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Chapter 23 Faraday’s Law Predicting Possible Field Configuration Is the following “curly” pattern of electric field possible? f DV = - ò E · dl dl i E is always parallel to dl dl dl DVAA = - ò E · dl = -El = -E2p R ¹ 0 This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges! Faraday’s Experimentation Set wire (connected to ammeter) around a solenoid. Can a current be observed in this wire? 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 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? “Current opposes change in B” Driving Current by Changing B ENC causes current to run along the ring What is emf and I? emf ENC dl ENC 2r2 I E NC 2r2 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 which does not enclose the solenoid! D Exercise Is there current in these circuits? Quantitative Relationship Between B and EMF Can observe experimentally: I=emf/R 1. 2. ENC~emf ENC~dB/dt 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 (wire loop) 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: Sign: given by right hand rule “Current opposes change in B” Michael Faraday (1791 - 1867) Various ways of making changing magnetic field Change current through the solenoid Time varying B can be produced by moving coil: by moving magnet: by rotating magnet: (or coil) Clicker ammeter + - Solenoid has circumference 10 cm. The current I1 is increasing causing increasing B1 as in the figure. Solenoid is surrounded by a wire with finite resistance. When length of the wire is changed from 30 to 20 cm what will happen to detected current? A. It will decrease B. Increase C. Does not change A Circuit Surrounding a Solenoid emf d mag dt mag B1 r12 Example: B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2. 4 2 0.6 T 3 10 m emf 9 10 4 V t 0.2 s What is the ammeter reading? (resistance of ammeter+wire is 0.5) I emf / R 1.8 103 A Clicker: A Circuit Not Surrounding a Solenoid If we increase current through solenoid what will be ammeter reading? A. positive current B. negative current C. zero 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 Exercise 1. A bar magnet is moved toward a coil. What is the ammeter reading (+/-)? 2. The bar magnet is moved away from the coil. What will ammeter read? emf d mag dt