Class 30: OutlineLast Time:Traveling WavesTraveling Sine WaveTraveling Sine WaveTraveling Sine WaveThis Time:Standing WavesStanding WavesStanding Waves: Who Cares?Standing Waves: BridgeGroup Work: Standing WavesLast Time:Maxwell’s EquationsMaxwell’s EquationsWhich Leads To…EM WavesElectromagnetic Radiation: Plane WavesTraveling E & B WavesProperties of EM WavesPRS Questions:Direction of PropagationHow Do Maxwell’s Equations Lead to EM Waves?Derive Wave EquationWave EquationWave EquationWave EquationWave Equation1D Wave Equation for E1D Wave Equation for E1D Wave Equation for BElectromagnetic RadiationAmplitudes of E & BGroup Problem: EM Standing WavesNext Time: How Do We Generate Plane Waves?1P30-Class 30: OutlineHour 1:Traveling & Standing WavesHour 2:Electromagnetic (EM) Waves2P30-Last Time:Traveling Waves3P30-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)4P30-Traveling Sine Wave()0siny y kx kvt=+00sin( ) sin( )yy kvt y tω=≡At x=0, just a function of time:Amplitude (y0)1Period ( )frequency ( )2angular frequency ( )Tfπω==5P30-Traveling Sine Wave0sin( )yy kx tω=−Wavelength: Frequency : 2Wave Number: Angular Frequency: 212Period: Speed of Propagation: Direction of Propagation: fkfTfvfkxλπλωππωωλ======+iiiiiii6P30-This Time:Standing Waves7P30-Standing WavesWhat happens if two waves headed in opposite directions are allowed to interfere?10sin( )EE kx tω=−20sin( )EEkxtω=+12 0Superposition: 2 sin( )cos( )EEE E kx tω=+=8P30-Standing Waves: Who Cares?Most commonly seen in resonating systems:Musical Instruments, Microwave Ovens02sin()cos()EE kx tω=9P30-Standing Waves: BridgeTacoma Narrows Bridge Oscillation:http://www.pbs.org/wgbh/nova/bridge/tacoma3.html10P30-Group Work: Standing WavesDo Problem 210sin( )EE kx tω=−20sin( )EEkxtω=+12 0Superposition: 2 sin( )cos( )EEE E kx tω=+=11P30-Last Time:Maxwell’s Equations12P30-Maxwell’s Equations0000(Gauss's Law)(Faraday's Law)0 (Magnetic Gauss's Law)(Ampere-Maxwell Law)( (Lorentz force Law)inSBCSEencCQddddtdddIdtqεµµε⋅=Φ⋅=−⋅=Φ⋅= +=+×∫∫∫∫∫∫EAEsBABsFEvB) 13P30-Which Leads To…EM Waves14P30-Electromagnetic Radiation: Plane Waveshttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html15P30-Traveling E & B Waves0ˆsin( )Ekxtω=−EEWavelength: Frequency : 2Wave Number: Angular Frequency: 212Period: Speed of Propagation: Direction of Propagation: fkfTfvfkxλπλωππωωλ======+iiiiiii16P30-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 ×EB17P30-PRS Questions:Direction of Propagation18P30-How Do Maxwell’s Equations Lead to EM Waves?Derive Wave Equation19P30-Wave Equation00Cddddtµε⋅= ⋅∫∫Bs EAStart with Ampere-Maxwell Eq:20P30-Wave Equation(,) ( ,)zzCd B xtl B x dxtl⋅= − +∫BsApply it to red rectangle:00 00yEddldxdt tµε µε∂⎛⎞⋅=⎜⎟∂⎝⎠∫EA00Cddddtµε⋅= ⋅∫∫Bs EAStart with Ampere-Maxwell Eq:00(,)(,)yzzEBxdxt Bxtdx tµε∂+−−=∂00yzEBxtµε∂∂−=∂∂So in the limit that dx is very small:21P30-Wave EquationCddddt⋅=− ⋅∫∫Es BANow go to Faraday’s Law22P30-Wave EquationCddddt⋅=− ⋅∫∫Es BA(,)(,)yyCd E x dxtl E xtl⋅= + −∫EszBddldxdt t∂−⋅=−∂∫BAFaraday’s Law:Apply it to red rectangle:(,)(,)yyzExdxt ExtBdx t+−∂=−∂yzEBxt∂∂=−∂∂So in the limit that dx is very small:23P30-1D Wave Equation for E00 yyzzEEBBxt x tµε∂∂∂∂=− − =∂∂ ∂ ∂Take x-derivative of 1st and use the 2nd equation220022yy yzzEEEBBxx x x t tx tµε∂∂ ∂⎛⎞∂∂∂∂∂⎛⎞ ⎛⎞==−=− =⎜⎟⎜⎟ ⎜⎟∂∂ ∂ ∂ ∂ ∂∂ ∂⎝⎠ ⎝⎠⎝⎠220022yyEExtµε∂∂=∂∂24P30-1D Wave Equation for E220022yyEExtµε∂∂=∂∂()yEfxvt=−This is an equation for a wave. Let:()()22222''''yyEfxvtxEvfxvtt∂=−∂∂=−∂2001vµε=25P30-1D Wave Equation for B00 yyzzEEBBtxx tµε∂∂∂∂=− =−∂∂∂ ∂Take x-derivative of 1st and use the 2nd equation2222001yyzz zEEBBBtt t t x xt xµε∂∂⎛⎞ ⎛⎞∂∂ ∂∂∂∂⎛⎞==−=− =⎜⎟ ⎜⎟⎜⎟∂∂ ∂ ∂ ∂ ∂ ∂ ∂⎝⎠⎝⎠ ⎝⎠220022zzBBxtµε∂∂=∂∂26P30-Electromagnetic RadiationBoth E & B travel like waves:222200 002222yyzzEEBBxtxtµε µε∂∂∂∂==∂∂∂∂00yyzzEEBBtx x tµε∂∂∂∂=− =−∂∂ ∂ ∂But there are strict relations between them:Here, Eyand Bzare “the same,” traveling along x axis27P30-Amplitudes of E & B()()00Let ;yzEEfxvt B Bfxvt=−=−yzEBtx∂∂=−∂∂()()00''vB f x vt E f x vt⇒− − =− −00vB E⇒=Eyand Bzare “the same,” just different amplitudes28P30-Group Problem: EM Standing WavesConsider EM Wave approaching a perfect conductor:If the conductor fills the XY plane at Z=0 then the wave will reflect and add to the incident wave1. What must the total E field (Einc+Eref) at Z=0 be?2. What is Ereflectedfor this to be the case?3. What are the accompanying B fields? (Binc& Bref)4. What are Etotaland Btotal? What is B(Z=0)?5. What current must exist at Z=0 to reflect thewave? Give magnitude and direction. incident 0ˆcos( )xE kz tω=−E()()()()()Recall: cos cos cos sin sinAB A B A B+= −29P30-Next Time: How Do We Generate Plane Waves?
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