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 6 electromagnetic induction 394 ElectromagneticInduction In our development thus far we have found the electric and magnetic fields to be uncoupled A net charge generates an electric field while a current is the source of a magnetic field In 1831 Michael Faraday experimentally discovered that a time varying magnetic flux through a conducting loop also generated a voltage and thus an electric field proving that electric and magnetic fields are coupled 6 1 6 1 1 FARADAY S LAW OF INDUCTION The Electromotive Force EMF Faraday s original experiments consisted of a conducting loop through which he could impose a dc current via a switch Another short circuited loop with no source attached was nearby as shown in Figure 6 1 When a dc current flowed in loop 1 no current flowed in loop 2 However when the voltage was first applied to loop 1 by closing the switch a transient current flowed in the opposite direction in loop 2 Positive current is induced to try to keep magnetic flux equal to a non zero constant Negative current is induced to try to keep magnetic flux equal to zero Figure 6 1 Faraday s experiments showed that a time varying magnetic flux through a closed conducting loop induced a current in the direction so as to keep the flux through the loop constant r Faraday sLaw of Induction 395 When the switch was later opened another transient current flowed in loop 2 this time in the same direction as the original current in loop 1 Currents are induced in loop 2 whenever a time varying magnetic flux due to loop 1 passes through it In general a time varying magnetic flux can pass through a circuit due to its own or nearby time varying current or by the motion of the circuit through a magnetic field For any loop as in Figure 6 2 Faraday s law is EMF Edl dl d I BdS 1 where EMF is the electromotive force defined as the line integral of the electric field The minus sign is introduced on the right hand side of 1 as we take the convention that positive flux flows in the direction perpendicular to the direction of the contour by the right hand rule 6 1 2 Lenz s Law The direction of induced currents is always such as to oppose any changes in the magnetic flux already present Thus in Faraday s experiment illustrated in Figure 6 1 when the switch in loop 1 is first closed there is no magnetic flux in loop 2 so that the induced current flows in the opposite direction with its self magnetic field opposite to the imposed field The induced current tries to keep a zero flux through 4 fB BdS f ndS dS E L dl d fB ds dt Figure 6 2 Faraday s law states that the line integral of the electric field around a closed loop equals the time rate of change of magnetic flux through the loop The positive convention for flux is determined by the right hand rule of curling the fingers on the right hand in the direction of traversal around the loop The thumb then points in the direction of positive magnetic flux 396 ElectromagneticInduction loop 2 If the loop is perfectly conducting the induced current flows as long as current flows in loop 1 with zero net flux through the loop However in a real loop resistive losses cause the current to exponentially decay with an LIR time constant where L is the self inductance of the loop and R is its resistance Thus in the dc steady state the induced current has decayed to zero so that a constant magnetic flux passes through loop 2 due to the current in loop 1 When the switch is later opened so that the current in loop 1 goes to zero the second loop tries to maintain the constant flux already present by inducing a current flow in the same direction as the original current in loop 1 Ohmic losses again make this induced current die off with time If a circuit or any part of a circuit is made to move through a magnetic field currents will be induced in the direction such as to try to keep the magnetic flux through the loop constant The force on the moving current will always be opposite to the direction of motion Lenz s law is clearly demonstrated by the experiments shown in Figure 6 3 When a conducting ax is moved into a magnetic field eddy currents are induced in the direction where their self flux is opposite to the applied magnetic field The Lorentz force is then in the direction opposite to the motion of the ax This force decreases with time as the currents decay with time due to Ohmic dissipation If the ax was slotted effectively creating a very high resistance to the eddy currents the reaction force becomes very small as the induced current is small Af 2nR B Figure 6 3 Lenz s law a Currents induced in a conductor moving into a magnetic field exert a force opposite to the motion The induced currents can be made small by slotting the ax b A conducting ring on top of a cdil is flipped off when a current is suddenly applied as the induced currents try to keep a zero flux through the ring I Faraday sLaw of Induction 397 When 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 that in the coil Instantaneously there is no z component of magnetic field through the ring so the flux must return radially This creates an upwards force f 27RIx B 2rRI Bri 2 which flips the ring off the coil If the ring is cut radially so that no circulating current can flow the force is zero and the ring does not move a Short Circuited Loop To be quantitative consider the infinitely long time varying line current I t in Figure 6 4 a distance r from a rectangular loop of wire with Ohmic conductivity o cross sectional area A and total length I 2 D d The magnetic flux through the loop due to I t is SD 2 D r d LoH r dr dz z DI2 r P olD 2 C H r 1 7 r ddr r r tolD r d 2r r i t 2r Cross sectional area A conductivity a dr V dt Pa Figure 6 4 A rectangular loop near a time varying line current When the terminals are …
View Full Document
Unlocking...