Lecture 22: TUE 13 APR 2010Lecture 22: TUE 13 APR 2010 Ch.32.1Ch.32.1––5: Maxwell5: Maxwell’’s equationss equationsCh.33.1Ch.33.1––3: Electromagnetic Waves3: Electromagnetic WavesJames Clerk Maxwell (1831-1879) Physics 2102Jonathan DowlingEXAM 03: 6PM THU 15 APR LOCKETT 6The exam will cover: Ch.28 (second half)through Ch.32.1-3 (displacement current, andMaxwell's equations).The exam will be based on: HW07 – HW10.The formula sheet for the exam can be found here:http://www.phys.lsu.edu/classes/spring2010/phys2102/formulasheet3.pdfYou can see examples of old exam IIIs here:http://www.phys.lsu.edu/classes/spring2009/phys2102/Test3.oldtests.pdfMaxwell I: GaussMaxwell I: Gauss’’ Law for E-Fields: Law for E-Fields:charges produce electric fields,field lines start and end in charges!=•SqdAE0/"SSSSMaxwell II: GaussMaxwell II: Gauss’’ law for B-Fields: law for B-Fields:field lines are closedor, there are no magnetic monopoles!=•SdAB 0SSSSSSMaxwell III: AmpereMaxwell III: Ampere’’s law:s law:electric currents produce magnetic fields!=•CidsB0µCMaxwell IV: FaradayMaxwell IV: Faraday’’s law:s law:changing magnetic fields produce (“induce”) electric fields!!•"=•SCdABdtddsEMaxwell Equations I Maxwell Equations I –– IV: IV:!=•SqdAE0/"!=•SdAB 0!=•CidsB0µ!!•"=•SCdABdtddsE!=•SdAB 0!=•CdsB 0!!•"=•SCdABdtddsEIn Empty Space with No Charge or CurrentIn Empty Space with No Charge or Current…very suspicious…NO SYMMETRY!?qq=0=0ii=0=0 E • dA = 0S!!MaxwellMaxwell’’s Displacement Currents Displacement CurrentIf we are charging a capacitor, there is acurrent left and right of the capacitor.Thus, there is the same magnetic field rightand left of the capacitor, with circular linesaround the wires.But no magnetic field inside the capacitor?With a compass, we can verify there isindeed a magnetic field, equal to the fieldelsewhere.But there is no current producing it! ?EEBBBBThe missingMaxwellEquation!Maxwell’s FixdtddtEAddtEdddAdtdVCdtCVddtdqiE!======000)()()("""We can write the current as:We calculate the magnetic field produced by thecurrents at left and at right using Ampere’s law :!=•CidsB0µq=CVV=EdC=ε0A/dΦE=∫E•dA=EA EEid=ε0dΦ/dt!!•=•SCdAEdtddsB00"µB !EiBiBDisplacement CurrentDisplacement CurrentMaxwell proposed it based onsymmetry and math — no experiment!!"•CdsB 0MaxwellMaxwell’’s Equations I s Equations I –– V:V:!=•SqdAE0/"!=•SdAB 0idAEdtddsBSC000µ!µ+•=•""!!•"=•SCdABdtddsEIIIIIIIIIIIIIVIVV!=•SdAE 0!=•SdAB 0!!•=•SCdAEdtddsB00"µ!!•"=•SCdABdtddsEMaxwell Equations in Empty Space:Maxwell Equations in Empty Space:Fields withoutsources?Changing E gives B.Changing B gives E.A solution to the Maxwell equations in empty space isa “traveling wave”…c =1µ0!0= 3 " 108m/sThe electric-magnetic wavestravel at the speed of light?Light itself is a wave of electricity and magnetism!Maxwell, Waves, and LightMaxwell, Waves, and Lightd2Edx2= !µ0"0d2Edt2# E = E0sin k(x ! ct)electric and magnetic fields can travel in EMPTY SPACE! !!•=•SCdAEdtddsB00"µ!!•"=•SCdABdtddsEFirst person to use electromagnetic waves for communications:Guglielmo Marconi (1874-1937), 1909 Nobel Prize (first transatlantic commercial wirelessservice, Nova Scotia, 1909)Electromagnetic wavesElectromagnetic wavesFirst person to prove that electromagnetic waves existed:Heinrich Hertz (1875-1894)Electromagnetic Waves: Electromagnetic Waves: One Velocity, Many Wavelengths!One Velocity, Many Wavelengths!with frequencies measured in “Hertz” (cycles per second)and wavelength in meters.http://imagers.gsfc.nasa.gov/ems/http://www.astro.uiuc.edu/~kaler/sow/spectra.htmlHow do E&M Waves Travel?How do E&M Waves Travel?Is there an “ether” they ride on? Michelsonand Morley looked and looked, and decidedit wasn’t there. How do waves travel???Electricity and magnetism are “relative”:Whether charges move or not depends onwhich frame we use…This was how Einstein beganthinking about his “theory ofspecial relativity”…We’ll leave that theory for later.A solution to Maxwell’s equations in free space:)sin( txkEEm!"=)sin( txkBBm!"=n.propagatio of speed ,ck=!c =EmBm=1µ0!0=299,462,954 ms!=!187,163 miles/secVisible light, infrared, ultraviolet,radio waves, X rays, Gammarays are all electromagnetic waves.Electromagnetic WavesElectromagnetic WavesRadio waves are reflected by the layer of the Earth’satmosphere called the ionosphere.This allows for transmission between two points which arefar from each other on the globe, despite the curvatureof the earth.Marconi’s experiment discovered the ionosphere! Expertsthought he was crazy and this would never work.Fig. 33-1The wavelength/frequency range in which electromagnetic (EM) waves(light) are visible is only a tiny fraction of the entire electromagneticspectrum.Maxwell’s RainbowFig. 33-2(33-2)An LC oscillator causes currents to flow sinusoidally, which in turn producesoscillating electric and magnetic fields, which then propagate through space asEM waves.Fig.33-3Oscillation Frequency:1LC!=Next slideThe Traveling Electromagnetic (EM) Wave, Qualitatively(33-3)c =EmBm=1µ0!0c =EmBm=1µ0!0Fig. 33-5Mathematical Description of Traveling EM WavesElectric Field:( )sinmE E kx t!= "Magnetic Field:( )sinmB B kx t!= "Wave Speed:0 01cµ !=Wavenumber:2k!"=Angular frequency:2!"#=Vacuum Permittivity:0!Vacuum Permeability:0µAll EM waves travel a c in vacuumAmplitude Ratio:mmEcB=Magnitude Ratio:( )( )E tcB t=EM Wave Simulation(33-5)Electromagnetic waves are able to transport energy from transmitterto receiver (example: from the Sun to our skin).The power transported by the wave and itsdirection is quantified by the Poynting vector.John Henry Poynting (1852-1914) 211|| EcEBS00==µµThe The Poynting Poynting Vector:Vector:Points in Direction of Power FlowPoints in Direction of Power FlowEBSUnits: Watt/m2For a wave, sinceE is perpendicular to B: BES!!!!=0µ1In a wave, the fieldschange with time.Therefore the Poyntingvector changes too!!The direction is constant,but the magnitude changesfrom 0 to a maximumvalue.____________222___)(sin11tkxEcEcSIm!µµ"===00The average of sin2 overone cycle is ½:221mEcI0=µ21rmsEcI0=µBoth fields have the same energy density. 2 22 201 1 1 1( )2 2 2 2E BB Bu E cB u! ! !! µ µ0 00 0 0= = = = =or, EM Wave Intensity, Energy DensityEM Wave Intensity, Energy DensityA better measure of the amount of energy in an EM wave is obtainedby averaging
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