Physics 2102 Spring 2007 Jonathan Dowling Physics 2102 Spring 2007 Lecture 17 Ch30 Induction and Inductance I QuickTime and a decompressor are needed to see this picture Faraday s Law A time varying magnetic FLUX creates an induced EMF Definition of magnetic flux is similar to definition of electric flux r B B n dA S d B EMF dt B n dA Take note of the MINUS sign The induced EMF acts in such a way that it OPPOSES the change in magnetic flux Lenz s Law Example When the N pole approaches the loop the flux into the loop downwards increases The loop can oppose this change if a current were to flow clockwise hence creating a magnetic flux upwards So the induced EMF is in a direction that makes a current flow clockwise If the N pole moves AWAY the flux downwards DECREASES so the loop has a counter clockwise current Example n A closed loop of wire encloses an 300 area of 1 m2 in which in a uniform r magnetic field exists at 300 to the B B n dA PLANE of the loop The magnetic S field is DECREASING at a rate of BA 0 1T s The resistance of the wire is BA cos 60 10 2 d B A dB What is the induced current EMF dt Is it clockwise or counterclockwise 2 dt EMF A dB i R 2 R dt 2 1m i 1T s 0 05 A 2 10 Example 3 loops are shown B 0 everywhere except in the circular region where B is uniform pointing out of the page and is increasing at a steady rate Rank the 3 loops in order of increasing induced EMF a III II I b III I II are same c ALL SAME I II III Just look at the rate of change of ENCLOSED flux III encloses no flux and it does not change I and II enclose same flux and it changes at same rate Example An infinitely long wire carries a constant current i as shown A square loop of side L is moving towards the wire with a constant velocity v What is the EMF induced in the loop when it is a distance R from the loop R v x L L Choose a strip of width dx located as shown Flux thru this strip 0iL dx BL dx 2 R x L 0iL dx B 2 R x 0 L 0iL ln R x 2 0 0iL R L ln 2 R d B EMF dt 0 Li d L ln 1 2 dt R Example continued R d B EMF dt 0 Li d L ln 1 2 dt R 0 Li dR R L R2 2 dt R L 2 0i L v 2 R L R v L x What is the DIRECTION of the induced current Magnetic field due to wire points INTO page and gets stronger as you get closer to wire So flux into page is INCREASING Hence current induced must be counter clockwise to oppose this increase in flux Example the Generator A square loop of wire of side L is rotated at a uniform frequency f in the presence of a uniform magnetic field B as shown Describe the EMF induced in the loop r B B n dA S 2 L B B BL cos d B 2 2 d EMF BL sin BL 2 f sin 2 ft dt dt Example Eddy Currents A non magnetic e g copper aluminum ring is placed near a solenoid What happens if There is a steady current in the solenoid The current in the solenoid is suddenly changed The ring has a cut in it The ring is extremely cold Another Experimental Observation Drop a non magnetic pendulum copper or aluminum through an inhomogeneous magnetic field What do you observe Why Think about energy conservation Pendulum had kinetic energy What happened to it Isn t energy conserved N S Some interesting applications MagLev train relies on Faraday s Law currents induced in nonmagnetic rail tracks repel the moving magnets creating the induction result levitation Guitar pickups also use Faraday s Law a vibrating string modulates the flux through a coil hence creating an electrical signal at the same frequency Example the ignition coil The gap between the spark plug in a combustion engine needs an electric field of 107 V m in order to ignite the air fuel mixture For a typical spark plug gap one needs to generate a potential difference 104 V But the typical EMF of a car battery is 12 V So how does a spark plug work spark 12V Breaking the circuit changes the current through primary coil Result LARGE change in flux thru secondary large induced EMF The ignition coil is a double layer solenoid Primary small number of turns 12 V Secondary MANY turns spark plug http www familycar com Classroom ignition htm Another formulation of Faraday s Law We saw that a time varying B n magnetic FLUX creates an induced EMF in a wire exhibited as a current Recall that a current flows in dA a conductor because of r r electric field d B E ds Hence a time varying dt magnetic flux must induce C an ELECTRIC FIELD Another of Maxwell s equations Closed electric field To decide SIGN of flux use right lines No potential hand rule curl fingers around loop flux thumb Example The figure shows two circular regions R 1 R2 with radii r1 1m r2 2m In R1 the magnetic field B1 points out of the page In R2 the magnetic field B2 points into the page Both fields are uniform and are DECREASING at the SAME steady rate 1 T s Calculate the Faraday integral for the two paths shown r r E ds R1 R2 I II C r r d B 2 r1 1T s 3 14V Path I E ds Path II dt C r r 2 2 E ds r1 1T s r2 1T s 9 42V C Example A long solenoid has a circular cross section of radius R The current through the solenoid is increasing at a steady rate di dt Compute the variation of the electric field as a function of the distance r from the axis of the solenoid R First let s look at r R Next let s look at r R dB E 2 r r dt r dB E 2 dt dB E 2 r R dt 2 2 2 R dB E 2r dt electric field lines magnetic field lines Example continued r dB E 2 dt 2 R dB E 2r dt E r r r R Summary Two versions of Faradays law A varying magnetic flux produces an EMF d B EMF dt A varying magnetic flux produces an electric field r r d B E ds dt C
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