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MIT 8 02X - Electricity and Magnetism

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Electricity and MagnetismMutual InductanceMutual InductanceMutual InductanceMutal InductanceExample: Two SolenoidsIn-Class Demo: Two CoilsIn-Class Demo: Marconi CoilsSelf InductanceExample: SolenoidSelf InductanceIn-Class Demo: Levitating CoilIn-Class Demo: Levitating CoilRL CircuitsRL CircuitsRL Circuits‘Back EMF’RL CircuitsEnergy Storage in InductorRLC circuitsSummary of Circuit ComponentsR,L,C in AC CircuitAC circuitFirst: Look at the componentsRLC circuitRLC circuitResonanceRLC circuitResonanceLC CircuitLC CircuitLC CircuitLC CircuitElectromagnetic OscillationsDisplacement CurrentDisplacement CurrentDisplacement CurrentMaxwell’s EquationsMaxwell’s EquationsElectromagnetic WavesReminder on WavesReminder on WavesReminder on WavesWave EquationWave PropertiesDifferential Form of M.E.Differential Form of M.E.Maxwell’s Equations in VacuumMaxwell’s Equations in VacuumElectromagnetic WavesElectromagnetic WavesPlane wavesPlane wavesE.M. Wave SummaryTypical E.M. wavelengthEnergy in E.M. WavesMay 6 2002Electricity and Magnetism•Review– Self and mutual inductance–Energy in B-Field–LR circuit – LRC circuits and Oscillations– AC circuits– Displacement current– Maxwell’s equations– EM wavesMay 6 2002Mutual Inductance• Transformer actionξs /ξp= Ns/NpB~IAC= I0sin(ωt)SecondaryPrimaryξp= - NpdΦB/dtξs= - NsdΦB/dtNssameNpFlux through single turnMay 6 2002Mutual Inductance•Transformer action• Transformers allow change of amplitude for AC voltage – ratio of secondary to primary windings• Constructed such that ΦBidentical for primary and secondary• What about general case of two coils?ξs /ξp= Ns/NpMay 6 2002Mutual Inductance~BΦB ~ BB ~ I1Def.:M12= N2ΦB/I1N1I1N2May 6 2002Mutal Inductance• Coupling is symmetric: M12 = M21= M• M depends only on Geometry and Material• Mutual inductance gives strength of coupling between two coils (conductors):•M relatesξ2and Ι1(or ξ1and Ι2)•Units:[M] = V/(A/s) = V s /A = H (‘Henry’)ξ2= - N2dΦB/dt = - M dI1/dtMay 6 2002Example: Two SolenoidsN1N2Area AQ: How big is M = N2ΦB/I1?A: M = µ0N1N2 A/lLength lMay 6 2002In-Class Demo: Two CoilsRadioSpeaker• Signal transmitted by varying B-Field• Coupling depends on Geometry (angle, distance)May 6 2002In-Class Demo: Marconi CoilsNSNPSecondarySpringPrimaryNP<< NS IronBatterySwitchMay 6 2002Self Inductance~IBCircuit sees flux generated by it selfDef.: L = N ΦB/ISelf-InductanceMay 6 2002Example: Solenoid~IBQ: How big is L ?A: L = µ0 N2A/LMay 6 2002Self Inductance• L is also measured in [H]• L connects induced EMF and variation in current:ξ = - L dI/dt• Remember Lenz’ Rule:Induced EMF will ‘act against’ change in current -> effective ‘inertia’• Delay between current and voltageMay 6 2002In-Class Demo: Levitating CoilIAC= I0sin(ωt)Iind~ -dΦB/dt ~-cos(ωt+φ)IACtWithout delay (φ = 0):No net forceIindMay 6 2002In-Class Demo: Levitating CoilIAC= I0sin(ωt)Iind~-dΦB/dt ~-cos(ωt+φ)IACtWith delay (φ >0):Net repulsion (currents areopposite most of the time)IindMay 6 2002RL CircuitsKirchoffs Rule: V0+ ξind= R I -> V0= L dI/dt + R IQ: What is I(t)? L dI/dtR IVV0L1R2May 6 2002RL CircuitsI(t)tτ = L/R63%I(t)=V0/R [1-exp(-t/τ)]ξ(t)ξ(t)=V0exp(-t/τ)tτ = L/R37%May 6 2002RL Circuits• Inductance leads to ‘delay’ in reaction of current to change of voltage V0• All practical circuits have some L and R–change in I never instantaneousMay 6 2002‘Back EMF’V0RL• What happens if we move switch to position 2? 12May 6 2002tτ = L/R63%I(t)tτ = L/R37%2τ = L/RΙ(t)=V0/R exp(-t/τ)ξ(t)1May 6 2002RL Circuits• L counteracts change in current both ways– Resists increase in I when connecting voltage source– Resists decrease in I when disconnecting voltage source–‘Back EMF’• That’s what causes spark when switching off e.g. appliance, lightMay 6 2002Energy Storage in Inductor•Energy in Inductor– Start with Power P = ξ I = L dI/dt I = dU/dt-> dU = L dI I-> U = ½ L I2• Where is the Energy stored?– Example: SolenoidU/Volume = ½ B2/µ0May 6 2002RLC circuits• Combine everything we know...• Resonance Phenomena in RLC circuits– Resonance Phenomena known from mechanics (and engineering)– Great practical importance–video...May 6 2002Summary of Circuit Components~VV(t)RVR = IRVL = L dI/dtLVC = 1/C IdtCMay 6 2002R,L,C in AC Circuit •AC circuit– I(t) = I0sin(ωt)– V(t) = V0sin(ωt + φ)• Relationship between V and I can be characterized by two quantities– Impedance Z = V0/I0– Phase-shift φsame ω!May 6 2002AC circuitI(t)=I0 sin(ωt)V(t)=V0 sin(ωt + φ)2π/ωI0φ/ωV0Impedance Z = V0/I0 Phase-shiftφMay 6 2002First: Look at the componentsZ = Rφ= 0 V and I in phaseR~I(t)V= I RC~I(t)V = Q/C = 1/C IdtZ = 1/(ωC)φ = - π/2 V lags I by 90oL~I(t)V= L dI/dtZ = ω Lφ= π/2I lags V by 90oMay 6 2002RLC circuitLRC~V(t)V –L dI/dt -IR -Q/C = 0L d2Q/dt2= -1/C Q – R dQ/dt + V2ndorder differential equationMay 6 2002RLC circuitLRC~V(t)V –L dI/dt -IR -Q/C = 0L d2Q/dt2= -1/C Q – R dQ/dt + VWaterSpringMass mFextm d2x/dt2= -k x – f dx/dt + Fext‘Inertia’‘Spring’‘Friction’May 6 2002ResonanceI0ω = (LC)1/2ωωφ−π/2π/2Like LLike CImax = V0/RLow Frequency High FrequencyMay 6 2002RLC circuitV0sin(ωt) = I0{[ωL -1/(ωC)] cos(ωt – φ) +R sin(ωt – φ)} Solution (requires two tricks):I0= V0/([ωL -1/(ωC)]2+ R2)1/2= V0/Ztan(φ) = [ωL-1/(ωC)]/R -> For ωL= 1/(ωC), Z is minimal and φ =0i.e. ω0= 1/(LC)1/2Resonance FrequencyMay 6 2002Resonance• Practical importance– ‘Tuning’ a radio or TV means adjusting the resonance frequency of a circuit to match the frequency of the carrier signalMay 6 2002LC Circuit• What happens if we open switch?– L dI/dt - Q/C = 0L d2Q/dt2+ Q/C = 0LCV0d2x/dt2+ ω02x = 0Harmonic Oscillator!May 6 2002LC Circuit1/2 L I2LC1/2 k x21/2 m v2Spring kMass m1/2 Q2/CPotential Energy Kinetic EnergyEnergy in E-Field Energy in B-FieldOscillationOscillationMay 6 2002LC CircuitLCSpring kMass md2Q/dt2+ 1/(LC) Q = 0ω02= 1/(LC) d2x/dt2+ k/m x = 0ω02= k/mMay 6 2002LC Circuit1/2 L I2L•Total energy U(t) is conserved:Q(t) ~ cos(ωt) dQ/dt ~ sin(ωt)UL~ (dQ/dt)2~ sin2UC~ Q(t)2~ cos2cos2(ωt) + sin2(ωt) = 1C1/2 Q2/CEnergy in E-Field Energy in B-FieldOscillationMay 6 2002Electromagnetic Oscillations• In an LC circuit, we see oscillations:• Q: Can we get oscillations without circuit?•A: Yes! – Electromagnetic


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