The story so far…Exam 2 resultsAmpere’s law“Ampere’s law” in electrostaticsGauss’ law in electrostaticsMagnetic fluxSlide 7Gauss’ law in magnetostaticsComparison with electrostaticsTime-dependent fieldsMeasuring the induced fieldCurrent but no battery?Faraday’s lawQuick quizSlide 15Lenz’s lawSlide 17Quick QuizSlide 19Motional EMFThe story so far… dIMagnetic field generated by current element: Biot-SavartAmpere’s lawdBr€ dB =μo4πIds ׈ r r2€ B • ds∫= μoIclosed pathsurface bounded by pathIMon. Mar. 31, 2008 Physics 208, Lecture 18 2Exam 2 resultsGrade cutoffs:A: 86AB: 79B: 66BC: 58C: 37D: 230510152025303510 20 30 40 50 60 70 80 90 100Phy208 Exam 2SCOREAve=69Mon. Mar. 31, 2008 Physics 208, Lecture 18 3Ampere’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 pathMon. Mar. 31, 2008 Physics 208, Lecture 18 4“Ampere’s law” in electrostatics € r E • dr s path∫= ? € WAB=r F Coulomb• dsAB∫= qr E • dsAB∫Work done by E-field =Sois work per unit charge to bring charge back to where it started. € r E • dr s path∫This is zero.Mon. Mar. 31, 2008 Physics 208, Lecture 18 5Gauss’ law in electrostaticsElectric flux through surface charge enclosedWhat about magnetic flux?Mon. Mar. 31, 2008 Physics 208, Lecture 18 6Magnetic fluxMagnetic 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 )Mon. Mar. 31, 2008 Physics 208, Lecture 18 7Magnetic fluxWhat is that magnetic flux through this surface?A. PositiveB. NegativeC. ZeroMon. Mar. 31, 2008 Physics 208, Lecture 18 8Gauss’ law in magnetostaticsNet 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 fieldMon. Mar. 31, 2008 Physics 208, Lecture 18 9Comparison with electrostaticsGauss’ lawAmpere’s lawMagnetostaticsElectrostaticsMon. Mar. 31, 2008 Physics 208, Lecture 18 10Time-dependent fieldsUp to this point, have discussed only magnetic and electric fields constant in time.E-fields arise from chargesB-fields arise from moving charges (currents)Faraday’s discoveryAnother source of electric fieldTime-varying magnetic field creates electric fieldMon. Mar. 31, 2008 Physics 208, Lecture 18 11Measuring the induced fieldA 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=IRMon. Mar. 31, 2008 Physics 208, Lecture 18 12Current 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. fluxMon. Mar. 31, 2008 Physics 208, Lecture 18 13Faraday’s law€ ε= E • ds∫= −ddtΦB= −ddtB∫• dAMagnetic flux through surface bounded by pathEMF around loop EMF no longer zero around closed loopMon. Mar. 31, 2008 Physics 208, Lecture 18 14Quick quizWhich of these conducting loops will have currents flowing in them?Constant II(t) increasesConstant IConstant vConstant IConstant vMon. Mar. 31, 2008 Physics 208, Lecture 18 15Faraday’s lawFaraday’s lawTime-varying B-field creates E-fieldConductor: E-field creates electric currentBiot-Savart lawElectric current creates magnetic fieldResultAnother magnetic field createdMon. Mar. 31, 2008 Physics 208, Lecture 18 16Lenz’s lawInduced current produces a magnetic field. Interacts with bar magnet just as another bar magnetLenz’s lawInduced 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.Mon. Mar. 31, 2008 Physics 208, Lecture 18 17Quick quizWhat direction force do I feel due to Lenz’ law when I push the magnet down?A. UpB. DownC. LeftD. RightCopperStrong magnetMon. Mar. 31, 2008 Physics 208, Lecture 18 18Quick QuizA 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 currentMon. Mar. 31, 2008 Physics 208, Lecture 18 19Quick 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.Mon. Mar. 31, 2008 Physics 208, Lecture 18 20LMotional EMFConductor moving in uniform magnetic field+ / - charges in conductor are moving.Magnetic field exerts force. € r v - € r v € r F BCharges pile up at endsStatic equilibrium: E-field generated canceling magnetic force€ qE = qvB€ EMF = vBLSolid
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