Understanding Electromagnetic Induction: Currents, Faraday's Law, and Magnetic Energy, Slides of Physics

Download Understanding Electromagnetic Induction: Currents, Faraday's Law, and Magnetic Energy and more Slides Physics in PDF only on Docsity! Short Version : 27. Electromagnetic Induction Docsity.com 27.1. Induced Currents 4 results from Faraday / Henry (1831) 1. Current induced in coil by moving magnet bar. 2. Moving the coil instead of the magnet gives the same result. v = 0, I = 0 v > 0, I > 0 v >> 0, I >> 0 v < 0, I < 0 Docsity.com Magnetic flux B d  B AMagnetic flux: Reminder: 0d  B A For a uniform B on a flat surface: cosB BA    B A Move magnet right  more lines thru loop Docsity.com Example 27.1. Solenoid A solenoid of circular cross section has radius R, consists of n turns per unit length, and carries current I. Find the magnetic flux through each turn of the solenoid. B A  20 n I R  B out of plane I B Docsity.com Example 27.2. Nonuniform Field A long, straight wire carries current I. A rectangular wire loop of dimensions l by w lies in a plane containing the wire, with its closest edge a distance a from the wire, and its dimension l parallel to the wire. Find the magnetic flux through the loop. 0 2 a w a I l dr r     B d  B A 02 a w a I l d r r      0 ln 2 I l a w a     Area element for integration Docsity.com Example 27.4. Changing Area Two parallel conducting rails a distance l apart are connected at one end by a resistance R. A conducting bar completes the circuit, joining the two rails electrically but free to slide along. The whole circuit is perpendicular to a uniform B, as show in figure. Find the current when the bar is pulled to the right with constant speed v. B l xB B A  Bd d t   E Let x = 0 be at the left end of rail. B l v  I R  E B l v R   CCW S x C I Docsity.com 27.3. Induction & Energy Direction of emf is to oppose magnet’s motion. Lenz’s law : Direction of induced emf is such that B created by the induced current opposes the changes in  that created the current. Magnet moving right RH rule: thumb // m. Loop ~ magnet with N to left. RH rule: thumb // m. Loop ~ magnet with S to left. Magnet moving left m I I m Docsity.com Motional EMF & Len’s Law Motional emf: induced emf due to motion of conductor in B. Square loop of sides L & resistance R pulled with constant speed v out of uniform B. Force on e:  e  F v B downward force  upward I (CW) Force on current carrying wire: mag I F L B ,mag net applied F F Docsity.com Example 27.5. Designing a Generator An electric generator consists of a 100-turn circular coil 50 cm in diameter. It’s rotated at f = 60 rev/s to produce standard 60 Hz alternating current. Find B needed for a peak output voltage of 170 V, which is the actual peak in standard 120 V household wiring. 1 costurn B A     2 cos 2 2 d dN B f t d t         Bd d t   E   2 cos 2 2 dB f t         2 22 sin 2 2 dN B f f t           2 2 0.5170 2 100 60 / 2 mV B s       2 22 2peak dN B f       E 323 10B T  Loop rotation Docsity.com Swiping a credit card. Patterns of magnetization on the strip induce currents in the coil. EM induction is basis of magnetic recording ( audio, video, computer disks, …). Modern hard disks: Giant magnetoresistance. Coil Iron Magnetic strip Card motion Information stored in magnetization pattern Docsity.com Eddy Currents Application: Metal Detectors Eddy current: current in solid conductor induced by changing . Usage: non-frictional brakes for rotating saw blades, train wheels, … Transmitter coil Receiver coil Current detector Nothing between coils Metal between coils AC Induced Current Strong I Weak I : alarm. Docsity.com Example 27.6. Solenoid A long solenoid of cross section area A and length l has n turns per unit length. Find its self-inductance. 1 turn B A  0B n IB of solenoid: 0 n I A 1 turnn l   2 0 n l I A 2 0L n l AI   Docsity.com B L I  Bd d t   E d IL d t   back emf Rapid switching of inductive devices can destroy delicate electronic devices. +direction = V  along I. d I /d t < 0 Docsity.com Inductor + (higher voltage) 0d I d t  E E 0d IL d t   E + (higher voltage)  (lower voltage)  (lower voltage) Docsity.com L < 0 ; | L |  2 2 0 d I d IR L d t d t    0 0LI R  E E 0 0 d II R L d t   E 0LL R d L d t   EE / 0 R t L L e  E E   00L t   E E  0 1 LI R  E E  /0 1 R t LeR   E Inductive time constant = L / R c.f. capacitive time constant = RC I   V = IR  But rate is  + _ Docsity.com Example 27.8. Firing Up a Electromagnet A large electromagnet used for lifting scrap iron has self-inductance L = 56 H. It’s connected to a constant 440-V power source; the total resistance of the circuit is 2.8 . Find the time it takes for the current to reach 75% of its final value.  /0 1 R t LI eR   E  /1 R t LI e   2.80.75 1 exp 56 t H          20ln 0.25t   28 s Docsity.com / 0 R t LI I e Short times: IL can’t change instantaneously. Long times: L = 0 ; inductor  wire. Switch at A, I . Switch at B, battery’s shorted out. I  exponentially. 0 0 d II R L d t   E  /0 1 R t LI eR   E 0d II R L d t    0 0t  0 t R    E 0 0I t  0 t   Docsity.com Magnetic Energy in an Inductor RL circuit: 0 0LI R  E E  20 0LI I R I  E E 2 0 0 d II I R L I d t   E Power from battery Power dissipated Power taken by inductor L d IP L I d t  U P d t  d IL I d t d t   2 1 2 L I  0 0U I   Energy stored in inductor: Docsity.com Example 27.10. MRI Disaster Superconducting electromagnets like solenoids in MRI scanners store a lot of magnetic energy. Loss of coolant is dangerous since current quickly decays due to resistance. A particular MRI solenoid carries 2.4 kA and has a 0.53 H inductance. When it loses superconductivity, its resistance goes abruptly to 31 m. Find (a)the stored magnetic energy, and (b) the rate of energy release at the instance the superconductivity is lost. 61.5 10 J    231 0.53 2.4 102 H A  21 2 U L I 2P I R    23 32.4 10 31 10A     51.8 10 W  In practice, Cu / Ag are incorporated into the superconducting wires to reduce R. Docsity.com Magnetic Energy Density Solenoid with length l & cross-section area A : 20L n A l (Eg. 27.6) 2 2 0 1 2 n A l I21 2 U L I 2 0 1 2 B A l   0B n I Magnetic Energy Density : 2 0 1 2B u B   c.f. electric energy density : 20 1 2E u E Docsity.com Conservative & Nonconservative Electric Fields For stationary charges (electrostatics) : 0d  E l E is conservative Induced fields (electromagnetics) : 0d  E l E is non-conservative W against E E does W I Docsity.com GOT IT? 27.7 The figure shows three resistors in series surrounding an infinitely long solenoid with a changing magnetic field; the resulting induced electric field drives a current counterclockwise, as shown. Two identical voltmeters are shown connected to the same points A and B. What does each read? Explain any apparent contradiction. Hint: this is a challenging question! VA  VB = IR 3R IE VA  VB = 2IR Docsity.com Diamagnetism Superconductor is a perfect diamagnet (Meissner effect). B  0 B = 0: net = 0 This e slows down. This e speeds up. net  0 Classical model of diamagnetism (not quite right) Docsity.com