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MIT 8 02T - Maxwell’s Equations

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1P28-Class 28: OutlineHour 1:Displacement CurrentMaxwell’s EquationsHour 2:Electromagnetic waves2P28-Finally:Bringing it All Together3P28-Displacement Current4P28-Ampere’s Law: CapacitorConsider a charging capacitor:IUse Ampere’s Law to calculate the magnetic field just above the top plate1) Red Amperian Area, Ienc= I2) Green Amperian Area, I = 0What’s Going On?0Ampere's law: encdIµ⋅=∫Bs5P28-Displacement Current0EdddQIdt dtεΦ=≡We don’t have current between the capacitor plates but we do have a changing E field. Can we “make” a current out of that?000EQEQEAAεεε= ⇒ ==ΦThis is called (for historic reasons) the Displacement Current6P28-Maxwell-Ampere’s Law0000()encl dCEencldIIdIdtµµµε⋅= +Φ=+∫Bs7P28-PRS Questions:Capacitor8P28-Maxwell’s Equations9P28-Electromagnetism Review• E fields are created by:(1) electric charges(2) time changing B fields• B fields are created by(1) moving electric charges(NOT magnetic charges)(2) time changing E fields• E (B) fields exert forces on (moving) electric chargesGauss’s LawFaraday’s LawAmpere’s LawMaxwell’s AdditionLorentz Force10P28-Maxwell’s Equations0000(Gauss's Law)(Faraday's Law)0 (Magnetic Gauss's Law)(Ampere-Maxwell Law)( (Lorentz force Law)inSBCSEencCQddddtdddIdtqεµµε⋅=Φ⋅=−⋅=Φ⋅= +=+×∫∫∫∫∫∫EAEsBABsFEvB) 11P28-Electromagnetic Radiation12P28-A Question of Time…http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/05-CreatingRadiation/05-pith_f220_320.html13P28-14P28-Electromagnetic Radiation: Plane Waveshttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html15P28-Traveling WavesConsider f(x) = x=0What is g(x,t) = f(x-vt)?x=0t=0x=vt0t=t0x=2vt0t=2t0f(x-vt) is traveling wave moving to the right!16P28-Traveling Sine WaveNow consider f(x) = y = y0sin(kx):xAmplitude (y0)2Wavelength ( )wavenumber ( )kπλ=What is g(x,t) = f(x+vt)? Travels to left at velocity vy = y0sin(k(x+vt)) = y0sin(kx+kvt)17P28-Traveling Sine WaveAmplitude (y0)1Period ( )frequency ( )2angular frequency ( )Tfπω==()0siny y kx kvt=+00sin( ) sin( )yy kvt y tω=≡At x=0, just a function of time:18P28-Traveling Sine Wave0sin( )yy kx tω=−Wavelength: Frequency : 2Wave Number: Angular Frequency: 212Period: Speed of Propagation: Direction of Propagation: fkfTfvfkxλπλωππωωλ======+iiiiiii19P28-Electromagnetic WavesRemember: fcλ=Hz20P28-Electromagnetic Radiation: Plane WavesWatch 2 Ways:1) Sine wave traveling to right (+x)2) Collection of out of phase oscillators (watch one position)Don’t confuse vectors with heights – they are magnitudes of E (gold) and B (blue)http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html21P28-PRS Question:Wave22P28-Group Work:Do Problem 123P28-Properties of EM Waves8001310mvcsµε== =×00EEcBB==Travel (through vacuum) with speed of lightAt every point in the wave and any instant of time, E and B are in phase with one another, withE and B fields perpendicular to one another, and to the direction of propagation (they are transverse):Direction of propagation = Direction of ×EB24P28-Direction of Propagation() ()00ˆˆˆˆsin();sin()Ek t Bk tωω=⋅−=⋅−EE pr BB pr()ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆzxyzxy⋅−−−−−−EB p prijkjk iki jji kkj iik jˆˆˆ×=EB p25P28-PRS Question:Direction of Propagation26P28-In Class Problem:Plane EM Waves27P28-Energy & the Poynting Vector28P28-Energy in EM Waves220011, 22EBuEu Bεµ==Energy densities:Consider cylinder:22001()2EBBdU u u Adz E Acdtεµ⎛⎞=+ = +⎜⎟⎝⎠What is rate of energy flow per unit area?002cEBcEBcεµ⎛⎞=+⎜⎟⎝⎠()200012EBcεµµ=+0EBµ=1 dUSAdt=22002cBEεµ⎛⎞=+⎜⎟⎝⎠29P28-Poynting Vector and Intensity0: Poynting vectorµ×=EBSunits: Joules per square meter per secDirection of energy flow = direction of wave propagationIntensity I:2200 0 000 022 2EBEcBIScµµ µ≡< >= = =30P28-Energy Flow: Resistor0µ×=EBSOn surface of resistor is INWARD31P28-PRS Questions:Poynting Vector32P28-Energy Flow: InductorOn surface of inductor with increasing current is INWARD0µ×=EBS33P28-Energy Flow: InductorOn surface of inductor with decreasing current is


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MIT 8 02T - Maxwell’s Equations

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