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The story so far… dI dB r Magnetic field generated by current element: Biot-Savart I closed path Ampere’s law B ds I o o Ids rˆ dB 4 r 2 surface bounded by path Exam 2 results Phy208 Exam 2 35 Ave=69 30 Count 25 20 15 10 5 0 10 20 30 Mon. Mar. 31, 2008 40 50 60 70 SCORE 80 Grade cutoffs: A: 86 AB: 79 B: 66 BC: 58 C: 37 D: 23 90 100 Physics 208, Lecture 18 2 Ampere’s law Sum up component of B around path Equals current through surface. Component of B along path B ds I o Ampere’s law B I closed path surface bounded by path Mon. Mar. 31, 2008 Physics 208, Lecture 18 3 “Ampere’s law” in electrostatics E ds ? path Work done by E-field = W AB B F Coulomb A So ds B qE ds A E ds path is work per unit charge to bring charge back to where it started. This is zero. Mon. Mar. 31, 2008 Physics 208, Lecture 18 4 Gauss’ law in electrostatics Electric flux through surface charge enclosed What about magnetic flux? Mon. Mar. 31, 2008 Physics 208, Lecture 18 5 Magnetic flux Magnetic flux is defined B B dA exactly as electric flux (Component of B surface) x (Area element) zero flux Maximum flux SI unit of magnetic flux is the Weber ( = 1 T-m2 ) Mon. Mar. 31, 2008 Physics 208, Lecture 18 6 Magnetic flux What is that magnetic flux through this surface? A. Positive B. Negative C. Zero Mon. Mar. 31, 2008 Physics 208, Lecture 18 7 Gauss’ law in magnetostatics Net magnetic flux through any closed surface is always zero: magnetic 0 Compare to Gauss’ law for electric field electric Qenclosed o No magnetic ‘charge’, so right-hand side=0 for mag. Mon. Mar. 31, 2008 Basic magnetic element is the dipole Physics 208, Lecture 18 8 Comparison with electrostatics Gauss’ law Ampere’s law Electrostatics Magnetostatics Mon. Mar. 31, 2008 Physics 208, Lecture 18 9 Time-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 field Mon. Mar. 31, 2008 Physics 208, Lecture 18 10 Measuring 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: IR Mon. Mar. 31, 2008 Physics 208, Lecture 18 11 Current but no battery? Electric currents require a battery (EMF) Faraday: Time-varying magnetic field creates EMF Faraday’s law: EMF around loop = - rate of change of mag. flux Mon. Mar. 31, 2008 Physics 208, Lecture 18 12 Faraday’s law d d E ds B dt dt EMF around loop B dA Magnetic flux through surface bounded by path EMF no longer zero around closed loop Mon. Mar. 31, 2008 Physics 208, Lecture 18 13 Quick quiz Which of these conducting loops will have currents flowing in them? I(t) increases Constant I Constant v Constant v Constant I Mon. Mar. 31, 2008 Constant I Physics 208, Lecture 18 14 Faraday’s law Faraday’s law Biot-Savart law Time-varying B-field creates E-field Conductor: E-field creates electric current Electric current creates magnetic field Result Another magnetic field created Mon. Mar. 31, 2008 Physics 208, Lecture 18 15 Lenz’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. Mon. Mar. 31, 2008 Physics 208, Lecture 18 16 Quick quiz What direction force do I feel due to Lenz’ law when I push the magnet down? A. Up B. Down Strong magnet C. Left D. Right Copper Mon. Mar. 31, 2008 Physics 208, Lecture 18 17 Quick 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? A. CCW y B. CW C. No induced current Mon. Mar. 31, 2008 XXXXXXXXXXXX XXXXXXXXXXXX X X X X X X X vX X X X X XXXXXXXXXXXX x Physics 208, Lecture 18 18 Quick 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? I A. CCW B. CW C. No induced current v B-field from wire into page at loop Loop moves to region of smaller B, so flux decreases Induced 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 19 Motional EMF Conductor moving in uniform magnetic field + / - charges in conductor are moving. Magnetic field exerts force. Charges pile up at ends FB L - Static equilibrium: E-field generated canceling magnetic force qE qvB v v Solid conductor Mon. Mar. 31, 2008 EMF vBL Physics 208, Lecture 18 20