Mutual Inductance. If current i 1 is time varying, then the Φ B2 flux is varying and this induces an emf ε 2 in coil 2, the emf is

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1 Mutua Inductance If we have a constant cuent i in coi, a constant magnetic fied is ceated and this poduces a constant magnetic fux in coi. Since the Φ B is constant, thee O induced cuent in coi. If cuent i is time vaying, then the Φ B fux is vaying and this induces an emf ε in coi, the emf is ε dφ B We intoduce a atio, caed mutua inductance, of fux in coi divided by the cuent in coi. Φ M B i

2 Mutua Inductance mutua inductance, Φ M B, can now be used in Faaday s eqn i M i Φ B di d di M Φ B M ε ; ε We can aso the vaying cuent i which ceates a changing fux Φ B in coi and induces an emf ε. This is given by a simia eqn. ε di M It can be shown (we do not pove hee) that, M M The units of mutua inductance is T m /A Webe/A Heny (afte the Joseph Heny, who missed Faaday s Law) M

3 Mutua Inductance The induced emf, has the foowing featues; ε di M The induced emf opposes the magnetic fux change The induced emf inceases if the cuents changes vey fast The induced emf depends on M, which depends ony the geomety of the two cois and not the cuent. Fo a few simpe cases, we can cacuate M, but usuay it is just measued.

4 Pobem 30. Two cois have mutua inductance of H. The cuent in the fist coi inceases at a unifom ate of 830 A/s. A) What is the magnitude of induced emf in the nd coi? Is it constant? B) suppose that the cuent is instead in the nd coi, what is the magnitude of the induced emf in the st coi? ε M di ( H)(830 A ) s 0.7V di ε M 0. 7V

5 Magnetic fied due to coi is B μ 0 n i μ 0 i / Mutua inductance is, A i A i i A B i M B 0 0 μ μ Φ The induced emf in coi fom coi is di A di M 0 μ ε Tesa Coi Exampe

6 icoai Tesa ( ) Bon in Coatia, gaduated fom Univesity of Pague. Aived in ew Yok with 4cents and went to wok fo Edison. Tesa invented poyphase atenating-cuent system, induction moto, atenating-cuent powe tansmission, Tesa coi tansfome, wieess communication, adio, and fuoescent ights. He set up a Tesa coi in Cooado Spings in 899, beow is a photo of this ab. He ighted amps 40Km away. He aso caimed to eceive messages fom anothe panet!! In hono of his contibutions to eectomagnetic phenomena, the Magnetic fied intensity was named in units of Tesa

7 Appications of Mutua Inductance Tansfomes Change one AC votage into anothe ε Aipot Meta Detectos Pused cuent pused magnetic fied Induces emf in meta Feomagnetic metas daw in moe B age mutua inductance age emf Emf cuent (how much, how ong it asts, depends on the esistivity of the mateia) Decaying cuent poduces decaying magnetic fied induces cuent in eceive cois Magnitude & duation of signa depends on the composition and geomety of the meta object. V V ion V V (pimay) (seconday)

8 Appications of Mutua Inductance Pacemakes It s not easy to change the battey! Instead, use an extena AC suppy. Atenating cuent atenating B atenating Ф B inside weae induces AC cuent to powe pacemake ~

9 Appications of Mutua Inductance

10 Sef Inductance We peviousy consideed induction between cois. ow we conside the situation whee a singe isoated coi induces emf on itsef. This is Caed back emf and if the cuent changes, thee is a sef induced emf that opposes the change in cuent. We fom the same atio, now caed Sef-Inductance, L, L Φ i B and we have the back emf, L ε di

11 Behavio of isoated coi in cicuits Resisto with cuent I has potentia dop, ViR fom a to b Coi with a) constant cuent i has O Votage dop b) di/>0, potentia deceases fom a to b, VLdi/ c) di/<0, potentia inceases fom a to b, V-L di/ Remembe, emf in coi opposes cuent change.

12 Sef inductance of ong soenoid tuns Long Soenoid: tuns tota, adius, Length I B 0 μ << Fo a singe tun, 0 I BA A π μ φ π The fux though a tun is given by: 0 I B π μ Φ Inductance of soenoid can then be cacuated as: 0 0 I L B π μ π μ Φ

13 Lectue 7, Act Conside the two inductos shown: Inducto has ength, tota tuns and has inductance L. Inducto has ength, tota tuns and has inductance L. What is the eation between L and L? tuns tuns (a) L < L (b) L L (c) L > L

14 Lectue 7, Act Conside the two inductos shown: Inducto has ength, tota tuns and has inductance L. Inducto has ength, tota tuns and has inductance L. What is the eation between L and L? (a) L < L (b) L L (c) L > L tuns tuns To detemine the sef-inductance L, we need to detemine the fux Φ B which passes though the cois when a cuent I fows: L Φ B / I. To cacuate the fux, we fist need to cacuate the magnetic fied B poduced by the cuent: B μ 0 (/)I i.e., the B fied is popotiona to the numbe of tuns pe unit ength. Theefoe, B B. But does that mean L L?

15 Lectue 7, Act To cacuate L, we need to cacuate the fux. SinceB B, the fux though any given tun is the same in each inducto tuns tuns Thee ae twice as many tuns in inducto ; theefoe the net fux though inducto is twice the fux though inducto! Theefoe, L L. Inductos in seies add (ike esistos): Leff L + L And inductos in paae add ike esistos in paae: + L L L eff

16 Sef Inductance of tooida soenoid The magnetic fied in a tooid was B μ0i π and the net mag. fux is μ i Appoximates B ΦB BA 0 A as vaue at π cente of coi. Hence the sef inductance is, Φ i L B μ0 μ A 0 A Exampe: i iπ π 00, A 5 cm, 0. m π L (5 0 m ) x0 H 0. mh π (0.m)