chapter5_81_94

chapter5_81_94 - AAE 590E Motional EMF In the last section,...

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Unformatted text preview: AAE 590E Motional EMF In the last section, we have listed several different sources for EMF. However, we have not discussed nor listed to most common source of EMF: Generator. Generators exploit "motional EMF" by moving a conductor in a magnetic field. Region with Magnetic Field B C x h R D A B ! v ! Fvert,mag q ! v A Loop is pulled through magnetic field to the right at a speed v, Charges in segment AB experience a magnetic force, whose vertical component is Fver,mag drives current clockwise around the loop, Segments BC and DA don't contribute to current, since force is perpendicular to the wire. ! ! ! ! = " fS ! d l = " fmag ! d l = vBh NOTE: The integral to calculate is taken at one instant of time (Snapshot)! Ch5 81 AAE 590E Motional EMF Earlier, we have established that magnetic forces do not perform work. Question: Who is supplying energy that heats the resistor? Answer: The person who's pulling the loop. ! fmag f h mag ! u ! w v fmag = vB l = d sin ! ! = uB d ! ! v Charges in segment AB have vertical velocity u and horizontal velocity v due to the u v ! motion of loop: total velocity w w, ! Due to u there is a horizontal component of magnetic force to the left: uB, u, ! Due to v there is vertical component of magnetic force: vB, v, fpull = uB Person pulling on the wire must exert a force per unit charge: fpull=uB, Force is transmitted to charge by structure of wire, h ! Charged particle covers a distance d in the direction of w d = w: cos! Work done by person pulling the wire loop: ! ! h = " fpull ! d l = f pull l = u B ! d sin # = u B ! sin # = u B !h tan # = vBh cos# Ch5 82 ! fpull ! AAE 590E Magnetic Flux ! ! !B " $ B # da ! ! !E " $ E # da Definition of Magnetic Flux: !Wb = Tm 2 # " $ Recall definition of Electric Flux: Example: Rectangular Loop Important: !B = Bh x Question: What happens as the loop moves? Answer: Flux decreases as the loop moves: NOTE: d!B dx = Bh = "B h v dt dt The minus sign reflects the "decreasing" area of the loop penetrated by the magnetic field. Flux Rule for Motional EMF: =! d"B dt The Flux Rule is applicable to any shape and direction of movement! Ch5 83 AAE 590E Faraday's Law In 1831, Michael Faraday conducted a series of experiments: R I ! v ! v R I I R ! B Classic Case of Motional EMF: ! B Loop is stationary, Charges are stationary, Stationary charges experience NO Magnetic Force. ! B Both loop and magnetic field are stationary. Magnetic Field changes with time. =! d"B dt =! d"B dt ! ! = " E # dl $ Question: In the case of a stationary loop what makes the charges move to induce the current? Faraday's Hypothesis: A changing magnetic field induces an electric field! Ch5 84 AAE 590E Faraday's Law Similar to Faraday's experiments, let's consider a copper loop in a uniform external magnetic field. The area penetrated by the magnetic field is fixed. We change now the magnetic field strength with time and observe a current in the copper loop. If there is a current in the loop, an electric field must be present along the ring! The electric field is needed to do work on moving the charges. Keep in mind, that this induce electric field is not based on Coulomb forces: ! ENC is due to change in magnetic field. NC Ch5 85 AAE 590E Faraday's Law ! ! d$ E ! dl = # B " dt ! ! $B !"E = # $t ! !"E = 0 Faraday's Law: Integral Form Differential Form Static Case where B=const.: by applying Stokes' Theorem Universal Flux Rule: d"B =! dt There are two totally different mechanisms: dA 1. Change in Area: = !B dt True motional EMF 2. Change in Magnetic Field: = !A EMF is due to induced electric field by changing a magnetic field. dB dt Ch5 86 AAE 590E Induced Electric Field Due to Faraday's discoveries, we distinguish between two kinds of Electric Fields: ! 1. E due to electric charges: static case, Coulomb's Law, ! 2. E due to changing magnetic fields: Faraday's and Ampere's Law. Ch5 87 AAE 590E Lenz's Rule Lenz: The direction of the induced current is such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current. Increasing d!B : dt Decreasing d!B : dt Think of it as the current induced will flow in such a direction that the flux it produces tends to cancel the change!! Further, keep in mind that it is the Change in Flux and NOT the flux itself, which is countered. Think of it as the loop likes to maintain a constant flux through it; it doesn't like change. So if flux changes, it will produce a change in flux to counter it. Ch5 88 AAE 590E Inductor & Inductance Inductors produce a desired magnetic field by current flowing through the windings. A long solenoid is a basic type of inductor. A current I through an inductor produces a magnetic flux through the central region of the inductor. We have determined the magnetic field of an ideal solenoid as: Magnetic Field: Magnetic Flux: B = 0 n I where n = N # of turns = l length !B = B A = 0 n I " # R 2 EMF of One Loop of Inductor: =! d"B d dI =! 0 n I # $ R 2 = ! 0$ nR 2 dt dt dt ( ) EMF of All Loops of Inductor: The generated flux due to current I is enclosed by all N turns of a solenoid. N 2R 2 dI 2 dI = !N " 0# nR = ! 0# dt l dt Ch5 89 AAE 590E Inductor & Inductance N 2R 2 where L = 0! and R=radius of inductor l We define Inductance L as a proportionally constant: dI = !L dt Inductance (analogous to capacitance) depends purely on geometric features of the inductor: d"B dI = !L = !N dt dt N!B N 2R 2 L= = 0 " I l Energy Stored in Magnetic Field: 1 E = L I2 2 Definition of EMF: ! dW dq = LI dW dW dt dW 1 = = =! dq dq dt dt I dI dt Ch5 90 dW = !I dt Minus stated that work has to be performed on the system against the induced EMF not the work done by EMF! AAE 590E 5.4 MAXWEL'S EQUATIONS Ch5 91 AAE 590E Maxwell's Equations Magnetostatics ! !"B=0 Electrostatics & Magnetostatics Electrostatics Gauss's Law ! # !"E = $0 ! !"E=0 Ampere's Law ! ! ! " B = 0 J Electrodynamics before Maxwell Gauss's Law Faraday's Law ! # !"E = $0 ! !"B=0 ! ! $B !"E=# $t Ampere's Law ! ! ! " B = 0 J Apply divergence to curl of electric and magnetic field, and check result: Electric Field: ! ! ! & %B ) % !" ! # E = !"($ =$ !"B = 0 %t ' %t + * ( ) ( ) Checks Out! Divergence of Curl = 0 =0 Magnetic Field: ! !" !# B ( ) = ! 0 ! " J ( ) Doesn't Check Out! Two Cases Possible!! " = 0 for steady currents # $ ! 0 for unsteady currents Divergence of Curl = 0 Maxwell's Fix for Ampre's Law: Due to the left had side of the above equation, right hand side should be ZERO, but it isn't. ! $% Continuity Equation ! " J = # $t ! % Gauss' Law !"E = &0 See Slide Ch5-64 ' ) ) ( ) ) * + ! ! ! , $E / $ !" J = # & ! " E = #! " . & 0 $t 0 - $t 1 0 ( ) Ampere's Law is expanded to take into account Maxwell's discovery: ! ! ! $E A changing electric field induces a magnetic field: ! " B = 0 J + 0#0 $t Ch592 AAE 590E Maxwell's Equations Differential Form ! # !"E = $0 ! !"B=0 Maxwell's Equations Gauss' Law ! # !"E = $0 ! !"B=0 ! ! #B !"E+ =0 #t ! ! ! %E ! " B # 0$0 = 0 J %t Integral Form ! ! q " E ! dA = #enc 0 Comments Relates net electric field to net enclosed electric charges. " B ! dA = 0 ! ! d$ E! dl = # B " dt ! ! d$ " B ! d l = 0#0 dtE + 0I enc Relates induced electric field to changing magnetic flux. Relates induced magnetic field to changing electric flux and current. ! ! Faraday's Law Ampere's Law with Maxwell's correction Lorentz Force ! ! $B !"E=# $t ! ! ! $E ! " B = 0 J + 0#0 $t ! ! ! ! F=q E+ v!B ( ) These equations emphasize that electric fields can be produced either by charges () or ! changing magnetic fields by ( !B !t ); and magnetic fields can be produced by either ! current ( J ) of changing ! electric fields ( !E !t ). NOTE: This is however ! misleading, since !B !t and All electromagnetic field are ultimately attributable to charges and currents. ! !E !t are themselves due to charges and currents. NOTE: Maxwell's Equations describe how charges produce fields. Lorentz Force relates how fields affect charges. Ch593 AAE 590E Summary NOTE: Maxwell's Equations describe how charges produce fields. Force Law relates how fields affect charges. Maxwell's Equations Gauss' Law ! # !"E = $0 ! !"B = 0 " A ! ! q enclosed E ! da = #0 ! # !"E = $0 ! !"B = 0 " A ! ! B ! da = 0 ! ! $B Faraday's Law ! " E = # $t ! ! ! $E Ampere's Law ! " B = 0 J + # 0 0 $t ! ! ! ! F= q E+v!B Lorentz Force " P ! ! $% E ! dl = # B $t ! ! $%E B ! d l = 0 Ienclosed + # 0 0 $t ! ! #B !"E+ =0 #t ! ! ! %E ! " B # $ 0 0 = 0 J %t Fields ! ! B& E Sources ! !& J " P ( ) Comment: These equations and currents emphasize that electric fields can be produced either by charges or by changing mag. fields; and mag. fields can be produced by either ! ! current or changing electric fields. !E !B This is however misleading, because and !t !t are themselves due to charges and currents. Comment: All electromagnetic fields are ultimately attributed to charges and currents. Ch5 94 ...
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