Chapter 10 INDUCTANCE Recommended Problems:

Transcription

1 Chaper 0 NDUCTANCE Recommended Problems: 3,5,7,9,5,6,7,8,9,,,3,6,7,9,3,35,47,48,5,5,69, 7,7.

2 Self nducance Consider he circui shown in he Figure. When he swich is closed, he curren, and so he magneic field, hrough he circui increases from zero o a specific value. The increasing magneic flux induces an emf. By enz's law, his induced emf opposes he change in flux. The effec of his induced emf is o reard he change of he original curren, ha is, reard is increasing. The same phenomena occurred when he swich is opened where he curren in his case decreases from a specific value o zero. The emf induced due o he decreasing of he magneic flux now ends o oppose he decreasing of he original curren. b S R in

3 This phenomena is called he self inducion since he changing flux hrough he circui arises from he circui iself. The emf induced due o his phenomena is called he self-induced emf. f he emf induced in a circui is due o he changing of he magneic flux se up by anoher circui we have he muual inducion phenomena. To obain a quaniaive descripion of he self inducion, we know from Faraday's law ha he induced emf is proporional o he ime rae of he magneic flux, i.e., d N m Bu m d B and The consan is called he self-inducance, or simply he inducance of he coil. The S uni of inducance is Henry (H), which, from he las equaion, is equivalen o HV.s A B

4 Now comparing he las wo equaions d N m d N m As i is clear, depends on he geomeric feaures of he coil. should be noed ha all elemens in a circui have some inducance bu i is oo small o be significan excep ha of a coil. A coil ha has significan inducance is called inducor, and is represened in he circuis by he symbol Example 3. Find he inducance of an ideal solenoid of N urns and lengh l. Soluion: Knowing ha, inside he solenoid B is uniform and given by N N B on o m BAcos0 o A l l N o N l A on l A

5 Tes Your Undersanding () A coil wih zero resisance has is ends labeled a and b. The poenial a a is higher han a b. Which of he following could be consisen wih his siuaion? a b a) The curren is increasing and is direced from a o b; b) The curren is decreasing and is direced from a o b; c) The curren is increasing and is direced from b o a; d) The curren is consan and is direced from b o a.

6 R Circuis Consider he R circui shown. Suppose ha he swich is hrown o poin a =0. Applying Kirchhoff's loop rule o he circui a ime we ge d R 0 is no difficul o verify ha he soluion of he differenial equaion given in above is and R e wih s he ime consan of he R circui is clear ha a =0, =0, while as, =. This means ha: ha is, he inducor acs as if i were an open circui a =0, and acs as if i were a wire wih negligible resisance a. R S R

7 f he baery is suddenly removed, by hrowing he swich o poin in he circui and applying Kirchhoff's rule again we ge d R 0 e The variaions of wih ime are ploed in in he figure shown. As i is clear from he graph (a), he curren akes some ime o reach is imum value. m q The graph of Figure (b) ells ha he curren akes some ime o reach i zero value. (a) n anoher word, he inducor has he effec o hinder he curren from reaching is final value for some ime. o (b)

8 f one plo he variaion of vs ime when closing or opening a swich in a circui wih and wihou an inducor we obain Closing he swich Opening he swich Red line represening vs wihou self inducion Blue line represening vs wih self inducion

9 Example 3.3 Consider he circui shown. Find he ime consan of he circui, he curren in he circui a =ms, and compare he P.D across he resisor wih ha across he inducor. Soluion S 6 V 30 mh a) The ime consan is given by he Equaion R ms b) The curren is e e A 6

10 c) The P.D. across he resisor is given by e R R V o R While he P.D. across he inducor is given by Re e d V o o V R V V o R

11 Tes Your Undersanding () For he circui shown in he figure, jus afer closing he swich S, across which of he following is he volage equal o he emf of he baery? (a) R. (b). S R (c) Boh R and. (d) Neiher R nor. Afer closing he S for a long ime, across which of he following is equal o he emf of he baery? (a) R. (b). (c) Boh R and. (d) Neiher R nor.

12 Tes Your Undersanding (3) The circui shown includes sinusoidal volage source such ha he magneic field in he inducor is consanly changing. The inducor is a simple aircore solenoid. The swich in he circui is closed and he lighbulb glows seadily. An iron rod is insered ino he inerior of he solenoid, which increases he magniude of he magneic field in he solenoid. As his happens, he brighness of he lighbulb (a) increases, (c) Goes off, (b) decreases, (d) unaffeced.

13 Example (problem 3) The swich in he figure is open for <0 and hen closed a ime =0. Find he currens if he circui a =0 and a long ime afer closing he swich. Soluion 0 V 4 S 4 A =0, he inducor reaed as an open circui H 0 3.5A 8 Afer a long ime he inducor reaed as a wire R eq A.0A and eq A

14 Energy in Magneic Field e us sar from he equaion of he R circui, i.e., d R 0 Muliplying he above equaion by R d 0 The s erm represens he power of he baery, while he nd erm represens he power delivered o he resisor he 3 rd erm represens he power delivered o he inducor, i.e., P du d du U m 0 d

15 Example 3.4 Consider once again he R circui shown wih he swich is hrown o poin afer being on posiion for a long ime. Show ha all he energy iniially sored in he magneic field of he inducor appears as inernal energy in he resisor as he curren decays o zero. Soluion: is known ha he curren decay as du R P R o e 0 du o e R R Roe R U R R o o S R

16 Oscillaions in an C Circui Consider he circui shown wih he capacior is charged wih Q. Afer closing S he charge will flow hrough he inducor. A some ime le he charge in he capacior o be q and he curren in he inducor o be. The oal energy in he circui a his ime is U oal UC U q C Deriving he above Eq. wih respec o ime du oal dq C d C S

17 bu du q dq d oal 0 0 C Bu dq d d q q d q and 0 C or d q q C 0 This differenial Eq.is he SHM equaion wih is soluion q Q cos To find he consan we know ha q Q a 0 0 q Q cos wih C

18 To find he curren we have dq Q sin sin wih Q Knowing ha T, Q q Q cos T sin T Q m m T T

19 , Q Q m m T T

20 e s go back o he energy expression U oal U C U Q C Subsiuing for Q and we ge U C U U m U C bu oal Q C cos Q sin Q C U U oal Q C cos Q C Q C sin cos sin U oal Q C

21 Example 3.3 Consider he Circui show. Firs S is open and S is closed such ha he capacior is charged. Now if S is opened o remove he baery and hen S is closed o connec he capacior wih he inducor. a) Find of he circui. b) Find Q and. c) Find () and Q(). V 9 pf.8 mh S S Soluion a) The frequency is given by C Hz b) The imum charge on he capacior is he iniial charge before opening S, i.e., Q C C

22 And for he curren we have 6 Q A c) Using he obained resuls we ge cos sin q( ) Q cos 0 q( ) sin 0

23 Tes Your Undersanding (4) A an insan of ime during he oscillaions of an C circui, he curren is a is imum value. A his insan, he volage across he capacior: a) is differen from ha across he inducor b) is zero c) has is imum value d) is impossible o deermine A an insan of ime during he oscillaions of an C circui, he curren is a is momenarily zero. A his insan, he volage across he capacior: a) is differen from ha across he inducor b) is zero c) has is imum value d) is impossible o deermine