" = #N d$ B. Electromagnetic Induction. v ) $ d v % l. Electromagnetic Induction and Faraday s Law. Faraday s Law of Induction
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1 Eletromgnet Induton nd Frdy s w Eletromgnet Induton Mhel Frdy ( ) dsoered tht hngng mgnet feld ould produe n eletr urrent n ondutor pled n the mgnet feld. uh urrent s lled n ndued urrent. The phenomenon s lled eletromgnet nduton. Eletromgnet Induton 1 Eletromgnet Induton 2 Frdy s w of Induton Frdy found tht the ndued emf s proportonl to the rte of hnge of the mgnet flux! pssng through the loop of re A. " = d In generl " = da For mgnet felds tht re onstnt " = A = Aos" Eletromgnet Induton 3 Frdy s w of Induton If the flux pss through N loops the ndued emf s " = N d The mnus sgn s neessry to ge the orret dreton the ndued emf ts. Eletromgnet Induton 4 enz s w An ndued emf lwys ges rse to urrent whose mgnet feld opposes the orgnl hnge n flux. " = A = Aos Note: An emf s ndued wheneer there s hnge n flux nd n e ndued n three wys: 1.) y hngng the mgnet feld 2.) y hngng the re A of the loop n the feld 3.) y hngng the orentton " the mgnet feld mkes wth the loop Eletromgnet Induton 5 Motonl emf F = q " E = F = q = E - q q l F (Eletr Feld n rod) E = "V "d ondutor " = l (Motonl emf ) For ondutor of ny shpe, mong n ny mgnet feld. " = ( ) d % l Eletromgnet Induton 6
2 emf Indued n Mong ondutor me Knd of Prolem Assume tht unform mgnet feld s perpendulr to the re ounded y U-shped ondutor nd mole rod restng on t. The rod s mde to moe t speed. The mgntude of the ndued emf s gen y = d ( " A ) = da E = d" dx = d( Aos" ) l da = l dx = ldx = l Eletromgnet Induton 7 l F = I l " F Enterng Mgnet Feld Eletromgnet Induton 8 me Knd of Prolem ottng ols F Flux Deresng eng Mgnet Feld F = I l " Eletromgnet Induton 9 Mxmum flux Zero flux Eletromgnet Induton 1 d" ottng ols " = A = Aos = daos = A dos" ne " = t (ssumng onstnt ngulr speed) d" = A dost = "Asnt Eletromgnet Induton 11 Indued Eletr Felds hngng -Feld ondute loop The hngng mgnet flux uses n ndued eletr feld n the loop. " = E d l = % d& I ndued There s fore tht mke the hrges moe round the loop. Eletromgnet Induton 12
3 Mxwell s Equtons 1.) Guss s w for Eletr Felds E " da = Q enlosed o 2.) Guss s w for Mgnet Felds " da = Mxwell s Equtons 3.) Ampere s w " d & d% l = µ o E ) ( ' o * 4.) Frdy s w E " d l = d% Eletromgnet Induton 13 Eletromgnet Induton 14 Mutul Indutne onsder two neghorng ols of wre. A urrent flowng n ol 1 produes mgnet flux through ol 2. Indutne 1 1 ol 1 ol 2 If the urrent 1 hnges then t wll ndue urrent n ol 2. If! 2 s the flux n ol 2 due to the urrent n ol 1 then the ndued emf wll e " 2 = N 2 d 2 Eletromgnet Induton 15 Eletromgnet Induton 16 Mutul Indutne If we ntrodue proportonlty onstnt M 21 lled the mutul ndutne of the two ols N 2 " 2 = M 21 1 where! 2 s the flux n sngle turn of ol 2. Then N 2 d" 2 = M 21 d 1 Mutul Indutne Ths n e repeted for the se where urrent 2 n ol 2 ndues n emf n ol 1 resultng n d " 1 d 1 = N 1 = "M 2 12 where! 1 s the flux n sngle turn of ol 1. It turns out tht M 21 s lwys equl to M 12 nd s wrtten s M, nd t ompletely hrterzes the ndued-emf nterton etween the two ols. The mutully ndued emf s re then " 2 = N 2 d 2 = "M 21 d 1 " 1 = M d 2 nd " 2 = M d 1 Eletromgnet Induton 17 Eletromgnet Induton 18
4 Mutul Indutne Mutul ndutne (M) desres the ouplng etween two ols n whh hngng urrent n one ol ndues n emf n n djent ol. M = N 2" 2 1 The I unt of mutul ndutne s lled the Henry (1 H). 1 H = 1 W A = N 1" 1 2 = 1 V " s A = 1 " s elf Indutne Any rut tht rres ryng urrent wll he n emf ndued n t y the rton n ts own mgnet feld. uh n emf s lled self-ndued emf. elf-ndutne () of rut s gen y: ne " = N d = N" = N" or d = N d" " = d (self-ndued emf) Eletromgnet Induton 19 Eletromgnet Induton 2 Indutors A rut element desgned to he prtulr ndutne s lled n ndutor (). The potentl dfferene V etween the termnls of the ndutor s equl n mgntude to the self-ndued emf. onstnt = d = nresng V = V > d s poste V = " = d Eletromgnet Induton 21 deresng d s negte V < Indutors nd Energy The totl energy U needed to estlsh fnl urrent I n n ndutor wth ndutne n lso e determned. P = V The energy du suppled to the ndutor durng tme nterl s: du = P I U = " d = 1 2 I2 = d Eletromgnet Induton 22 = q ptors ersus Indutors ptors so = 1 " tores energy n the form of n eletr feld. Indutors = d tores energy n the form of mgnet feld. U = 1 2 V 2 U = 1 2 I 2 esst hnges n oltge. esst hnges n urrent. ruts (urrent Growth) E " " = E " " d = d = E " Eletromgnet Induton 23 Eletromgnet Induton 24
5 d = E " ruts (urrent Growth) d = E " d = " " E & ' = " " E d t " E = " ln " E & % ' ( ) * = " t ln " E & % ' (" ln " E & % ( = " ' t " E & ' ln " E = " t " E & ' " E = e " t " E = " E e" t = E " E e" t (t) = E 1" e" t & ' Eletromgnet Induton 25 ruts (urrent Growth) The urrent n n rut res ordng to: (t ) = E 1" e" t & ' / s lled the tme onstnt (% ) nd s the tme t tkes the urrent to eome 63.2%of ts fnl lue. The oltge ross res ordng to: V = " d = "Ee " t Eletromgnet Induton 26 ruts (urrent Growth) (t ) = E 1" e" t & ' V = "Ee " t I = / / - V Eletromgnet Induton 27 t t ruts (urrent Dey) I = / = " d = d = " = " = = Eletromgnet Induton 28 ruts (urrent Dey) I = / = = " d d = " = = Eletromgnet Induton 29 ruts (urrent Dey) d = " d t " = " I ln( )] I = " t ln( ) " ln( I ) = " t " ln % ' = ( & t I = e " I (t) = I e " t t I = E & ' Eletromgnet Induton 3
6 ruts (urrent Dey) (t ) = I e " t I = E & ' I I /e t / Eletromgnet Induton 31 ersus ruts Unhrged Unmgnetzed = short = open " rut " rut hrged Mgnetzed " open " = rut = hrged ptor hs sme oltge s the dee tht s eletrlly prllel to t. short rut Mgnetzed ndutor hs sme urrent s the dee tht s eletrlly n seres wth t. Eletromgnet Induton 32 ruts (Eletrl Osllton) ruts (Eletrl Osllton) V m Q -Q -V m -Q Q V m Q -Q I m I m m m t = t = T/4 t = T/2 t = 3T/4 t = T E U E U E U E U E U Eletromgnet Induton 33 Eletromgnet Induton 34 ruts In n rut wth no energy losses the hrge on the ptor oslltes k nd forth. q -q " d " q = d 2 q 2 1 q = ruts Ths equton hs extly the sme form s tht for smple hrmon moton. whose soluton ws where d 2 q 1 2 q = d 2 x 2 k m x = x = Aos ("t ) " = k m Eletromgnet Induton 35 Eletromgnet Induton 36
7 ruts d 2 q 2 1 q = In the nlogous eletrl stuton the ptor hrge q s gen y q = Qos ("t ) nd the ngulr frequeny of the osllton s gen y " = 1 Eletromgnet Induton 37
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