MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.chapter 6electromagneticinduction394 Electromagnetic InductionIn our development thus far, we have found the electricand magnetic fields to be uncoupled. A net charge generatesan electric field while a current is the source of a magneticfield. In 1831 Michael Faraday experimentally discoveredthat a time varying magnetic flux through a conducting loopalso generated a voltage and thus an electric field, provingthat electric and magnetic fields are coupled.6-1 FARADAY'S LAW OF INDUCTION6-1-1 The Electromotive Force (EMF)Faraday's original experiments consisted of a conductingloop through which he could impose a dc current via a switch.Another short circuited loop with no source attached wasnearby, as shown in Figure 6-1. When a dc current flowed inloop 1, no current flowed in loop 2. However, when thevoltage was first applied to loop 1 by closing the switch, atransient current flowed in the opposite direction in loop 2.Positive current is inducedto try to keep magnetic fluxequal to a non-zero constantNegative current is inducedto try to keep magnetic fluxequal to zeroFigure 6-1 Faraday's experiments showed that a time varying magnetic flux througha closed conducting loop induced a current in the direction so as to keep the fluxthrough the loop constant.-·r· -·-Faraday'sLaw of Induction 395When the switch was later opened, another transient currentflowed in loop 2, this time in the same direction as the originalcurrent in loop 1. Currents are induced in loop 2 whenever atime varying magnetic flux due to loop 1 passes through it.In general, a time varying magnetic flux can pass through acircuit due to its own or nearby time varying current or by themotion of the circuit through a magnetic field. For any loop,as in Figure 6-2, Faraday's law isdl dEMF= Edl=-I BdS (1)where EMF is the electromotive force defined as the lineintegral of the electric field. The minus sign is introduced onthe right-hand side of (1) as we take the convention thatpositive flux flows in the direction perpendicular to the direc-tion of the contour by the right-hand rule.6-1-2 Lenz's LawThe direction of induced currents is always such as tooppose any changes in the magnetic flux already present.Thus in Faraday's experiment, illustrated in Figure 6-1, whenthe switch in loop 1 is first closed there is no magnetic flux inloop 2 so that the induced current flows in the oppositedirection with its self-magnetic field opposite to the imposedfield. The induced current tries to keep a zero flux through4 =fBBdSndS = dSf#E dl d=---fB*dsL dtFigure 6-2 Faraday's law states that the line integral of the electric field around aclosed loop equals the time rate of change of magnetic flux through the loop. Thepositive convention for flux is determined by the right-hand rule of curling the fingerson the right hand in the direction of traversal around the loop. The thumb then pointsin the direction of positive magnetic flux.396 Electromagnetic Inductionloop 2. If the loop is perfectly conducting, the induced cur-rent flows as long as current flows in loop 1, with zero net fluxthrough the loop. However, in a real loop, resistive lossescause the current to exponentially decay with an LIR timeconstant, where L is the self-inductance of the loop and R isits resistance. Thus, in the dc steady state the induced currenthas decayed to zero so that a constant magnetic flux passesthrough loop 2 due to the current in loop 1.When the switch is later opened so that the current in loop1 goes to zero, the second loop tries to maintain the constantflux already present by inducing a current flow in the samedirection as the original current in loop 1. Ohmic losses againmake this induced current die off with time.If a circuit or any part of a circuit is made to move througha magnetic field, currents will be induced in the directionsuch as to try to keep the magnetic flux through the loopconstant. The force on the moving current will always beopposite to the direction of motion.Lenz's law is clearly demonstrated by the experimentsshown in Figure 6-3. When a conducting ax is moved into amagnetic field, eddy currents are induced in the directionwhere their self-flux is opposite to the applied magnetic field.The Lorentz force is then in the direction opposite to themotion of the ax. This force decreases with time as the cur-rents decay with time due to Ohmic dissipation. If the ax wasslotted, effectively creating a very high resistance to the eddycurrents, the reaction force becomes very small as theinduced current is small.Af, = 2nR BFigure 6-3 Lenz's law. (a) Currents induced in a conductor moving into a magneticfield exert a force opposite to the motion. The induced currents can be made small byslotting the ax. (b) A conducting ring on top of a cdil is flipped off when a current issuddenly applied, as the induced currents try to keep a zero flux through the ring.·___ I,Faraday'sLaw of Induction 397When the current is first turned on in the coil in Figure 6-3b,the conducting ring that sits on top has zero flux through it.Lenz's law requires that a current be induced opposite to thatin the coil. Instantaneously there is no z component ofmagnetic field through the ring so the flux must return radi-ally. This creates an upwards force:f = 27RIx B= 2rRI#Bri, (2)which flips the ring off the coil. If the ring is cut radially sothat no circulating current can flow, the force is zero and thering does not move.(a) Short Circuited LoopTo be quantitative, consider the infinitely long time varyingline current I(t) in Figure 6-4, a distance r from a rectangularloop of wire with Ohmic conductivity o', cross-sectional areaA, and total length I = 2(D+d). The magnetic flux throughthe loop due to I(t) isSD/2 r+dD. = LoH,(r') dr' dz=z--DI2 rP olD •r+ddr' =tolD r+d27-r r' 2r rC i(t)H. 1r'( 2r'Cross sectional area A:conductivity adrV,= dt~PaFigure 6-4 A rectangular loop near a time varying line current. When the terminalsare short circuited the
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