Lecture 25Chapter 31Induction and InductanceReview• Magnetic flux• Faraday’s law• Lenz’s law – induced emf gives rise to a current whose B field opposes the change in flux that produced it• Changing B field produces an E field• Restate Faraday’s law• Inductance, L defined ∫•=Φ AdBBrrdtdBΦ−=EdtdsdEBΦ−=•∫rriNLBΦ=Review• Inductor – device produces known Bfield• Solenoid is an inductor with inductance per unit length of• Self-induce emf, EEEELappears in any coil in which the current is changing• Direction of EEEELfollows Lenz’s law and opposes the change in currentdtdiLL−=EAnlL20µ=Review• RL circuit – resistor and inductor in series• Time dependence on current in RL circuit• Initially inductor acts to oppose changes in current through it• Long time later, inductor acts like simple wire• Rise of current• Decay of current• Inductive time constant()LteRτ−−= 1EiRLL=τLLtteieRττ−−==0EiInductance (39)• Mutual induction –current in one coil induces emf in other coil• Distinguish from self-induction• Mutual inductance, M21 of coil 2 with respect to coil 1 is 121221iNMΦ=Inductance (40)• Rearrange equation•Vary i1with time• Faraday’s law• Induced emf in coil 2 due to i in coil 1 is• Obeys Lenz’s law (minus sign)121221iNMΦ=212121Φ= NiMdtdNdtdiM212121Φ=dtdN212Φ−=2EdtdiM121−=2EInductance (41)• Reverse roles of coils• What is induced emf in coil 1 from a changing current in coil 2?• Same game as beforedtdiM212−=1E212112iNMΦ=Inductance (42)• The mutual inductance terms are equal• Rewrite emfs as• Notice same form as self-induced emfdtdiM1−=2EdtdiLL−=EdtdiM2−=1EMMM ==1221iNLBΦ=Inductance (43)• Generators – convert mechanical energy to electrical energy• External agent rotates loop of wire in B field– Hydroelectric plant– Coal burning plant• Changing ΦBinduces an emf and current in an external circuitInductance (44)• Alternating current (ac) generator– Ends of wire loop are attached to slip rings which rotate with loop– Stationary metal brushes are in contact with slip rings and connected to external circuit– emf and current in circuit alternate in timeEEEEtInductance (45)• Calculate emf for generator with N turns of area A and rotating with constant angular velocity, ω• Magnetic flux is • Relate angular dis-placement to angular velocityθcosBAAdBB=•=Φ∫rrtωθ=tBABωcos=Φ• Flux through one loop isInductance (46)• Faraday’s law says• SubstitutetBABωcos=ΦdtdNBΦ−=E()tdtdNBAωcos−=EEEEEttNBAωωsin=E• Maximum emf is when ωt = 90 or 270 degrees• Emf is 0 when ωt = 0 or 180 degreesωNBA=maxEInductance (47)• Direct current (dc) generator– Ends of loop are connected to a single split ring– Metal brush contacts to split ring reverse their roles every half cycle– Polarity of induced emfreverses but polarity of split ring remains the sameEEEEt• Not suitable for most applications– Can use to charge batteries• Commercial dc gen. use out of phase coilsInductance (48)• Motors – converts electrical energy to mechanical energy– Generator run in reverse– Current is supplied to loop and the torque acting on the current-carrying loop causes it to rotate– Do mechanical work by using the rotating armature– As loop rotates, changing B field induces an emf– Induced emf (back emf) reduces the current in the loop – remember Lenz’s law– Power requirements are greater for starting a motor and for running it under heavy loadsInductance (49)• Instead of a loop of wire, what happens when a bulk piece of metal moves through a B field?• Free electrons in metal move in circles as if caught in a whirlpool called eddy currents• A metal plate swinging through a B field will generate eddy currentsInductance (50)• Eddy currents will oppose the change that caused them – Lenz’s law• Induced eddy currents will always produce a retarding force when plate enters or leaves B field causing the plate to come to rest• Cutting slots in metal plate will greatly reduce the eddy currentsInductance (51)• Induction and eddy currents are used for braking systems on some subways and rapid transit cars • Moving vehicle has electromagnet (e.g. solenoid) which is positioned near steel rails• Current in electromagnet generates B field• Relative motion of B field to rails induces eddy currents in rails• Direction of eddy currents produce a drag force on the moving vehicle • Eddy currents decrease steadily as car slows giving a smooth stopInductance (52)• Eddy currents often undesirable since they dissipate energy in form of heat• Moving conducting parts often laminated – Build up several thin layers separated by nonconducting material– Layered structure confines eddy currents to individual layers• Used in transformers and motors to minimize eddy currents and improve efficiencyInductance (53)• How much energy is stored in a B field?• Conservation of energy expressed in loop rule• Multiply each side by iiRdtdiL +=ERidtdiLi2+=Ei•EEEEi is rate at which emfdevice delivers energy to rest of circuit• i 2R is rate at which energy appears as thermal energy in resistorInductance (54)• Middle term represents the rate dUB/dt at which energy is stored in the B fieldRidtdiLi2+=Ei• Integrating gives• Energy stored in B field• Similar to UEdtdiLidtdUB=LididUB=∫∫=iUBLididUB00221LiUB=CqUE221=Inductance (55)• What is the energy density of B field?• Energy density, uBis energy per unit volume• Volume is area x lengthAlUuBB=221LiUB=• Substituting UBgives• For a solenoid • Energy density is AlLiuB22=AnlL20µ=22021inuBµ=• Substituting B gives magnetic energy density• Similar to electric energy densityInductance (56)• Remember B field from a solenoid is 0221µBuB=inBoµ=22021inuBµ=2021EuEε=Inductance (57)• Place coil C at center of long solenoid which has a steadily decreasing current. What is the magnitude of the induced emf in coil C?• Solenoid generates uniform B field of • Current is decreasing so B field decreasesinBoµ=Inductance (58)•Since B field decreases the flux decreases and an emf is induced in coil C (Faraday’s law)• The current is decreasing at a steady rate so flux also decreases at steady rate and write∫•=Φ AdBBrrdtdNBΦ−=EttdtdiBfBBB∆Φ−Φ=∆∆Φ=Φ,,Inductance (59)• Need to find initial and final flux• Current decreases to
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