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Induction March 29, 2006 Induction - Spring 2006 1 Calendar… Today we finish up some material from the last chapter and begin the chapter on induction. Friday – Quiz on LAST chapter (30) Next Friday … still likely date for the next exam. If you have a problem with this date please email me with reason and we will try to figure out how to deal with it. Induction - Spring 2006 2 Let’s Finish Some Details Displacement Current Induction - Spring 2006 3 Magnetic Flux B dA Like Gauss - Open Surface For a CLOSED Surface we might expect this to be equal to some constant times the enclosed poles … but there ain’t no such thing! B d A 0 4 Induction - Spring 2006 Examples S 5 N Induction - Spring 2006 Consider the poor little capacitor… i i CHARGING OR DISCHARGING …. HOW CAN CURRENT FLOW THROUGH THE GAP?? 6 Induction - Spring 2006 Through Which Surface Do we measure the current for Ampere’s Law? I=0 7 Induction - Spring 2006 In the gap… DISPLACEMENT CURRENT The FLUX through S2 EA q 0 d 1 dq dt 0 dt Let dq I d dt 0 8 Displaceme nt Current d Id dt Induction - Spring 2006 Induction - Spring 2006 9 From The Demo .. Induction - Spring 2006 10 Faraday’s Experiments ? ? Induction - Spring 2006 11 Insert Magnet into Coil Induction - Spring 2006 12 Remove Coil from Field Region Induction - Spring 2006 13 That’s Strange ….. These two coils are perpendicular to each other Induction - Spring 2006 14 Definition of TOTAL ELECTRIC FLUX through a surface: d surface Total Flux of the Electric Field LEAVING a surface is E n out dA Induction - Spring 2006 15 Magnetic Flux: THINK OF MAGNETIC FLUX as the “AMOUNT of Magnetism” passing through a surface. Don’t quote me on this!!! Induction - Spring 2006 16 Consider a Loop xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx Magnetic field passing through the loop is CHANGING. FLUX is changing. There is an emf developed around the loop. A current develops (as we saw in demo) Work has to be done to move a charge completely around the loop. Induction - Spring 2006 17 Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx For a current to flow around the circuit, there must be an emf. (An emf is a voltage) The voltage is found to increase as the rate of change of flux increases. Induction - Spring 2006 18 Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx Faraday' s Law d emf dt We will get to the minus sign in a short time. Induction - Spring 2006 19 Faraday’s Law (The Minus Sign) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx Using the right hand rule, we would expect the direction of the current to be in the direction of the arrow shown. Induction - Spring 2006 20 Faraday’s Law (More on the Minus Sign) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx The minus sign means that the current goes the other way. This current will produce a magnetic field that would be coming OUT of the page. The Induced Current therefore creates a magnetic field that OPPOSES the attempt to INCREASE the magnetic field! This is referred to as Lenz’s Law. Induction - Spring 2006 21 How much work? xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx Work/Unit Charge d W / q V E ds dt A magnetic field and an electric field are intimately connected.) Induction - Spring 2006 22 The Strange World of Dr. Lentz Induction - Spring 2006 23 MAGNETIC FLUX B B dA This is an integral over an OPEN Surface. Magnetic Flux is a Scalar The UNIT of FLUX is the weber 1 weber = 1 T-m2 Induction - Spring 2006 24 We finally stated FARADAY’s LAW d emf V E ds dt Induction - Spring 2006 25 From the equation Lentz d emf V E ds dt B B dA Induction - Spring 2006 26 Flux Can Change B B dA If B changes If the AREA of the loop changes Changes cause emf s and currents and consequently there are connections between E and B fields These are expressed in Maxwells Equations Induction - Spring 2006 27 Maxwell’s Equations (Next Course .. Just a Preview!) Gauss Faraday Induction - Spring 2006 28 Another View Of That damned minus sign again …..SUPPOSE that B begins to INCREASE its MAGNITUDE INTO THE PAGE xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx The Flux into the page begins to increase. An emf is induced around a loop A current will flow That current will create a new magnetic field. THAT new field will change the magnetic flux. Induction - Spring 2006 29 Lenz’s Law Induced Magnetic Fields always FIGHT to stop what you are trying to do! i.e... Murphy’s Law for Magnets Induction - Spring 2006 30 Example of Nasty Lenz The induced magnetic field opposes the field that does the inducing! Induction - Spring 2006 31 Induction - Spring 2006 32 Don’t Hurt Yourself! The current i induced in the loop has the direction such that the current’s magnetic field Bi opposes the change in the magnetic field B inducing the current. Induction - Spring 2006 33 Let’s do the Lentz Warp again ! Induction - Spring 2006 34 Lenz’s Law An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current. (The result of the negative sign!) … OR The toast will always fall buttered side down! Induction - Spring 2006 35 An Example The field in the diagram creates a flux given by B=6t2+7t in milliWebers and t is in seconds. (a) What is the emf when t=2 seconds? (b) What is the direction of the current in the resistor R? Induction - Spring 2006 36 This is an easy one … B 6t 7t 2 d emf 12t 7 dt at t 2 seconds emf 24 7 31mV Direction? B is out of the screen and increasing. Current will produce a field INTO the paper (LENZ). Therefore current goes clockwise and R to left in the resistor. Induction - Spring 2006 37 Figure 31-36 shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x >> R. Consequently, the magnetic field due to the current i in the larger loop is nearly constant throughout the smaller loop. Suppose that x is increasing at the constant rate of dx/dt = v. (a) Determine the magnetic flux through the area bounded by the smaller loop as a function of x. (Hint: See Eq. 30-29.) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current. v Induction - Spring 2006 38 B is assumed to be constant through the center of the small loop and caused by the large one. q Induction - Spring 2006 39 The calculation of Bz dBz dB cos q cos q cos q R R 0 ids 4 R 2 x 2 1/ 2 x2 q 0 ids R dBz 4 R 2 x 2 R 2 x 2 ds Rd 2 Bz 0iR 2 2 R x 2 Induction - Spring 2006 1/ 2 2 3/ 2 40 More Work In the small loop: Bz A r 2 Bz r 2 0iR 2 2R x For x R (Far Away as prescribed ) 2 r 2 0iR 2 3 2x d 3r 2 0iR 2 V emf 4 2x dt Induction - Spring 2006 2 3/ 2 dx/dt=v 41 Which Way is Current in small loop expected to flow?? B q Induction - Spring 2006 42 What Happens Here? Begin to move handle as shown. Flux through the loop decreases. Current is induced which opposed this decrease – current tries to re-establish the B field. Induction - Spring 2006 43 moving the bar Flux BA BLx Dropping the minus sign... d dx emf BL BLv dt dt emf BLv i R R Induction - Spring 2006 44 Moving the Bar takes work F BiL BL BLv R or v B 2 L2 v F R dW d POWER Fx Fv dt dt B 2 L2 v P v R B 2 L2 v 2 P R Induction - Spring 2006 45 What about a SOLID loop?? Energy is LOST BRAKING SYSTEM METAL Induction - Spring 2006 Pull 46 Back to Circuits for a bit …. Induction - Spring 2006 47 Definition Current in loop produces a magnetic field in the coil and consequently a magnetic flux. If we attempt to change the current, an emf will be induced in the loops which will tend to oppose the change in current. This this acts like a “resistor” for changes in current! Induction - Spring 2006 48 Remember Faraday’s Law d emf V E ds dt Lentz Induction - Spring 2006 49 Look at the following circuit: Switch is open NO current flows in the circuit. All is at peace! Induction - Spring 2006 50 Close the circuit… After the circuit has been close for a long time, the current settles down. Since the current is constant, the flux through the coil is constant and there is no Emf. Current is simply E/R (Ohm’s Law) Induction - Spring 2006 51 Close the circuit… When switch is first closed, current begins to flow rapidly. The flux through the inductor changes rapidly. An emf is created in the coil that opposes the increase in current. The net potential difference across the resistor is the battery emf opposed by the emf of the coil. Induction - Spring 2006 52 Close the circuit… d emf dt Ebattery V (notation) d V iR 0 dt Induction - Spring 2006 53 Moving right along … Ebattery V (notation) d V iR 0 dt The flux is proportion al to the current as well as to the number of turns, N. For a solonoid, i Li N B d di L dt dt di V iR L 0 dt Induction - Spring 2006 54 Definition of Inductance L N B L i UNIT of Inductance = 1 henry = 1 T- m2/A B is the flux near the center of one of the coils making the inductor Induction - Spring 2006 55 Consider a Solenoid l B ds i 0 enclosed Bl 0 nli or n turns per unit length Induction - Spring 2006 B 0 ni 56 So…. N B nlBA nl 0 niA L i i i or L 0 n 2 Al or inductance 2 L/l n A unit length Depends only on geometry just like C and is independent of current. Induction - Spring 2006 57 Inductive Circuit i Switch to “a”. Inductor seems like a short so current rises quickly. Field increases in L and reverse emf is generated. Eventually, i maxes out and back emf ceases. Steady State Current after this. Induction - Spring 2006 58 THE BIG INDUCTION As we begin to increase the current in the coil The current in the first coil produces a magnetic field in the second coil Which tries to create a current which will reduce the field it is experiences And so resists the increase in current. Induction - Spring 2006 59 Back to the real world… Switch to “a” i sum of voltage drops 0 : di E iR L 0 dt same form as the capacitor equation q dq E R 0 C dt Induction - Spring 2006 60 Solution E Rt / L i (1 e ) R time constant L R Induction - Spring 2006 61 Switch position “b” E0 di L iR 0 dt E t / i e R Induction - Spring 2006 62 Max Current Rate of increase = max emf VR=iR ~current Induction - Spring 2006 63 E (1 e Rt / L ) R L (time constant) R i Induction - Spring 2006 64 IMPORTANT QUESTION Switch closes. No emf Current flows for a while It flows through R Energy is conserved (i2R) WHERE DOES THE ENERGY COME FROM?? Induction - Spring 2006 65 For an answer Return to the Big C E=0A/d +dq +q -q We move a charge dq from the (-) plate to the (+) one. The (-) plate becomes more (-) The (+) plate becomes more (+). dW=Fd=dq x E x d Induction - Spring 2006 66 The calc q dW (dq) Ed (dq) d (dq) d 0 0 A d d q2 W qdq 0 A 0 A 2 or 1 2 Ad 1 2 1 2 W (A) 0 2 Ad 0 E Ad 2 0 A 2 0 2 0 2 d 2 energy 1 2 u 0 E unit volum e 2 Induction - Spring 2006 The energy is in the FIELD !!! 67 What about POWER?? di E L iR dt i : di 2 iE Li i R dt power to circuit power dissipated by resistor Must be dWL/dt Induction - Spring 2006 68 So dWL di Li dt dt 1 2 WL L idi Li 2 1 2 WC CV 2 Induction - Spring 2006 Energy stored in the Capacitor 69 WHERE is the energy?? l B ds i 0 enclosed 0l Bl 0 nil B 0 ni or B 0 Ni l BA Induction - Spring 2006 0 Ni l A 70 Remember the Inductor?? N L i N Number of turns in inductor i current. Φ Magnetic flux throu gh one turn. ????????????? Induction - Spring 2006 71 So … N i N i L 1 1 N 1 W Li 2 i 2 N i 2 2 i 2 0 NiA l 1 0 NiA 1 2 2 2 A W Ni 0 N i 2 l 2 0 l L Induction - Spring 2006 72 1 A W N i 2 0 l 2 0 2 2 From before : B 0 Ni l 1 A 1 2 W Bl B V (volume) 2 0 l 2 0 2 2 or W 1 2 u B V 2 0 ENERGY IN THE FIELD TOO! Induction - Spring 2006 73 IMPORTANT CONCLUSION A region of space that contains either a magnetic or an electric field contains electromagnetic energy. The energy density of either is proportional to the square of the field strength. Induction - Spring 2006 74 Induction - Spring 2006 75