Apr 29 2005 web.mit.edu/8.02x/wwwWaves Roger Ramjet/www.surfcam.comApr 29 2005 web.mit.edu/8.02x/wwwReminder on wavesPosition xD(X)At a moment in time:Wavelength !AmplitudeTime tD(t)At a point in space:Period T = 1/fAmplitudeApr 29 2005 web.mit.edu/8.02x/wwwWave Equation• Wave equation:• Speed of propagation: v = ! f•How can we derive a wave equation fromMaxwells equations?Couples variation in time and spaceApr 29 2005 web.mit.edu/8.02x/wwwWave properties• What do we want to know about waves:– Speed of propagation?– Transverse or longitudinal oscillation?– What is oscillating?– What are typical frequencies/wavelengths?Apr 29 2005 web.mit.edu/8.02x/wwwBack to Maxwell’s equation• Wave equation is differential equation• M.E. (so far) describe integrals of fieldsTransform into differential equ’sApr 29 2005 web.mit.edu/8.02x/wwwGauss TheoremDivergence Flux/Unit VolumeApr 29 2005 web.mit.edu/8.02x/wwwStokes TheoremCurl Loop Integral/Unit Area of LoopApr 29 2005 web.mit.edu/8.02x/wwwDifferential Form of M.E. Gauss, StokesApr 29 2005 web.mit.edu/8.02x/wwwDifferential Form of M.E.Loop Integral/Unit Area Charge Density Flux/Unit Volume Current DensityApr 29 2005 web.mit.edu/8.02x/wwwDifferential Form of M.E.• Q: Do we need " and j to understand E.M.waves?Apr 29 2005 web.mit.edu/8.02x/wwwDifferential Form of M.E.• Q: Do we need " and j to understand E.M.waves?• A: No! Light travels from sun to earth, i.e. invacuum (no charge, no current)!• There’s no ‘medium’ involved!?– unlike waves on water or sound wavesApr 29 2005 web.mit.edu/8.02x/wwwMaxwell’s Equations in Vacuum• Look at Maxwell’s Equations without charges,currents Now completely symmetric!Apr 29 2005 web.mit.edu/8.02x/wwwMaxwell’s Equations in Vacuum Solve for a simple geometry z x y Allow variations only in z-direction:I.II.III.VI.Apr 29 2005 web.mit.edu/8.02x/wwwIllustration Roger Ramjet/www.surfcam.comXZ2-D wave:x,z,D(x,z,t)Apr 29 2005 web.mit.edu/8.02x/wwwMaxwell’s Equations in Vacuum Solve for a simple geometry z x y Allow variations only in z-direction:I.II.III.VI.Apr 29 2005 web.mit.edu/8.02x/wwwElectromagnetic Waves• Note: (Ex, By) and (Ey, Bx) independent:Ex, ByEy, BxApr 29 2005 web.mit.edu/8.02x/wwwElectromagnetic Waves• We found wave equations: same for Ex, Bx v = cE and B are oscillating!Apr 29 2005 web.mit.edu/8.02x/wwwElectromagnetic Waves• Note: (Ex, By) and (Ey, Bx) independent:Ex, ByEy, BxE BApr 29 2005 web.mit.edu/8.02x/wwwPlane waves• Example solution: Plane waves• We can express other functions as linearcombinations of sin,cos– ‘White’ light is combination of waves of differentfrequency– Demo...Apr 29 2005 web.mit.edu/8.02x/wwwPlane waves• Example solution: Plane wavesCheckApr 29 2005 web.mit.edu/8.02x/wwwPlane waves• Example solution: Plane wavesApr 29 2005 web.mit.edu/8.02x/wwwPlane waves• Example solution: Plane waves-Apr 29 2005 web.mit.edu/8.02x/wwwE.M. Wave Summary• E B and perpendicular to direction ofpropagation• Transverse waves• Speed of propagation v = c = ! f• |E|/|B| = c• E.M. waves travel without
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