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Clicker Question: Non-uniform B field A current loop is oriented so that its magnetic dipole is oriented along direction “2”. In which direction is the force on the loop? A) B) C) D) E) In direction 1 In direction 2 In direction 3 In direction 4 No force Maxwell’s Equations (incomplete) E nˆdA q inside 0 Gauss’s law for electricity Gauss’s law for magnetism B nˆA 0 Incomplete version of Faraday’s law E dl 0 Ampere’s law B d l I 0 inside_ path (Incomplete Ampere-Maxwell law) First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence Chapter 23 Faraday’s Law Changing Magnetic Field Solenoid: inside outside B 0 NI B0 d Constant current: there will be no forces on charges outside (B=0, E=0) What if current is not constant in time? Let B increase in time E~dB/dt E~1/r Non-Coulomb ENC ! Two Ways to Produce Electric Field 1. Coulomb electric field: produced by charges E1 2. Non-Coulomb electric field: using changing magnetic field Field outside of solenoid Same effect on charges: F21 q2 E1 1 q1 rˆ 2 40 r 𝑑𝐵1 1 𝐸1 ~ ~ 𝑑𝑡 𝑟 Direction of the Curly Electric Field Right hand rule: dB Thumb in direction of 1 dt fingers: E NC Exercise: Magnetic field points down from the ceiling and is increasing. What is the direction of E? Lenz’s Rule The direction of the induced magnetic field due to a change in flux is such it attempts to keep the flux constant. • The direction of the induced current (and 𝐸𝑁𝐶 ) is what is needed to produce the induced magnetic field. 𝐸𝑁𝐶 𝐵X 𝐵𝑖𝑛𝑑 𝐵𝑜𝑢𝑡 𝑎𝑛𝑑 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝐸𝑁𝐶 𝐵 𝐵𝑖𝑛𝑑 𝐵𝑜𝑢𝑡 𝑎𝑛𝑑 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 Driving Current by Changing B ENC causes current to run along the ring What is the surface charge distribution? What is emf and I? emf ENC dl ENC 2r2 E NC 2r2 I R Ring has resistance, R Effect of the Ring Geometry emf ENC dl ENC 2r2 1. Change radius r2 by a factor of 2. ENC ~ 1 / r2 L 2r2 emf does not depend on radius of the ring! 2. One can easily show that emf will be the same for any circuit surrounding the solenoid Round-Trip Not Encircling the Solenoid emf ENC dl =0 =0 B C D A emf ENC dl ENC dl ENC dl ENC dl A B C D emf ENC dl ENC dl B A + C _ emf 0 for any path not Enclosing solenoid! D Exercise Is there current in these circuits? Quantitative Relationship Between B and EMF Can observe experimentally: I=emf/R ENC~emf 1. ENC~dB/dt 2. ENC~ cross-section of a solenoid d emf B1 r12 dt Magnetic Flux d emf B1 r12 dt B r 1 2 1 - magnetic flux mag on the area encircled by the circuit Magnetic flux on a small area A: B nˆA B A Definition of magnetic flux: mag B nˆdA BdA This area does not enclose a volume! Faraday’s Law emf d mag dt Faraday’s law cannot be derived from the other fundamental principles we have studied Formal version of Faraday’s law: 𝑑 𝐸𝑁𝐶 ∙ 𝑑𝑙 = − 𝑑𝑡 𝐵 ∙ 𝑛𝑑𝐴 Sign: given by right hand rule Michael Faraday (1791 - 1867) Including Coulomb Electric Field 𝑑 𝐸𝑁𝐶 ∙ 𝑑𝑙 = − 𝑑𝑡 𝐵 ∙ 𝑛𝑑𝐴 Can we use total E in Faraday’s law? Etotal dl ENC EC dl ENC dl EC dl =0 𝑑 𝐸 ∙ 𝑑𝑙 = − 𝑑𝑡 𝐵 ∙ 𝑛𝑑𝐴 A Circuit Surrounding a Solenoid emf d mag dt mag B1 r12 𝐸𝑁𝐶 Example: B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2. 𝑒𝑚𝑓 = − ∆Φ𝑚𝑎𝑔 Δ𝑡 =− 0.6T 3×10−4 m2 0.2s = −9 × 10−4 V What is the ammeter reading? (resistance of ammeter+wire is 0.5) 𝐼 = 𝑒𝑚𝑓/𝑅 = −1.8 × 10−3 V 𝑑𝐵 − 𝑑𝑡 downwards A Circuit Not Surrounding a Solenoid If we increase current through solenoid what will be ammeter reading? The EMF for a Coil With Multiple Loops Each loop is subject to similar magnetic field emf of loops in series: emf N d mag dt Moving Coils or Magnets Time varying B can be produced by moving coil: by moving magnet: by rotating magnet: (or coil) Exercise 1. A bar magnet is moved toward a coil. What is the ammeter reading (+/-)? + 𝐵 𝐵 2. The bar magnet is moved away from the coil. What will ammeter read? _ 3. The bar magnet is rotated. What will ammeter read? _ emf d mag 𝑑𝐵 − 𝑑𝑡 dt ? Faraday’s Law and Motional EMF ‘Magnetic force’ approach: Ftot qE qv B E vB emf vB L 𝐿 I Use Faraday law: emf d mag 𝑣∆𝑡 dt mag B A B Lvt emf lim t 0 mag t vB L I 𝐿 Example R L emf d mag I B1 dt v B2 mag B2 A B1A B2 B1 A mag B2 B1 Lvt emf lim t 0 mag t Lvt vLB2 B1 I emf vLB2 B1 R R Faraday’s Law and Generator emf d mag dt B nˆA Bwh cos t Bwh cos t emf d mag dt d Bwh cost dt emf Bwh sin t I Two Ways to Produce Changing emf d mag dt Two ways to produce curly electric field: 1. Changing B 2. Changing A d mag dt d dB dA B A A B dt dt dt