Physics 1402: Lecture 22 Today s Agenda

Transcription

1 Physics 142: ecure 22 Today s Agenda Announcemens: R - RV - R circuis Homework 6: due nex Wednesday Inducion / A curren Inducion Self-Inducance, R ircuis X X X X X X X X X long solenoid Energy and energy densiy 1

2 ε on ε off ε/r /R 2/R ε/r /R 2/R I I ε V V -ε harging R 2R ε ε Discharging R 2R q q ε/r I I - ε/r 2

3 3

4 Muual Inducance Suppose you have wo coils wih muliple urns close o each oher, as shown in his cross-secion We can define muual inducance M 12 of coil 2 wih respec o coil 1 as: oil 1 oil 2 B N 1 N 2 I can be shown ha : 4

5 Inducors in Series Wha is he combined (equivalen) inducance of wo inducors in series, as shown? a a Noe: he induced EMF of wo inducors now adds: 1 2 b b eq Since: And: Inducors in parallel Wha is he combined (equivalen) inducance of wo inducors in parallel, as shown? a a Noe: he induced EMF beween poins a and be is he same! 1 b 2 b eq Also, i mus be: We can define: And finally: 5

6 onsider he and R series circuis shown: ircuis R Suppose ha he circuis are formed a = wih he capacior charged o a value Q. laim is ha here is a qualiaive difference in he ime developmen of he currens produced in hese wo cases. Why?? onsider from poin of view of energy! In he R circui, any curren developed will cause energy o be dissipaed in he resisor. In he circui, here is NO mechanism for energy dissipaion; energy can be sored boh in he capacior and he inducor! Q i R R/ ircuis Q i R: curren decays exponenially i : curren oscillaes -i 1 6

7 Oscillaions (qualiaive) Energy ransfer in a resisanceless, nonradiaing circui. The capacior has a charge Q max a =, he insan a which he swich is closed. The mechanical analog of his circui is a block spring sysem. 7

8 Oscillaions (quaniaive) Wha do we need o do o urn our qualiaive knowledge ino quaniaive knowledge? Wha is he frequency ω of he oscillaions (when R=)? (i ges more complicaed when R finie and R is always finie) Begin wih he loop rule: Oscillaions (quaniaive) Q i Guess soluion: (jus harmonic oscillaor!) remember: where: ω deermined from equaion φ, Q deermined from iniial condiions Procedure: differeniae above form for Q and subsiue ino loop equaion o find ω. 8

9 Review: Oscillaions Guess soluion: (jus harmonic oscillaor!) Q i where: ω deermined from equaion φ, Q deermined from iniial condiions which we could have deermined from he mass on a spring resul: The energy in circui conserved! When he capacior is fully charged: When he curren is a maximum (I o ): The maximum energy sored in he capacior and in he inducor are he same: A any ime: 9

10 ecure 22, AT 1 A = he capacior has charge Q ; he resuling oscillaions have frequency ω. The maximum curren in he circui during hese oscillaions has value I. Wha is he relaion beween ω and ω 2, he 1A frequency of oscillaions when he iniial charge = 2Q? (a) ω 2 = 1/2 ω (b) ω 2 = ω (c) ω 2 = 2 ω ecure 22, AT 1 A = he capacior has charge Q ; he resuling oscillaions have frequency ω. The maximum curren in he circui during hese oscillaions has value I. 1B Wha is he relaion beween I and I 2, he maximum curren in he circui when he iniial charge = 2Q? (a) I 2 = I (b) I 2 = 2 I (c) I 2 = 4 I 1

11 Summary of E&M J.. Maxwell (~186) summarized all of he work on elecric and magneic fields ino four equaions, all of which you now know. However, he realized ha he equaions of elecriciy & magneism as hen known (and now known by you) have an inconsisency relaed o he conservaion of charge! Gauss aw Gauss aw For Magneism Faraday s aw Ampere s aw I don expec you o see ha hese equaions are inconsisen wih conservaion of charge, bu you should see a lack of symmery here! Ampere s aw is he ulpri! Gauss aw: Symmery: boh E and B obey he same kind of equaion (he difference is ha magneic charge does no exis!) Ampere s aw and Faraday s aw:! If Ampere s aw were correc, he righ hand side of Faraday s aw should be equal o zero -- since no magneic curren. Therefore(?), maybe here is a problem wih Ampere s aw. In fac, Maxwell proposes a modificaion of Ampere s aw by adding anoher erm (he displacemen curren) o he righ hand side of he equaion! ie 11

12 Displacemen curren Remember: I in Φ E I ou changing elecric flux 12

13 Maxwell s Displacemen urren an we undersand why his displacemen curren has he form i does? onsider applying Ampere s aw o he curren shown in he diagram. If he surface is chosen as 1, 2 or 4, he enclosed curren = I If he surface is chosen as 3, he enclosed curren =! (ie here is no curren beween he plaes of he capacior) circui Big Idea: The Elecric field beween he plaes changes in ime. displacemen curren I D = ε (dφ E /d) = he real curren I in he wire. 13

14 Maxwell s Equaions These equaions describe all of Elecriciy and Magneism. They are consisen wih modern ideas such as relaiviy. They even describe ligh 14