Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook