Phys-272 Lecture 18. Mutual Inductance Self-Inductance R-L Circuits

Transcription

1 Phys-7 ectue 8 Mutual nductance Self-nductance - Cicuits

2 Mutual nductance f we have a constant cuent i in coil, a constant magnetic field is ceated and this poduces a constant magnetic flux in coil. Since the Φ B is constant, thee O induced cuent in coil. f cuent i is time vaying, then the Φ B flux is vaying and this induces an emf in coil, the emf is dφ B We intoduce a atio, called mutual inductance, of flux in coil divided by the cuent in coil. Φ M B i

3 Mutual nductance mutual inductance, Φ M B, can now be used in Faaday s eqn. i M i Φ B di d di M Φ B M ; We can also the vaying cuent i which ceates a changing flux Φ B in coil and induces an emf. This is given by a simila eqn. di M t can be shown (we do not pove hee) that, M M The units of mutual inductance is T m /A Webe/A Heny (afte Joseph Heny, who missed Faaday s aw) M

4 Mutual nductance The induced emf, has the following featues; di M The induced emf opposes the magnetic flux change (lenz s aw) The induced emf inceases if the cuent changes vey fast The induced emf depends on M, which depends only the geomety of the two coils and not the cuent. Fo a few simple cases, we can calculate M, but usually it is just measued.

5 Poblem 3. Two coils have mutual inductance of H. The cuent in the fist coil inceases at a unifom ate of 83 A/s. A) What is the magnitude of induced emf in the nd coil? s it constant? B) suppose that the cuent is instead in the nd coil, what is the magnitude of the induced emf in the st coil? M di (3.5 4 H)(83 A ) s.7v di M. 7V

6 Magnetic field due to coil is B µ n i µ i / l Mutual inductance is, l A l i A i i A B i M B µ µ Φ The induced emf in coil fom coil is di l A di M µ Tesla Coil Example

7 icolai Tesla ( ) Bon in Coatia, gaduated fom Univesity of Pague. Aived in ew Yok with 4cents and went to wok fo Edison. Tesla invented polyphase altenating-cuent system, induction moto, altenating-cuent powe tansmission, Tesla coil tansfome, wieless communication, adio, and fluoescent lights. He set up a Tesla coil in Coloado Spings in 899, below is a photo of this lab. He lighted lamps 4Km away. He also claimed to eceive messages fom anothe planet!! n hono of his contibutions to electomagnetic phenomena, the Magnetic field intensity was named in units of Tesla

8 Applications of Mutual nductance Tansfomes Change one AC voltage into anothe Aipot Metal Detectos Pulsed cuent pulsed magnetic field nduces emf in metal Feomagnetic metals daw in moe B lage mutual inductance lage emf Emf cuent (how much, how long it lasts, depends on the esistivity of the mateial) Decaying cuent poduces decaying magnetic field induces cuent in eceive coils Magnitude & duation of signal depends on the composition and geomety of the metal object. V V V (pimay) ion V (seconday)

9 Applications of Mutual nductance Pacemakes ot easy to change the battey! nstead, use an extenal AC supply. Altenating cuent altenating B altenating Ф B inside weae induces AC cuent to powe pacemake ~

10 Self nductance We peviously consideed induction between coils. ow we conside the situation whee a single isolated coil induces emf on itself. This is Called back emf and if the cuent changes, thee is a self induced emf that opposes the change in cuent. We fom the same atio, now called Self-nductance,, Φ i B and we have the back emf, di

11 Behavio of isolated coil in cicuits esisto with cuent has potential dop, Vi fom a to b Coil with a) constant cuent i has O Voltage dop b) di/>, potential deceases fom a to b, Vdi/ c) di/<, potential inceases fom a to b, V- di/ emembe, emf in coil opposes cuent change.

12 Self inductance of long solenoid l tuns ong Solenoid: tuns total, adius, ength l l B l µ << Fo a single tun, l BA A π µ φ π The flux though a tun is given by: l B π µ Φ nductance of solenoid can then be calculated as: l l l B π µ π µ Φ

13 Clicke: Two simple pieces of wie A and B ae shaped into almost complete loops. The loose ends of each loop ae connected to identical batteies. Assume the loops have the same total esistance, and that they do not intefee with each othe. ) Which loop has the geatest flux though it (assume the loops have the same cuent in them)? a) A b) B c) same 3) Which loop has the geatest self inductance? a) A b) B c) same ecall the B-field of cuent loop (at the cente): B µ /() The aea of the loop is A π Flux is Φ B ds The flux though the loop inceases with. Self inductance is defined as so if is the same, A > B. B Φ B The flux though loop A is bigge than the flux though loop B.

14 Clicke poblem Conside the two inductos shown: nducto has length l, total tuns and has inductance. nducto has length l, total tuns and has inductance. What is the elation between and? l tuns l tuns (a) < (b) (c) >

15 Clicke poblem Conside the two inductos shown: nducto has length l, total tuns and has inductance. nducto has length l, total tuns and has inductance. What is the elation between and? (a) < (b) (c) > l tuns l tuns To detemine the self-inductance, we need to detemine the flux Φ B which passes though the coils when a cuent flows: Φ B /. To calculate the flux, we fist need to calculate the magnetic field B poduced by the cuent: B µ (/l) i.e., the B field is popotional to the numbe of tuns pe unit length. Theefoe, B B. But does that mean?

16 How to combine inductos To calculate, we need to calculate the flux. Since B B, the flux though any given tun is the same in each inducto l tuns l tuns Thee ae twice as many tuns in inducto ; theefoe the net flux though inducto is twice the flux though inducto! Theefoe,. nductos in seies add (like esistos): eff + And inductos in paallel add like esistos in paallel: + eff

17 Self nductance of tooidal solenoid The magnetic field in a tooid was B and the net mag.flux is Φ B BA Hence the self inductance is, Φ i µ i π B µ i π A µ i A iπ µ π Example 3.3, A5cm,.m 7 4π 4 5 4µH π (.) A

18 Enegy of an nducto How much enegy is stoed in an inducto when a cuent is flowing though it? Stat with loop ule: + Multiply this equation by : + Fom this equation, we can identify P, the ate at which enegy is being stoed in the inducto: du P We can integate this equation to find an expession fo U, the enegy stoed in the inducto when the cuent : U U du d d d a b U d

19 Magnetic field Enegy in a tooid Conside a tooid magnet, the B field is, B µ /π (ex.8.). The enegy is, Substituting the B field into the Eqn., we have, A U π µ ( ) ( ) µ π µ µ π µ π B A U density Enegy B volume U A U µ π

20 Enegy in Electic Fields and Magnetic Fields n chapte 4.3, we discussed enegy in a paallel plate with aea A and sepaation d, The electic field enegy in the capacito was U CV C U Ad A d ( ) ( ) ( ) Ed enegy volume E Ed Ad ow we find the magnetic field enegy in the tooid magnet is U Aπ B, E enegy volume µ The fields ae popotional to the enegy density B E

21 Announcements - Cicuits (many iclicke examples) Hints fo Homewok X Some small inductos nducto in a powe supply Chaging Mat uses induction

22 nductos in Cicuits Geneal ule: inductos esist change in cuent Attached to a cuent souce nitially, the inducto behaves like an open switch. Afte a long time, the inducto behaves like an ideal wie. Disconnected fom a cuent souce nitially, the inducto behaves like a cuent souce.

23 Multi-pat clicke At t the switch is thown fom position b to position a in the cicuit shown: a b A What is the value of the cuent a long time afte the switch is thown? (a) (b) / (c) / B What is the value of the cuent immediately afte the switch is thown? (a) (b) / (c) / Afte a long time the switch is opened. C What is the value of the cuent just afte the switch is opened? (a) (b) / (c) /

24 Multi-pat clicke a b At t the switch is thown fom position b to position a in the cicuit shown: What is the value of the cuent a long time A afte the switch is thown? (a) (b) / (c) / A long time afte the switch is thown, the cuent appoaches an asymptotic value: as t, d/. As d/, the voltage acoss the inducto. /. Theefoe,

25 Multi-pat clicke At t the switch is thown fom position b to position a in the cicuit shown: B What is the value of the cuent immediately afte the switch is thown? a b (a) (b) / (c) / Just afte the switch is thown, the ate of change of cuent is as lage as it can be (we had been assuming it was!) The inducto limits d/ to be initially equal to /. The voltage acoss the inducto ; the cuent, then, must be! Anothe way: the moment the switch is thown, the cuent ties to geneate a huge B-field. Thee is a huge change in flux though coil an emf is geneated to oppose this. nitially, then, no cuent flows though no voltage dop acoss the esistos.

26 Multi-pat clicke Afte a long time the switch is C opened. What is the value of the cuent just afte the switch is opened? (a) (b) / (c) / a b Just afte the switch is thown, the inducto induces an emf to keep cuent flowing: emf d/ (can be much lage than ) Howeve, now thee s no place fo the cuent to go chages build up on switch contacts high voltage acoss switch gap f the electic field exceeds the dielectic stength (~3 kv/cm in ai) beakdown SPAK!

27 Cicuits, Quantitative At t, the switch is closed and the cuent stats to flow. a b oop ule: d ote that this equation is identical in fom to that fo the C cicuit with the following substitutions: Q dq C: C C : Theefoe, τ C C τ Q C

28 Cicuits To find the cuent as a function of time t, we need to choose an exponential solution which satisfies the bounday condition: a b d ( t ) ( t ) We theefoe wite: ( t / e ) The voltage dop acoss the inducto is given by: d t / V e τ

29 Cicuit ( on) Cuent Max / ( t / e ) / / / Sketch cuves! 63% Max at t/ t Voltage on V d e t / Max / V 37% Max at t/ t

30 Cicuits Afte the switch has been in position a fo a long time, edefined to be t, it is moved to position b. oop ule: + d a b The appopiate initial condition is: The solution then must have the fom: V e t / ( t d e ) t /

31 Cicuit ( off) Cuent Max / e t / / / / Sketch cuves! 37% Max at t/ Voltage on V d e t / V t Max - 37% Max at t/ - t

32 on off / / / / / / ( t / e ) e t / t t V V d e t/ V V d e t/ t - t

33 Clicke At t, the switch is thown fom position b to position a as shown:»et t be the time fo cicuit to each / of its asymptotic cuent. a a b et t be the time fo cicuit to each / of its asymptotic cuent. What is the elation between t and t? b (a) t < t (b) t t (c) t > t

34 Clicke At t, the switch is thown fom position b to position a as shown: et t be the time fo cicuit to each / of its asymptotic cuent. et t be the time fo cicuit to each / of its asymptotic cuent. What is the elation between t and t? a b a (a) t < t (b) t t (c) t > t We must detemine the time constants of the two cicuits by witing down the loop equations. b : : d d t d d d t d t τ τ This confims that inductos in seies add!

35 Anothe inducto example At t hs the switch is opened, abuptly disconnecting the battey fom the cicuit. What will happen to all the enegy stoed in the solenoid? Enegy stoed in the inducto: U / When the switch is opened, this enegy is dissipated in the esisto. An inducto doesn t like change!!! When the switch is opened, the inducto will ty to maintain the cuent that was flowing though it befoe the switch is opened. Since the battey is disconnected fom the cicuit, the enegy which is necessay to keep cuent flowing though the esisto is povided by the inducto.