- #1
sandy.bridge
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Homework Statement
Hey guys. I have an electromagnetic wave traveling 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.
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?