Faraday s Law. Faraday s Law. Faraday s Experiments. Faraday s Experiments. Magnetic Flux. Chapter 31. Law of Induction (emf( emf) Faraday s Law

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1 Faaday s Law Faaday s Epeiments Chapte 3 Law of nduction (emf( emf) Faaday s Law Magnetic Flu Lenz s Law Geneatos nduced Electic fields Michael Faaday discoeed induction in 83 Moing the magnet induces a cuent eesing the diection eeses the cuent eesing the magnet eeses the cuents The induced cuent is set up by an induced EMF Faaday s Epeiments Magnetic Flu d/ EMF Changing the cuent in the ight-hand coil induces a cuent in the left-hand coil The induced cuent does not depend on the size of the cuent in the ight-hand coil; t depends on d/ These ey diffeent appeaing cases can be united by the concept of magnetic flu n the easiest case, with a constant and a flat suface of aea aea, the magnetic flu is = Φ Units : tesla m = webe Magnetic Flu Faaday s Law When is not constant, o the suface is not flat, one must do an integal eak the suface into bits d The flu though one bit is dφ = d = d cos dd them: d Φ = d = cos d Moing the magnet changes the flu Φ Change cuent changes the flu Φ nd changing the flu induces an emf: The emf induced aound a loop i emf = - dφ / i di/ EMF Faaday s law equals the ate of change of the flu though that loop

2 Lenz s Law Faaday s law gies the diection of the induced emf and theefoe of any induced cuent Lenz s law is a simple way to get the diections staight with less effot Lenz s Law: The induced emf is diected so that any esulting induced cuent flow will oppose the change in magnetic flu which causes the induced emf Lenz s Law Faaday s law gies the diection of the induced emf and theefoe of any induced cuent Lenz s law is a simple way to get the diections staight with less effot Lenz s Law: The induced emf is diected so that any esulting induced cuent flow will oppose the change in magnetic flu which causes the induced emf Lenz s Law: obody wants to be FLUXE with Eample of Faaday s Law Eample of Faaday s Law Conside a coil of adius 5 cm with = 50 tuns magnetic field though it changes at the ate of d/ = 06 T/s The total esistance of the coil is 8 Ω What is the induced cuent? d Lenz s law: d Use Lenz s law to detemine the diection of the induced cuent pply Faaday s law to find the emf and then the cuent nduced Eample of Faaday s Law d Lenz s law: The change in is inceasing the upwad flu though the coil o the induced cuent will hae a magnetic field whose flu (and theefoe field) is down Hence the induced cuent must be clockwise when looked at fom aboe ow use Faaday s law to get the magnitude of the induced emf and cuent nduced d The induced EMF is emf = - dφ / Theefoe emf = - (π ) d / n tems of : Φ = () = (π ) emf = - (50) (π 0005 )(06T/s) = -8 V (V=Tm /s) Cuent = emf / = (8V) / (8 Ω) = 047

3 Type of Poblems with Faaday s Law Φ = = cos() and emf = -dφ / Fo the emf to be non-zeo one (o moe) of thee things must be changing in time:,, o Thus, thee ae thee types of poblems n the peious eample was changing in time, d/ changing in time f d / = ω, then the esult is an C geneato 3 nd, finally the ea,, can change in time Motional EMF Up until now we hae consideed fied loops The flu though them changed because the magnetic field changed with time ow ty moing the loop in a unifom and constant magnetic field This changes the flu, too points into sceen Eample: Find the cuent in the esisto Φ = d Φ = = y Φ = y Eample: Find the diection of the cuent y Whee, = 5 Ω y = 5 m = 36 m = 5 m/s = 3 T dφ d d = y = y dφ = y dφ emf = = y emf y i = = = 5 Whee, = 5 Ω y = 5 m = 36 m = 5 m/s = 3 T Eample: Find the foce equied to moe the segment at elocity, F Powe = Whee, = 5 Ω y = 5 m = 36 m = 5 m/s = 3 T y Powe = i F = Powe i F = = 05 ut the foce is on the moing wie, not the esisto F = il = iy??? emembe what is happening when a conducto is moed though a field: Wie is like a cylinde full of fee electons: Motional EMF V

4 Motional EMF emembe what is happening when a conducto is moed though a field: Wie is like a cylinde full of fee electons: V E F V= Motional EMF emembe what is happening when a conducto is moed though a field: ow E = E, E=E/ (E is electic field, E is emf) We know F=eE V so, F=e=eE / E= E F V= ow use Faaday s Law: Faaday s Law: The flu is Φ = = This changes in time: dφ / = d()/ = d/ = - Hence by Faaday s law thee is an induced emf and cuent What diection is it? Lenz s law: thee is less inwad flu though the loop Hence the induced cuent gies inwad flu o the induced cuent is clockwise ow Faaday s Law dφ / = - E gies the EMF: E = n a cicuit with a esisto, this gies E = = : = / Thus moing a cicuit in a magnetic field poduces an emf eactly like a battey This is the pinciple of an electic geneato otating Loop - The Geneato Conside a loop of aea in a egion of space in which thee is a unifom magnetic field otate the loop with an angula fequency ω

5 otating Loop - The Geneato Conside a loop of aea in a egion of space in which thee is a unifom magnetic field otate the loop with an angula fequency ω The flu changes because angle changes with time: = ωt Hence: Φ = = cos() = cos(ωt) dφ / = d(cos(ωt))/ = d(cos(ωt))/ = - ω sin(ω t) dφ / = - ω sin(ω t) 3 Then by Faaday s Law this motion causes an emf E = - dφ / = ω sin(ω t) This is an C (altenating cuent) geneato new souce of EMF f we hae a conducting loop in a magnetic field, we can ceate an EMF (like a battey) by changing the alue of This can be done by changing the aea, by changing the magnetic field, o both We can use this souce of EMF in electical cicuits in the same way we used batteies emembe we hae to do wok (kinetic enegy) to moe the loop o change, to geneate EMF (othing is fo fee) Eample: cicula UHF TV antenna has a diamete of cm The magnetic field of a TV signal is nomal to the plane of the loop, and at any instant in time its magnitude is changing at the ate of 57 mt/s What is the EMF? Eample: cicula UHF TV antenna has a diamete of cm The magnetic field of a TV signal is nomal to the plane of the loop, and at any instant in time its magnitude is changing at the ate of 57 mt/s What is the EMF? Eample: a 0 tun coil (= 8 cm, = 53Ω ) is placed outside a solenoid,(=6cm, n=0/cm, i=5) The cuent in the solenoid is educed to 0 in 06s What cuent appeas in the coil? Magnetic flu: Φ = d = d cos d ( π ) nduced EMF: E d = [ Φ d ( 05 ] d = [ π ) = ( 05π ) = ( )[ 04π ( 0m) ]( 057T / s) 55 V

6 Eample: a 0 tun coil (= 8 cm, = 53Ω ) is placed outside a solenoid,(=6cm, n=0/cm, i=5) The cuent in the solenoid is educed to 0 in 06s What cuent appeas in the coil? Cuent induced in coil i c EMF d ic = =( ) Φ = = µ 0 Φ d ni s s d( µ 0nis s ) dis i0 ic = = µ 0ns = µ 0n t i = 597m c Eddy Cuents f you moe any conducto though a magnetic field, then you induce a cuent in the conducto Ciculating cuents F ae called eddy cuents ecause the esistance is small, these cuents can be lage Cuents dissipate enegy in conducto Motion of cuent is such as to poduce foce opposing motion Can be used as a fail-safe baking system Can be undesiable Laminations in coes of electic motos minimize eddy cuents and educe heat buildup nduced electic fields nduced electic fields Conside a conducto in a timeaying magnetic field When we induce a cuent in the conducto, Conside a conducto in a timeaying magnetic field When we induce a cuent in the conducto, the fee chages in the conducto must then epeience a foce The foce on a chage is qe This field is called induced electic field What kind of field is this? Wok done in inducing field must be Edl Thus E=- d Φ d = d = E d " The induced electic field is not conseatie nduced electic fields Conside a conducto in a timeaying magnetic field - with no conducto pesent: - is thee now an electic field? nduced electic fields Conside a conducto in a timeaying magnetic field - with no conducto pesent: - is thee now an electic field? d " - YE - the field is only felt by the chages in a conducto, not caused by them - d Φ = d d = E d "

7 emf = - dφ / Faaday s law dφ E d" = - Eample: magnetic field diected into a cicula egion of the boad is gien by = Csin(ωt), whee C = 3 T, ω = 30 ad/s and = 04 m electic field at P, =0 m at t=s? b What is the distance,, if E = E? with E d " = emf Eample: magnetic field diected into a cicula egion of the boad is gien by = Csin(ωt), whee C = 3 T, ω = 30 ad/s and = 04 m electic field at P, =0 m at t=s? b What is the distance,, if E = E? a pply Faaday s law d E d" = - Φ y symmety, E is constant and cicula at a fied adius E d " = E π Eample: magnetic field diected into a cicula egion of the boad is gien by = Csin(ωt), whee C = 3 T, ω = 30 ad/s and = 04 m electic field at P, =0 m at t=s? b What is the distance,, if E = E? E d " = E π ole the H -dφ / = π cos( ω ) t Φ fist, then dφ /, then finish E = Φ = = π dφ d d ( Csin( ωt) ) = π = π = π Eample: magnetic field diected into a cicula egion of the boad is gien by = Csin(ωt), whee C = 3 T, ω = 30 ad/s and = 04 m electic field at P, =0 m at t=s? b What is the distance,, if E = E? E = ads ads E = 0 m(3t)30 cos(30 s) = 5 b pply Faaday s law at P s s dφ E d" = - Φ = π d E π = -π = -π E = - C Eample: magnetic field diected into a cicula egion of the boad is gien by = Csin(ωt), whee C = 3 T, ω = 30 ad/s and = 04 m electic field at P, =0 m at t=s? b What is the distance,, if E = E? E = cos( ω ) t E = - = = = 0 = 4 m 0 08 m