Electricity and MagnetismLC CircuitLC CircuitElectromagnetic OscillationsMaxwell’s Equations (almost)Displacement CurrentDisplacement CurrentMaxwell’s EquationsElectromagnetic WavesReminder on wavesReminder on wavesReminder on wavesReminder on wavesElectromagnetic WavesBack to Maxwell’s equationDifferential Form of M.E.Differential Form of M.E.Differential Form of M.E.Differential Form of M.E.May 3 2002Electricity and Magnetism•Reminder– LC circuits / Oscillations– Displacement current–Maxwell’s equations•Today– More on Maxwell’s equations– Electromagnetic wavesMay 3 2002LC CircuitLCSpring kMass md2Q/dt2+ 1/(LC) Q = 0ω02= 1/(LC) d2x/dt2+ k/m x = 0ω02= k/mMay 3 2002LC Circuit½L 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) = 1C½Q2/CEnergy in E-Field Energy in B-FieldOscillationMay 3 2002Electromagnetic Oscillations• In an LC circuit, we see oscillations:• Q: Can we get oscillations without circuit, i.e. when we have just the fields?•A: Yes! – Electromagnetic WavesEnergy in E-Field Energy in B-FieldMay 3 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 3 2002Displacement Current• Ampere’s Law broken – How can we fix it?I IQ = C VDisplacement Current ID= ε0dΦE/dtChanging field inside C also produces B-Field!May 3 2002Displacement Current• Example calculation: B(r) for r > RDI IRQ = C V-> B(r) = R2/(2rc2) dV/dtMay 3 2002Maxwell’s Equations•M.E.’s predictelectromagnetic waves, moving with speed of light• Major triumph of science1/c2May 3 2002Electromagnetic Waves• Until end of semester:– What are electromagnetic waves?– How does their existence follow from Maxwellsequations?– What are the properties of E.M. waves?• Prediction was far from obvious– No hint that E.M. waves exist– Involves quite a bit of mathMay 3 2002Reminder on waves• Examples of waves– Mechanical waves – Pressure waves– E.M. waves• In-Class Demo...May 3 2002Reminder on wavesAt a moment in time:Wavelength λAmplitudeD(X)Position xAt a point in space:Time t Period T = 1/fAmplitudeD(t)May 3 2002Reminder on waves• Types of waves– Transverse– Longitudinal• compression/decompressionMay 3 2002Reminder on waves• For a travelling wave (sound, water)Q: What is actually moving?• -> Energy!• Speed of propagation: v = λ f•Wave equation:Couples variation intime and spaceMay 3 2002Electromagnetic Waves• Is light an electromagnetic wave?– Check speed and see if we can predict thatLaserMirror∆tMay 3 2002Back to Maxwell’s equation• Wave equation is differential equation• M.E. (so far) describe integrals of fieldsTransform into differential eqn’sMay 3 2002Differential Form of M.E.• Need two theorems: Gauss and Stokes–GaussFlux/Unit VolumeDivergenceMay 3 2002Differential Form of M.E.• Need two theorems: Gauss and Stokes– StokesLoop Integral/Unit Area of LoopCurlMay 3 2002Differential Form of M.E.Gauss, StokesMay 3 2002Differential Form of M.E.In Vacuumno
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