Electricity and MagnetismAC CircuitAC CircuitFirst: Look at the componentsRLC CircuitRLC CircuitIn-Class Demo (on scope)ResonanceResonanceLC CircuitLC CircuitLC CircuitElectromagnetic OscillationsMaxwell’s Equations (almost)The missing pieceDisplacement CurrentMaxwell’s EquationsMaxwell’s EquationsMaxwell’s EquationsMay 1 2002Electricity and Magnetism•Reminder–RLC Circuits– Resonance•Today– LC circuits / Oscillations– Displacement current– Maxwell’s equationsMay 1 2002AC 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 1 2002AC CircuitI(t)=I0 sin(ωt)V(t)=V0 sin(ωt+φ)2π/ωI0φ/ωV0Impedance Z = V0/I0 Phase-shiftφMay 1 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 1 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’‘Drag’May 1 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 1 2002In-Class Demo (on scope)LRC~V(t)VR(t) ~ I(t)May 1 2002ResonanceI0ω0= (LC)1/2ωωφ−π/2π/2Like LLike CImax = V0/RLow Frequency High FrequencyMay 1 2002Resonance• Practical importance– ‘Tuning’ a radio or TV means adjusting the resonance frequency of a circuit to match the frequency of the carrier signalMay 1 2002LC Circuit• What happens if we open switch?–L dI/dt -Q/C = 0L d2Q/dt2+ Q/C = 0LCV0d2x/dt2+ ω02x = 0Harmonic Oscillator!May 1 2002LC CircuitLCSpring kMass md2Q/dt2+ 1/(LC) Q = 0ω02= 1/(LC) d2x/dt2+ k/m x = 0ω02= k/mMay 1 2002LC Circuit½L I2LC½k x2½m v2Spring kMass m½Q2/CPotential Energy Kinetic EnergyEnergy in E-Field Energy in B-FieldOscillationOscillationMay 1 2002Electromagnetic Oscillations• In an LC circuit, we see oscillations:• Q: Can we get oscillations without circuit?•A: Yes! – Electromagnetic WavesEnergy in E-Field Energy in B-FieldMay 1 2002Maxwell’s Equations (almost)Charges are the source of Electric Flux through close surfaceChanging magnetic field createsan electric fieldThere are no magnetic monopolesMoving charges create magnetic field• Connection between electric and magnetic phenomena• But not symmetric• -> James Clerk Maxwell (~1860)May 1 2002The missing pieceA2A2A1A1II=0!May 1 2002Displacement Current• Ampere’s Law broken – How can we fix it?I IQ = C VDisplacement Current ID= ε0dΦE/dtMay 1 2002Maxwell’s Equations• Symmetry between E and B– although there are no magnetic monopoles• Basis for radio, TV, electric motors, generators, electric power transmission, electric circuits etcMay 1 2002Maxwell’s Equations• M.E.’s predictelectromagnetic waves, moving with speed of light• Major triumph of science1/c2May 1 2002Maxwell’s Equations• Symmetry between E and B– although there are no magnetic monopoles• Basis for radio, TV, electric motors, generators, electric power transmission, electric circuits
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