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Em wave and complex number

  1. Jul 16, 2011 #1
    why electromagnetic waves are represented by complex numbers?
     
  2. jcsd
  3. Jul 16, 2011 #2
    My answer is a bit general but I think its pretty relevant:

    Waves and harmonic oscillators are represented by sinusoidal functions. Using Euler's theorem you can rewrite them as the (real part) of an imaginary exponential, where the exponent is i*(arg), where the argument is the same one you would use for an oscillator(wt +phase) or a wave (kx - or +wt + phase).

    Its a bit more convenient to work with imaginary exponentials since they're more compact, taking their time derivatives to get velocities for example.

    Something worth trying to illustrate that example: show that the total energy (T+V) of a harmonic oscillator is proportional to the square of the amplitude. You can do this either way, but I think its more compact if you use y(t) = Re{Ae^(iwt)} instead of Acoswt as your starting point.
     
  4. Jul 17, 2011 #3
    EM wave usually are of sinusoidal nature. It is easier to represent harmonic wave ( sinusoidal) in cosine wave:

    [tex]\vec E =E_0 cos\;(\omega t -\vec k\cdot \vec R)\;=\; \Re e [E_0 e^{j\omega t}e^{-j\vec k \cdot \vec R}][/tex]


    And then use phasor form where [itex] \tilde E = E_0 e^{-j\vec k \cdot \vec R} \;\hbox { and }\;\vec E = \Re e [\tilde E \;e^{j\omega t}][/itex]


    The solution of homogeneous harmonic wave equation is something like:

    [tex] \nabla ^2 E +\delta^2 \vec E = 0 \;\hbox { is } E^+ e^{-\delta \vec k \cdot \vec R} +E^- e^{\delta \vec k \cdot \vec R} \;\hbox { where } \delta = \alpha + j\beta[/tex]


    It is not as common in Physics than in RF and microwave Electronics. In RF, we deal with transmission lines where we can assume the direction of propagation in z direction which really simplify the calculation tremendously. We avoid all the differential equations, PDE, integration and differentiation. In fact I learn in reverse order. I have been using phasor calculation to design filters, matching networks for years before I really start learning EM!!!!
     
    Last edited: Jul 17, 2011
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