Electricity and Magnetism-Travelling, Standing and E.M Waves.

1 P30-Class 30: OutlineHour 1:Traveling & Standing WavesHour 2:Electromagnetic (EM) Waves2 P30-Last Time:Traveling Waves3 P30-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)4 P30-Traveling Sine Wave()0sinyykxkvt=+00sin()sin()yykvtytω=≡At x=0, just a function of time:Amplitude (y0)1Period ()frequency ()2angular frequency ()Tfπω==5 P30-Traveling Sine Wave0sin()yykxtω=−Wavelength: Frequency : 2Wave Number: Angular Frequency: 212Period: Speed of Propagation: Direction of Propagation: fkfTfvfkxλπλωππωωλ======+iiiiiii6 P30-This Time:Standing Waves7 P30-Standing WavesWhat happens if two waves headed in opposite directions are allowed to interfere?10sin()EEkxtω=−20sin()EEkxtω=+120Superposition: 2sin()cos()EEEEkxtω=+=8 P30-Standing Waves: Who Cares?Most commonly seen in resonating systems:Musical Instruments, Microwave Ovens02sin()cos()EEkxtω=9 P30-Standing Waves: BridgeTacoma Narrows Bridge Oscillation:http://www.pbs.org/wgbh/nova/bridge/tacoma3.html10 P30-Group Work: Standing WavesDo Problem 210sin()EEkxtω=−20sin()EEkxtω=+120Superposition: 2sin()cos()EEEEkxtω=+=11 P30-Last Time:Maxwell’s Equations12 P30-Maxwell’s Equations0000(Gauss's Law)(Faraday's Law)0(Magnetic Gauss's Law)(Ampere-Maxwell Law)((Lorentz force Law)inSBCSEencCQddddtdddIdtqεµµε⋅=Φ⋅=−⋅=Φ⋅=+=+×∫∫∫∫∫∫EAEsBABsFEvB)􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁇􀁷􀁶􀁷􀁶13 P30-Which Leads To…EM Waves14 P30-Electromagnetic Radiation: Plane Waveshttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/07-EBlight/07-EB_Light_320.html15 P30-Traveling E & B Waves0ˆsin()Ekxtω=−EE􀁇Wavelength: Frequency : 2Wave Number: Angular Frequency: 212Period: Speed of Propagation: Direction of Propagation: fkfTfvfkxλπλωππωωλ======+iiiiiii16 P30-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􀁇􀁇17 P30-PRS Questions:Direction of Propagation18 P30-How Do Maxwell’s Equations Lead to EM Waves?Derive Wave Equation19 P30-Wave Equation00Cddddtµε⋅=⋅∫∫BsEA􀁇􀁇􀁇􀁇􀁶Start with Ampere-Maxwell Eq:20 P30-Wave Equation(,)(,)zzCdBxtlBxdxtl⋅=−+∫Bs􀁇􀁇􀁶Apply it to red rectangle:0000yEddldxdttµεµε∂⎛⎞⋅=⎜⎟∂⎝⎠∫EA􀁇􀁇00Cddddtµε⋅=⋅∫∫BsEA􀁇􀁇􀁇􀁇􀁶Start with Ampere-Maxwell Eq:00(,)(,)yzzEBxdxtBxtdxtµε∂+−−=∂00yzEBxtµε∂∂−=∂∂So in the limit that dxis very small:21 P30-Wave EquationCddddt⋅=−⋅∫∫EsBA􀁇􀁇􀁇􀁇􀁶Now go to Faraday’s Law22 P30-Wave EquationCddddt⋅=−⋅∫∫EsBA􀁇􀁇􀁇􀁇􀁶(,)(,)yyCdExdxtlExtl⋅=+−∫Es􀁇􀁇􀁶zBddldxdtt∂−⋅=−∂∫BA􀁇􀁇Faraday’s Law:Apply it to red rectangle:(,)(,)yyzExdxtExtBdxt+−∂=−∂yzEBxt∂∂=−∂∂So in the limit that dxis very small:23 P30-1D Wave Equation for E00 yyzzEEBBxtxtµε∂∂∂∂=−−=∂∂∂∂Take x-derivative of 1st and use the 2nd equation220022yyyzzEEEBBxxxxttxtµε∂∂∂⎛⎞∂∂∂∂∂⎛⎞⎛⎞==−=−=⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠220022yyEExtµε∂∂=∂∂24 P30-1D Wave Equation for E220022yyEExtµε∂∂=∂∂()yEfxvt=−This is an equation for a wave. Let:()()22222''''yyEfxvtxEvfxvtt∂=−∂∂=−∂2001vµε=25 P30-1D Wave Equation for B00 yyzzEEBBtxxtµε∂∂∂∂=−=−∂∂∂∂Take x-derivative of 1st and use the 2nd equation2222001yyzzzEEBBBttttxxtxµε∂∂⎛⎞⎛⎞∂∂∂∂∂∂⎛⎞==−=−=⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠220022zzBBxtµε∂∂=∂∂26 P30-Electromagnetic RadiationBoth E & B travel like waves:222200002222yyzzEEBBxtxtµεµε∂∂∂∂==∂∂∂∂00yyzzEEBBtxxtµε∂∂∂∂=−=−∂∂∂∂But there are strict relations between them:Here, Eyand Bzare “the same,” traveling along x axis27 P30-Amplitudes of E & B()()00Let;yzEEfxvtBBfxvt=−=−yzEBtx∂∂=−∂∂()()00''vBfxvtEfxvt⇒−−=−−00vBE⇒=Eyand Bzare “the same,” just different amplitudes28 P30-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. incident0ˆcos()xEkztω=−E􀁇()()()()()Recall: coscoscossinsinABABAB+=−29 P30-Next Time: How Do We Generate Plane Waves? http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/light/09-planewaveapp/09-planewaveapp320.html