VIII. Magnetic Induction

VIII. Magnetic Induction VIII. Magnetic Induction A. Dynamos & Generators Dr. Bill Pezzaglia B. Faraday s Law C. Inductance Updated 00Mar6 Schedules/Reading 3 Michael Faraday (79-867) 4 Faraday s Law, Chapter 5.-5.4 Transformer 6. RL circuit 6.5, and lab notes. 8 Creates first motor 83 Creates dynamo, a DC generator Skip 5.5-5.8 for now 83 Law of induction QUIZ covers: Chapter 4, 5.-5.4, 6., 6.5 (VII Magnetostatics & VIII Induction) 846 Discovers Diamagnetism A. Dynamos & Generators 5. Dynamo Rule 6 ) Dynamo Rule ) Motional EMF 3) Generating Power (a) Faraday (83) Move a wire so that it cuts magnetic field lines will generate current r r r Recall F = q v B Positive charges in wire will move one direction, negative the other

.(a).ii Planetary Dynamos 7.(a).iii Planetary Dynamos 8 Moon Io moving orbiting through Jupiters magnetic field generates BIG current The electric currents generate so much heat the moon has active volcanoes! (b) Magnetohydrodynamic Generators (MHD) 9 (b).ii MHD Generators 0 83 Faraday s Fluid Generator experiment at Waterloo Bridge: attempts to measure current generated from velocity of Thames cutting through earth s magnetic field (didn t work too well). r r r F = q v B - + (c). Faraday s Disk Dynamo (83). EMF (Electromotive Force) This was the first practical DC generator. It gives high current, but low voltage. I (a) Definition of EMF (ξ) Misnomer: its not really a force, it s a voltage (i.e. ENERGY per charge) Chemical EMF: A battery is like a pump. When charges pass through, their energy is increased. Change in voltage= EMF Work ξ = q

.(b). Motional EMF 3.(c). AC Generator 4 Magnetic ( Dynamo ) Work done by moving a length L wire through magnetic field B with velocity v generates an EMF 89 Tesla (working for Edison) invents AC Generator. Edison hates the idea (he is using DC dynamos), and so Tesla sells it to Westinghouse. Edison goes on a campaign to convince people AC is DANGEROUS while DC is safe. ξ = Work FL ( qvb L = = ) q q q General Result: (i.e. only part of vxb which is parallel to wire will count) r v v ξ = ( v B) L http://www.youtube.com/v/i-j-jgd8.(c).ii AC Generator Details 5 3. Generator Power 6 Rectangle area: A=ab a Velocity: v = ω EMF: r r v ξ = ( v B) L = vblsinθ ωa ξ = B(b) sinθ ξ = ωbasinθ θ = ωt (a) Electric Power Output: (b) Current created will experience force from magnetic field, which is in opposite direction as v r r r F = I L B P = ξ I (c) Mechanical Work: by Newton s 3 rd law, we must PUSH the wire through with force F, or power: r r P = F v = ( ILB) v = I( LBV ) = Iξ B. Faraday s Law 7 83 Faraday s Three Experiments 8 ) Magnetic Flux Generated current by: Moving a coil in and out of a magnetic field ) Faraday s Law 3) Lenz s Law Moving magnet in and out of coil When current turned on or off in a coil, current is generated in a nearby coil. He explained all of these effects with one single law 3

. Magnetic Flux (a) Definition: Magnetic Flux is the Magnetic Field B (aka Magnetic Intensity Vector ) times area it flows through Φ B A Units: Weber=Tesla m 9 (b) Lambert s Law (Orientation matters!) Sunlight coming in at a low altitude angle will have its energy spread out over more area. Lambert s Law (760) Intensity is reduced by cosine of angle of incidence Flux is the dot product of the electric field vector with the area vector (which is normal to the surface) r r Φ B A = BA cosθ 44 (c) Magnetic Flux is Conserved. Faraday s Law of Induction (83) Because there are no magnetic monopoles, there are no sources of magnetic field lines. Possibly done 830 by Henry (unpublished) The EMF generated in a loop of wire is equal to the change in magnetic flux through the loop (with respect to time) Magnetic Field Lines must be continuous (i.e. continue through magnet) Gauss s law for magnetism: total magnetic flux through a closed surface is ZERO. You can get a change of flux in ways r r r r = ( ΔB) A + B ( ΔA) (a) Change the Area of Loop 3 (b) Or, change the orientation of loop 4 Consider sliding wire in constant magnetic field Consider AC generator, where we twist the loop ΔA = B ξ = ( )BvL This is equivalent to Dynamo Rule (motional emf) ΔA = vl Φ = BAcosθ Δθ = BAsinθ ξ = ωbasinθ This is equivalent to Dynamo Rule (motional emf) 4

(c) Change Magnetic Field 5 3. Lenz s Law (834) 6 You can NOT explain this one by dynamo rule, as no wires move! (a) The law: Direction of induced current is such as to oppose the cause producing it. ΔB = A However, it makes perfect sense because motion is relative. Whether you think the magnet is moving with the coil still, or the magnet still with the coil moving merely depends upon the motion of the observer! It s the minus sign in Faraday s Law Heinrich Lenz 804-864 (b). Eddy Currents 7 (c). Magnetic Levitation 8 84, François Arago discovers when conductor is exposed to changing magnetic field, small circular eddy currents (also known as Foucault currents) are generated. 855, J.B.L. Foucault rotates a copper disc with rim between poles of magnet and discovers that the induced eddy currents in the metal cause: Inductive Braking: the force required for the rotation of a copper disc becomes greater with magnet (even though copper is not attracted by a magnet) For good conductors (superconductors), the eddy currents are so strong that the induced magnetic field can levitate the object. Lenz s law is usually demonstrated by Elihu Thomson's jumping ring (887? 897?). Inductive Heating: the disc becomes heated by the eddy current (i.e. friction ), because of resistance of metal. Braking (tubes) http://www.youtube.com/watch?v=splawcxvkmg http://www.magnet.fsu.edu/education/community/slideshows/eddycurrents/index.html Jumping Ring: http://www.youtube.com/watch?v=pl7kyvijie C. Inductance 9. Mutual Induction 30 ) Mutual Induction (Transformers) (a) 83 Faraday notes when current turned on or off in a coil, voltage is generated in a nearby coil. ) Self Inductance & RL circuits 3) Energy in Inductors Define Mutual Inductance M (units of Henries=ohm sec) ξ = M Δ t If first coil is solenoid, then can show M = μ N N 0 A l 5

(a).ii Coefficient of Coupling 3 (b) Ruhmkorff Induction Coil 3 Loosely Coupled: if coils are far apart, not all of magnetic flux from first coil goes through the second (flux leakage). Early transformers were very inefficient because of this. Tightly Coupled: either have coils wrapped around each other, or share same iron core so that nearly all flux from one goes through other. Probably first invented by invented by Nicholas Callan in 836. 85 Ruhmkorff shows if secondary coil has many more windings than primary, then a BIG voltage can be generated from a small one. DC current in primary creates magnetic field Reciprocity: If make coils same length with same area, and tightly coupled, then mutual inductance is same both ways ( L is self inductance to be discussed in next section) Current is periodically interrupted by a vibrating switch, causing field to collapse BIG voltage is generated in secondary by Faraday s law M = = M L L This was how early Cathode Ray, X-ray and neon signs were powered. (c) Transformer Equation Nikola Tesla pioneered the idea of using AC current, transmitting power at high voltages (low current) and then using transformers to step it back down to low voltage for user. 884 Closed Flux Transformer invented: the flux through both coils is the same: V = N V = N Divide the equations to get the transformer rule: V N = V N I = I Note that power is conserved! V = I VI 33. Self Inductance (a) Probably done first by Joseph Henry 830, a year before Faraday. Current in coil makes a magnetic field. Change in current changes field, which by Faraday s law creates a voltage in the coil (by Lenz law a back emf which opposes change). Definition of (self) inductance: Self Inductance L is a function only of geometric (and magnetic permeability) of coil. Generally goes like square of N. For solenoid: ΔB L = N = μn A l V = L ΔB L N 34 b. Inductors in Circuits 35 (b).ii. RL Circuits (incomplete) 36 Rules for networks of inductors is similar to resistors (hence opposite of capacitors) No notes here, as we did this in detail in lab. See lab notes (photos of board have been posted), and also 6.5 in book Inductors in series ADD Inductors in parallel: = + L L L L = L + L or L L L = L + L In brief, RL circuits behave complementary to RC circuits. Inductors try to keep the current flowing, so if there is an abrupt loss of current, inductors will attempt to supply any voltage necessary to keep the current flowing, the current will decay exponentially with time constant: L τ = R RC circuit: capacitors try to keep the voltage the same, so if there is an abrupt loss of voltage, the capacitor will attempt supply any current necessary to keep the voltage up. The voltage will decay exponentially with time. 6

3. Energy Stored in Inductor 37 b. Energy Density 38 a) Energy Formula Energy in Inductor: U = LI The energy stored in capacitor is PE, while KE of current is stored in an inductor Divide by volume (of a solenoid coil) u = U vol = B μ In other words, the energy is stored in the magnetic field itself! This energy creates a magnetic pressure that tries to push a coil apart c. Forces on Cores etc. If you insert a core, you increase the inductance by factor K m : μ Kmμ 0 Hence, increase the energy. Hence inductors will want to push cores out (doorbell or induction gun). 39 References 83 Faraday Law of Induction Neat video: http://www.youtube.com/watch?v=yahqtxbv5va&feature=related http://www.youtube.com/watch?v=vpxdlzpcc8&feature=related Magnetic flux and orientation: http://www.youtube.com/watch?v=kxfxurbwp-c&feature=related AC generator http://www.youtube.com/watch?v=gqa3woounea&feature=related Even better AC generator: http://www.youtube.com/watch?v=i-j-jgd8&nr= Variable AC generator: http://www.youtube.com/watch?v=mcvxa_vvfh4&feature=related Simulations: http://www.esjd.pt/recursos_educativos/phet.0/new/simulations/index057.html?cat=all_sims 40 By inserting core, you change inductance, so more correct formula for voltage (with possible sign errors): ΔL V = L + I 7