1The story so far…Magnetic fieldgenerated by currentelement: Biot-SavartAmpere’s lawdIdBr! dB =µo4"Ids #ˆ r r2! B • ds"=µoIclosed pathsurface bounded by pathIThur. Oct. 30, 2008 Physics 208, Lecture 18 2Exam 2 resultsGradeboundaries:A:83AB: 76B: 67BC: 57C: 43D: 230510152025303520 40 60 80 100Phy208 Exam 2CountScoreAve=70Thur. Oct. 30, 2008 Physics 208, Lecture 18 3Magnetic field from a current loop One loop:field still loops around the wire. Many loops: same effectThur. Oct. 30, 2008 Physics 208, Lecture 18 4Building a solenoidThur. Oct. 30, 2008 Physics 208, Lecture 18 5Ampere’s law for the solenoid ! r B " dr s = BL# ! r B " dr s =µo#Ithrough=µoNI! Bsolenoid=µoNIL=µonIThur. Oct. 30, 2008 Physics 208, Lecture 18 6Ampere’s lawSum up component of B around pathEquals current through surface. Ampere’s law! B • ds"=µoIclosed pathsurface bounded by pathI ! r B Component of Balong path2Thur. Oct. 30, 2008 Physics 208, Lecture 18 7Gauss’ law in electrostatics Electric flux through surface∝ charge enclosedWhat about magnetic flux?Thur. Oct. 30, 2008 Physics 208, Lecture 18 8Magnetic flux Magnetic flux is definedexactly as electric flux (Component of B ⊥ surface) x (Area element)! "B= B • dA#zero fluxMaximum fluxSI unit of magnetic flux is the Weber ( = 1 T-m2 )Thur. Oct. 30, 2008 Physics 208, Lecture 18 9Magnetic fluxWhat is that magnetic flux through thissurface?A. PositiveB. NegativeC. ZeroThur. Oct. 30, 2008 Physics 208, Lecture 18 10Gauss’ law in magnetostatics Net magnetic flux through any closedsurface is always zero:! "magnetic= 0No magnetic ‘charge’, so right-hand side=0 for mag.Basic magnetic element is the dipole! "electric=Qenclosed#oCompare to Gauss’ lawfor electric fieldThur. Oct. 30, 2008 Physics 208, Lecture 18 11Time-dependent fields Up to this point, have discussed only magneticand electric fields constant in time. E-fields arise from charges B-fields arise from moving charges (currents)Faraday’s discovery Another source of electric field Time-varying magnetic field creates electric fieldThur. Oct. 30, 2008 Physics 208, Lecture 18 12Measuring the induced field A changing magnetic flux produces anEMF around the closed path. How to measure this? Use a real loop of wire for the closed path.The EMF corresponds to a current flow: ! "= IR3Thur. Oct. 30, 2008 Physics 208, Lecture 18 13Current but no battery? Electric currents require a battery (EMF) Faraday:Time-varying magnetic field creates EMFFaraday’s law:EMF around loop = - rate of change of mag. fluxThur. Oct. 30, 2008 Physics 208, Lecture 18 14Faraday’s law! "= E • ds#= $ddt%B= $ddtB#• dAMagnetic flux through surface bounded by pathEMF around loop EMF no longer zero around closed loopThur. Oct. 30, 2008 Physics 208, Lecture 18 15Quick quizWhich of these conducting loops willhave currents flowing in them?Constant II(t) increasesConstant IConstant vConstant IConstant vA.C.B.D.Thur. Oct. 30, 2008 Physics 208, Lecture 18 16Faraday’s law Faraday’s law Time-varying B-field creates E-field Conductor: E-field creates electric current Biot-Savart law Electric current creates magnetic field Result Another magnetic field createdThur. Oct. 30, 2008 Physics 208, Lecture 18 17Thur. Oct. 30, 2008 Physics 208, Lecture 18 18Lenz’s law Induced current produces a magnetic field. Interacts with bar magnet just as another bar magnet Lenz’s law Induced current generates a magnetic fieldthat tries to cancel the change in the flux. Here flux through loop due to bar magnet is increasing.Induced current produces flux to left. Force on bar magnet is to left.4Demonstration: Faraday & LenzThur. Oct. 30, 2008 Physics 208, Lecture 18 20Quick quizWhat direction force do I feel due to Lenz’ law when Ipush the magnet down?A. UpB. DownC. LeftD. RightCopperStrong magnetThur. Oct. 30, 2008 Physics 208, Lecture 18 21Quick Quiz A conducting rectangular loop moves with constantvelocity v in the +x direction through a region ofconstant magnetic field B in the -z direction as shown. What is the direction of the induced loop current?X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XvxyA. CCWB. CWC. No induced currentThur. Oct. 30, 2008 Physics 208, Lecture 18 22Quick Quiz•Conducting rectangular loop moves with constantvelocity v in the -y direction away from a wire with aconstant current I as shown. What is the direction of the induced loop current?A. CCWB. CWC. No induced currentIvB-field from wire into page at loopLoop moves to region of smaller B, so flux decreasesInduced loop current opposes this change, so must create a field in samedirection as field from wire -> CW current.Thur. Oct. 30, 2008 Physics 208, Lecture 18 23AC GeneratorsThe AC generator consists ofa loop of wire rotated by someexternal means in a magnetic field N turns of same area rotating in auniform B ΦB = BA cos θ = BA cos ωt emfεmax=NABωIn USA & Canada:f=ω/(2π)=60 HzIn Europe f=50
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