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UW-Madison PHYSICS 208 - Lecture 18 Notes

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The story so far…Exam 2 resultsMagnetic field from a current loopBuilding a solenoidAmpere’s law for the solenoidAmpere’s lawGauss’ law in electrostaticsMagnetic fluxSlide 9Gauss’ law in magnetostaticsTime-dependent fieldsMeasuring the induced fieldCurrent but no battery?Faraday’s lawQuick quizSlide 16PowerPoint PresentationLenz’s lawDemonstration: Faraday & LenzSlide 20Quick QuizSlide 22AC GeneratorsThe story so far… Magnetic field generated by current element: Biot-SavartAmpere’s lawdIdBr€ dB =μo4πIds ׈ r r2€ B • ds∫= μoIclosed pathsurface bounded by pathIThur. Oct. 30, 2008 Physics 208, Lecture 18 2Exam 2 resultsGrade boundaries:A:83AB: 76 B: 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 B along pathThur. 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 defined exactly as electric flux(Component of B  surface) x (Area element)€ ΦB= 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 this surface?A. PositiveB. NegativeC. ZeroThur. Oct. 30, 2008 Physics 208, Lecture 18 10Gauss’ law in magnetostaticsNet magnetic flux through any closed surface is always zero:€ Φmagnetic= 0No magnetic ‘charge’, so right-hand side=0 for mag.Basic magnetic element is the dipole€ Φelectric=QenclosedεoCompare to Gauss’ law for electric fieldThur. Oct. 30, 2008 Physics 208, Lecture 18 11Time-dependent fieldsUp to this point, have discussed only magnetic and 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 an EMF 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: € ε=IRThur. 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€ ε= E • ds∫= −ddtΦB= −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 will have 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 field that 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.Demonstration: Faraday & LenzQuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.Thur. Oct. 30, 2008 Physics 208, Lecture 18 20Quick quizWhat direction force do I feel due to Lenz’ law when I push the magnet down?A. UpB. DownC. LeftD. RightCopperStrong magnetThur. Oct. 30, 2008 Physics 208, Lecture 18 21Quick QuizA conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant 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 constant velocity v in the -y direction away from a wire with a constant 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 same direction as field from wire -> CW current.Thur. Oct. 30, 2008 Physics 208, Lecture 18 23AC GeneratorsThe AC generator consists of a loop of wire rotated by some external means in a magnetic fieldN turns of same area rotating in a uniform B B = BA cos  = BA cos temfmax=NABIn USA & Canada:f=/(2)=60 HzIn Europe f=50


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UW-Madison PHYSICS 208 - Lecture 18 Notes

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