1ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLecture 13Electromagnetic Waves in Free SpaceIn this lecture you will learn:• Electromagnetic wave equation in free space• Uniform plane wave solutions of the wave equation• Energy and power of electromagnetic wavesECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityBasic Wave MotionvλxConsider a wave moving in the +x-direction:The wave travels a distance equal to one wavelength in one time periodBasic relation for wave motion:vf=λv = velocity of wave propagationλ = wavelength of the wavef = frequency of the waveT = period = 1/f1-D wave equation:()()22222,1,ttxavxtxa∂∂=∂∂()txa ,λλfTv ====timedistance velocity2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityElectromagnetic Wave Motion - I0..=∇=∇HEoorrµρεtEJHtHEoo∂∂+=×∇∂∂−=×∇rrrrrεµ Time varying electric and magnetic fields are coupled - this coupling is responsible for the propagation of electromagnetic wavesElectromagnetic Wave Equation:Assume free space: 0==⇒ Jrρ()22 tEtHtHEoooo∂∂−=∂×∇∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂×−∇=×∇×∇rrrrεµµµ()22 tEEoo∂∂−=×∇×∇⇒rrεµ()2221 tEcE∂∂−=×∇×∇⇒rrm/s10318×≈=oocµεECE 303 – Fall 2007 – Farhan Rana – Cornell University()2221 tEcE∂∂−=×∇×∇⇒rrEquation for a wave traveling at speed c in free spaceElectromagnetic Wave Motion - II()22221.tEcEE∂∂−=∇−∇∇⇒rrr022221tEcE∂∂=∇⇒rr0. ==∇ρεEor()()()222222222222222222222222222131211tEczEyExEtEczEyExEtEczEyExEzzzzyyyyxxxx∂∂=∂∂+∂∂+∂∂∂∂=∂∂+∂∂+∂∂∂∂=∂∂+∂∂+∂∂Wave equation is essentially three equations stacked together – one for each component of the E-fieldWave must also satisfy:00. =∂∂+∂∂+∂∂⇒=∇zEyExEEzyxorεUse the vector Identity:()()FFFrrr2. ∇−∇∇=×∇×∇3ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityElectromagnetic Wave Motion - IIIThe H-field also satisfies a similar wave equationtEJHtHEoo∂∂+=×∇∂∂−=×∇rrrrrεµ Start from Maxwell’s equations:Assume free space: 0==⇒ Jrρ()22 tHtEtEHoooo∂∂−=∂×∇∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂×∇=×∇×∇rrrrεµεε()22 tHHoo∂∂−=×∇×∇⇒rrεµ()2221 tHcH∂∂−=×∇×∇⇒rr()22221.tHcHH∂∂−=∇−∇∇⇒rrr00. =∇ Horµ22221tHcH∂∂=∇⇒rrECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityGeneral Solutions of Electromagnetic Wave EquationAssume only x-component of the E-field:xExEˆ=r00. =∂∂⇒=∇xEExorεThe x-component cannot have x-dependenceSo assume:()tzExEx,ˆ=rAnd plug it into the wave equation:22221tEcE∂∂=∇rrTo obtain:()()22222,1,ttzEcztzExx∂∂=∂∂Solution is:()()tczgtzEx±=,Any function g whose dependence on co-ordinate z and time t is in the form of (z±ct ) will satisfy the above equation Example:()()tczEtzEox−−=αexp,zc4ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySinusoidal Solutions of Electromagnetic Wave Equation - I()()22222,1,ttzEcztzExx∂∂=∂∂()ctzExEx−=⇒ˆrThe most commonly used solutions are sinusoids, for example:() ()⎟⎠⎞⎜⎝⎛−=−= tczExctzExEoxλπ2cosˆˆrThis solution represents a wave that:i) Has electric field pointing in x-directionii) Has wavelength λiii) Has frequency f = c/λiv) Is moving in the +z-directionλzzxcIn space the electric field looks as shown here (remember that the density of field lines correspond to the strength of the E-field)coEECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySinusoidal Solutions of Electromagnetic Wave Equation - II⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛−= ztcExcztcExEooλπλπλπ22cosˆ2cosˆr() ()⎟⎠⎞⎜⎝⎛−=−= tczExctzExEoxλπ2cosˆˆrThe sinusoidal solution,can also be written as:Define:λπ2=kandfcπλπω22==To get:()zktExEo−=ωcosˆrω= angular frequency (units: radians/sec)k = wave-vector (units: 1/m) ck=ωNote that:A dispersion relationvf=λ⇒5ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySinusoidal Solutions of Electromagnetic Wave Equation - IIIWhat about the magnetic field?Recall the E and H-fields are coupledtEHtHEoo∂∂=×∇∂∂−=×∇rrrrεµ So an E-field must be accompanied by an H-field which can be calculated from the equation:Plug in the following solution for the E-field:()zktExEo−=ωcosˆr()()()zktEyHzktEkyHzktEkyEtHooooooo−=⇒−=⇒−−=×∇−=∂∂ωηωωµωµµcosˆcosˆsinˆ 1rrrr⎟⎟⎠⎞⎜⎜⎝⎛∂∂=×∇∂∂−=×∇tEHtHEoorrrrεµ from even or ooooockηεµµωµ===Ω≈=377space free of impedanceoηECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySinusoidal Solutions of Electromagnetic Wave Equation - IVzxy()zktExEo−=ωcosˆr()zktEyHoo−=ωηcosˆrEH• These solutions of the wave equation are called uniform plane waves• The E-field (and the H-field as well) is constant over any infinite plane that is parallel to the x-y plane – in more technical terms, the surfaces of constant phase are infinite planes• The pattern shown above moves with a velocity equal tocck=ωCartoon depiction:EHxˆyˆzˆ6ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySinusoidal Solutions of Electromagnetic Wave Equation - V()zktExEo−=ωcosˆr()zktEyHoo−=ωηcosˆrConsider the plane wave:If a person takes a snapshot of the wave in space at any time, say at t = 0, he will see E-field look like:λz()⎟⎠⎞⎜⎝⎛==zExzkExEooλπ2cosˆcosˆrIf a person sits at one location, say z = 0, he will see an oscillating E-field in time that looks like:Tt()⎟⎠⎞⎜⎝⎛==tTExtExEooπω2cosˆcosˆrECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityPlane Waves in 3D - ISo far we have found one solution:EHWhat about plane waves with E-field pointing in direction and traveling in some arbitrary direction in 3D ?nˆEHThe answer is:()()rktEntrEorrrr.cosˆ, −=ωzzyyxxrkkkkkkzkykxkkzyxzyxˆˆˆ.ˆˆˆ2222++=++==⇒++=rrrrLets see if this solution satisfies the wave equation:()[]()[]()ckckckkrktEncrktEnkktEcEoo=⇒==⇒−−=−−⇒∂∂=∇ωωωωω2222222222..cosˆ.cosˆ.1rrrrrrrrrrThe solution can only be correct if: ω= k cxˆyˆzˆnˆ7ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityPlane Waves in 3D - IIThe solution must also satisfy:The solution for a plane wave in 3D is:()()rktEntrEorrrr.cosˆ,