fig explains a sysem? 68. Explain he meaning of he srucure of socks (recangles) and flows (hick arrows) in he sysem dynamics diagram of Fig Why is here only a single sock? Why are here wo flows conneced o he sock for Q? QUESTIONS AND EXERISES 7

8 Tank Q Pump alve R 1 IQ 1 IQ 2 Figure 2.13: A sysem dynamics model diagram for a sysem equivalen o a hydraulic windkessel. R 1 ~ U power supply U R1 U D U U R2 69. In Fig. 2.13, UD symbolizes he volage across he diode which we se o a consan value independen of he curren. How is UR1 calculaed? 70. To wha percenage of he final level does he level on he righ in he diagram of Fig rise in one ime consan? U Figure 2.14: Discharging or charging of capaciors having consan capaciance hrough resisors having consan leads o exponenially changing funcions. The iniial rae of change is used o define he ime consan of he sysem. R 1 U U τ τ 71. Esimae he ime consan of he circui in Fig Measure he ime consans in Fig alculae he capaciance of he capacior for boh deerminaions of ime consans. 73. A 1.0 mf capacior discharges hrough a 1000 Ω resisor. a. How long does i ake for he charge o reduce o 1/e of he iniial value? b. How long does his ake for he volage of he capacior and he curren in he circui o reduce o 1/ of heir iniial values? c. How long does i ake unil only half of he charge remains in he capacior? 8 PHYSIS AS A SYSTEMS SIENE

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