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Homework Help: Polarization of Electromagnetic Wave and Faraday's Law of Induction

  1. Nov 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Hey guys. I have an electromagnetic wave travelling in the z direction and polarized in the x direction. The frequency is 1 MHz and average power density is 1 W/m^2. An antenna in the shape of a circular wire is in the xy-plane centred at the origin. I would like to use Faraday's Law of Induction to estimate the amplitude of the emf induced on antenna from the wave passing through it. Assume the radius is 1cm, and that the wavelength of the electromagnetic wave is much larger than this.

    3. The attempt at a solution
    Since it is polarized in the x-direction I can assume [itex]\vec{E}=E_oe^{j(ωt-kz)}\vec{x}[/itex].

    Therefore, [itex]\vec{H}=\frac{\vec{∇}×\vec{E}}{-jμ_oω}=\frac{E_o}{120π}e^{j(ωt-kz)}\vec{y}[/itex]

    Since [itex]<\vec{S}>=0.5Re[\vec{E}×\vec{H^*}]=1W/m^2\vec{z}[/itex], I get [itex]E_o=\sqrt{2}[/itex]

    From here I get that the induced emf is [itex]-μ_o \int_S ∂\vec{H}/∂t . d\vec{S}[/itex]

    which I crunched to be [itex]\frac{-μ_o \sqrt{2}jω(0.0001π)e^{j(ωt-kz)}}{120π}[/itex]

    I have never encountered this when the magnetic field has complex components. Would I merely take the real component (hence the sine term), or have I messed up somewhere?
  2. jcsd
  3. Nov 6, 2013 #2
    I found some errors in this. Eo should have came out to sqrt(240pi) and the magnetic wave should have been sqrt(240pi)/(120pi)e^(j(wt-kz)) in the y direction.

    My question is this:
    since the magnetic field component of the electromagnetic wave is in the y direction and the ring of wire is situated in the xy plane, would the induced emf not be zero?
  4. Nov 6, 2013 #3

    rude man

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    Homework Helper
    Gold Member

    Yes. The H field needs to point in the direction of the loop normal or at least have a component along the normal. Since H has only a y component, the loop normal should also point along the y direction. So it can't be in the x-y plane since then the loop normal is along z.

    Otherwise your approach is OK. Use Poynting to compute H. There should be no complex numbers involved. Then use faraday' s induction law.
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