Brown University PHYS 0060 INDUCTANCE

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1 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide INTODUCTION INDUCTANCE Anyone who has ever grabbed an auomobile spark-plug wire a he wrong place, wih he engine running, has an appreciaion of he abiliy of a changing curren in (par of) a coil of wire o induce an emf in he coil. Wha happens is ha he breaker conacs open, suddenly inerruping he curren, and causing a sudden large change in he magneic field hrough he coil; according o Faraday s law, his resuls in a (large) induced emf. In general, he producion of an emf in a coil by a changing magneic field due o a curren in ha same coil is called self-inducion; and he abiliy of a coil o produce an emf in his way is commonly measured by is self-inducance L, usually called more briefly is inducance. A coil used in his way is more formally called an inducor. The ransmission of elecric signals by elevision, radio, and elephone depends on ime-varying currens and fields o represen he appearance of picures and he sound of voices; and so, as you can well imagine, capaciors and inducors plan an imporan role in he circuis of such devices. You already know ha a capacior can sore energy; so can an inducor. If an inducor carrying a curren is conneced o a resisor, is energy is dissipaed as hea in he resisor, much as for a charged capacior. Bu now suppose you connec i insead o a capacior; he inducor will ry o give is energy o he capacior and vice versa bu he iniial energy is no quickly dissipaed from he elecrical circui. Wha do you suppose happens? If you do no already know, can you guess, before sudying his module? PEEQUISITES Before you begin his module, you should be able o: *Use Ampere s law o calculae B inside oroids and long solenoids (needed for Objecives 1 his module) *elae he emf induced in such oroids and solenoids o he ime rae of change of B or φ B (needed for Objecive 1 of his module) *Find he power dissipaed by a resisor (needed for Objecives 2 of his module) *Add volages around an C circui verify he exponenial ime dependences of he curren and volage (needed for Objecive 2 of his Locaion of Prerequisie Conen Ampère s Law Module Faraday s Law Module Ohm s Law Module Direc-Curren Circuis Module

2 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide module) *Find he energy sored in a capacior (needed for Objecive 3 of his module) *elae he moion of a mechanical oscillaor o he mahemaical expression for is displacemen (needed for Objecive 3 of his module) Capaciors Module Simple Harmonic Moion Module LEANING OBJECTIVES afer you have masered he conen of his module, you will be able o: 1. Inducance Apply he definiion of inducance, Ampere s law, and Faraday s law o oroids and long solenoids o (a) find he inducance L; and (b) relae he induced emf o he rae of change of curren of flux. 2. L Circuis Deermine currens, volages, sored energies, and power dissipaions in simple L circuis. (This includes adding up volages around he circui o find a differenial equaion and deermine he ime dependence.) 3. LC Circuis Deermine charges, volages, currens, and sored energies in simple LC Circuis. (This includes using he principle of energy conservaion o find imum values, as well as o obain a differenial equaion and deermine he ime dependence.) 2

3 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide GENEAL COMMENTS Once you know i (), i is relaively simple maer o find any oher quaniy, such as he volage across he resisor, V = i, or he induced emf ε L = di/d. A summary of he resuls for ypical L circuis is presened in Figure L Circuis Suppose you are given he circui shown in Figure 1. Before he swich is closed, he curren is zero. When he swich is closed, he curren sars o rise bu only a a finie rae, since he inducor will no allow any sudden change in he curren. [The induced emf L (di/d) would be infinie!] Furhermore, he curren will no rise indefiniely, because of he opposing volage V = i across he resisor. Therefore, he behavior of he curren has he appearance of Figure 2. Since he emf induced in L vanishes as he curren approaches is final unchanging value i, we see ha V he swich is closed. Figure 1 L i = V B Nex, adding volages around he circui, much as you did in he module Direc-Curren Circuis for circuis conaining only resisors and baeries, leads o he equaion V B L d dd = This differenial equaion gives a mahemaical descripion of he behavior of he curren i afer Anoher possibiliy is he circui show in Figure 3. Iniially, he swich is closed; and we imagine ha he curren i has reached is seady-sae value, so ha he emf induced in L is zero, Therefore, he volage difference across is zero, and he curren hrough he inducor is Figure 2 = V B 1 while he swich is closed. When he swich is opened, he emf induced in he inducor again prevens any insananeous change in i; is iniial value is herefore jus i. The curren now flows hrough he only pah open o i, namely, hrough ; as a resul, he resisor heas up. Evidenly, his supply of hea is no unlimied (or we would use i o hea houses!); he curren mus fall oward zero, as in Figure 4. Incidenally, since he resisor does hea up, we have seen ha an inducor sores energy when a curren is flowing hrough i. i() i 3

4 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide i Figures 2 and 4 should recall he behavior of he charge and curren in he C-circuis ha you sudied in he module Direc-Curren Circuis. A synopsis of he L V B resuls for such circuis is represened in Figure 5. If you look back, you will find, for example, ha he Figure 3 volage across he capacior obeys he equaion V C = V B 1 e τ when he swich is moved o he charge posiion, and i() i V C = V B e τ Figure 4 4

5 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide when he swich is moved back o discharge afer he capacior has become fully charged. In fac, all he quaniies indicaed in Figure 5 can be expressed as y = consd f(d) or y = consd g(d) where f(d) = 1 e τ and g(d) = e τ Noe ha f () = and f ( ) = 1, g () = 1 and g ( ) = 1 Figure 5: C Circuis (τ = C) Charge i V V Discharge q C V C Noe proporionaliies: V C = q ; V C =. Swich is moved a = q or V C i or V CHAGE f() g() a) c) C C q or V C -i or -V DISCHAGE b) g() d) g() C C In each case, cons is jus he imum value of he quaniy in quesion, which is eiher he limiing value for large imes, or he iniial value a =. Of course, we are no really ineresed in re-solving C circuis in his module! The poin of 5

6 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide all his is ha we need o find he soluion of he differenial equaion (2). The similariy beween Figures 2 and 5(a) boh curve rise from zero o an asympoic value sugges ha we ry a soluion similar o Eq. (4). (Acually, if you checked he differenial equaions, you should find ha hey are similar, oo.) Tha is, we se (d) = Af(d) = A 1 e τ, where he consan A is deermined by he condiion, from Eq. (1), ha ( ) = = V B = A(1 ) = A Subsiuion of he expression (8) ino he differenial equaion (2) leads o = V B A 1 eτ L A τ e τ = V B V B + V B LV B e τ τ when he value A = V B / is insered, from Eq. (9). This equaion is saisfied if and only if τ = L Wih hese paricular values, Eq. (8) becomes (d) = V B 1 e τ ; we have found he needed soluion o Eq. (1) for he curren i() in he circui of Figure 1! (I can also be shown ha his soluion is unique.) Oher L circui problems can be analyzed in his same way; as above, he seps are: a. Deermine he qualiaive behavior of he curren as a funcion of ime, including is imum value b. Add volages around he circui o find he appropriae differenial equaion. c. Try a soluion o (b) of he form An f () or A g (), where f () and g () are defined in Eq. (7), depending on wheher (a) was increasing or decreasing. [The consan A is equal o he imum value found in par (a).] The resuling equaion gives he correc value of τ Once you know i (), i is relaively simple maer o find any oher quaniy, such as he volage across he resisor, V = i, or he induced emf ε L = di/d. A summary of he resuls for ypical L circuis is presened in Figure 6. 6

7 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide Figure 6: L Circuis (τ = L/) V Energize i Deenergize ε L L V di/d Noe proporionaliies: ε L = L dd dd ; V = Swich is moved a = ENEGIZE (Corresponds o Charge ) DEENEGIZE (Corresponds o Discharge ) i or V C i or V a) f() c) g() L/ L/ ε L or dd dd ε L or dd dd b) g() d) g() L/ L/ 2. LC Circuis As he erm LC implies, we are assuming idealized inducors and capaciors, wih negligible resisive or oher dissipaive effecs; he circui in Figure 7 is consruced from such idealized componens. Wih he swich in posiion a, he capacior acquires he charge q = CV B As you learned in he module Capaciors, his implies ha amoun of energy 7

8 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide U C = 1 2 q 2 = U C is sored in he capacior When he swich is moved o posiion b as in Figure 8, posiive charge sars o flow from he upper plae of C hrough L. Evenually, he capacior becomes discharged (q = ), a which ime U C = Bu he energy U ha was originally sored in he capacior mus have gone somewhere. I could no have been convered o inernal ( hea ) energy, since here are no resisors in he circui; herefore i mus have gone ino he inducor. Tha is, he curren mus have he value I = i such ha he energy sored in inducor is V B a b q C Figure 7 a b L U L = 1 2 L 2 = U Of course, he inducor will no le he curren sop abruply; he capacior hus proceeds o charge up again, bu wih negaive charge on he op plae. The curren does sop, however, when all he energy has been ransferred back o he capacior, i.e., when U C = U and U L = Nex, he poenial difference across he charged capacior V B q C Figure 8 L plaes again sars a curren flowing hrough he inducor, bu in he opposie direcion from before and so on. Thus, here is a coninual ransfer of energy back and forh beween he capacior and inducor, in such a way ha he oal energy is consan: U L + U C U C (d) + U L (d) = U See Figure 9. This ransfer of energy back and forh is very nicely porrayed by he upper circular diagram on p.1[fig. 11(a)] U C U L Figure 9 i or V x q or x Figure 1 8

9 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide Figures 11(a) and (b) visualize he energy ransfer ha occurs during one cycle of an elecrical oscillaor [Figure 11(a)] and of a mechanical oscillaor [Figure 11 (b)]. Noe he amazingly similar behavior of hese apparenly dissimilar devices. The diagrams on his page [Figure 11(a) and 11(b)] have been reprined from Fundamenals of Physics, by David Halliday and ober esnick (Wiley, New York, 197; revised prining, 1974), wih permission of he publisher. In he ex hey are Figures 34-1 and 7-4 respecively. The lower circular diagram [Fig. 11(b)] shows he analogous siuaion in a mechanical oscillaor; he spring poenial energy is he analog of U C, and he kineic energy of he moving mass is he analog of U L. The oscillaion of he posiive charge beween he upper and lower plaes of c is very similar o he back and forh moion of he mass in he mechanical case; In fac, we can represen he charge q and he displacemen x (or he curren I and he velociy v x ) by he same graph, as Figure 1, provided we mach up he ampliudes, frequencies, and phases. The curves in Figure 1 were drawn o be sinusoidal; how can we check his claimed behavior? Simple enough: a. Wrie he equaion of energy conservaion 1 2 q2 C L2 = U b. Differeniae wih respec o ime 9

10 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide q dq + L d C dd dd = c. Use i = dq/d, cancel a facor, and rearrange: LC d2 q dd2 + q = Anoher differenial equaion! Bu do no despair; we are merely going o check he claimed sinusoidal behavior hus we se q(d) = q m cos(ωd + φ), (where q m, ω, and φ are consans o be deermined) and subsiue his expression ino Eq. (19) o see wheher or no i is a soluion. The subsiuion leads o LCω 2 q m cos(ωd + φ) + q m cos(ωd + φ) = which is saisfied provided ω = 1 LC The sinusoidal behavior (2) is his verified, and we have found he frequency f = ω 2π = 1 2π 1 LC As in he mechanical case, he values of q and φ are deermined by iniial condiions. For example, if he swich in Figure 7 is moved o b a =, hen q(d = ) = CV B and dq (d = ) = ; dd applying hese condiions o Eq. (2) yields q m = CV B and φ = ADDITIONAL LEANING MATEIALS Auxiliary eading Sanley Williams, Kenneh Brownsein, and ober Gray, Suden Sudy Guide wih I and II, by David Halliday and ober esnick (Wiley, New York, 197). Objecive 1: Secions 31-1 and 31-2; Objecive 2: Secions 31-4 and 31-7 hrough 31-9; Objecive 3: Secion 33-1 Various Texs Frederick J. Bueche, Inroducion o Physics for Scieniss and Engineers (McGraw-Hill, New York, 1975), second ediion: Secions 25.3 hrough 25.5 and

11 Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide David Halliday and ober esnick, Fundamenals of Physics (Wiley, New York, 197; revised prining, 1974): Secions 32-1 hrough 32-4 and 34-1 hrough 34-3 Francis Weson Sears and Mark W. Zemansky, Universiy Physics (Addison-Wesley, eading, Mass, 197), fourh ediion: Secions 33-9 hrough ichard T. Wediner and ober L. Sells, Elemenary Classical Physics (Allyn and Bacon, Boson, 1973), second ediion, Vol. 2: Secions 32-1 hrough 32-3 and POBLEM A(1) An inducor has been wound on a long cylindrical form wih a square cross secion measuring 1. cm by 1. cm. The winding has been pained over, so ha i is impossible o coun he urns; however, you are able o deermine ha he flux hrough he cener is 1. µt when he curren is 4. A. 11

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