Given a plane wave with $\overrightarrow E$ field parallel to $z$ axis and a square metal plate in $xy$ plane. Suppose we know the power of the incident wave and the size of the plate, how can we calculate the total amount of power absorbed by the plate, knowing its electrical parameters? The thikness of the plate is about 1 mm and the frequency of the incident wave is about 14GHz. Thanks
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$\begingroup$ Possibly you have made a mistake here. A wave with the E-field polarised in the z-direction is travelling parallel to the surface of the plate and is not "incident" upon it. $\endgroup$– ProfRobCommented Oct 17, 2017 at 9:35
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$\begingroup$ @RobJeffries: Sorry...$\overrightarrow E$ in $y$ direction $\endgroup$– Riccardo.AlestraCommented Oct 17, 2017 at 12:23
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2 Answers
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If $\vec{E}$ is parallel to the z-axis, then the wave is propagating in the $xy$-plane, and thus no power is incident on the plate's large face (that is, the one parallel to the $xy$-plane). The wave would be incident on the side of the plate, so you would need to know that area.
For an electromagnetic wave, we determine from the magnitude of the electric field vector the power per unit area imparted by the wave, not the power itself. This is given by (source):
$$ P/A = \epsilon_0 |\vec{E}|^2 c $$
You can then just multiply by the area of the side of the plate that the wave is incident on.
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$\begingroup$ This is the power per unit area of the e.m. wave, not the power adsorbed by the plate $\endgroup$ Commented Oct 17, 2017 at 12:41
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$\begingroup$ Yes. As I said just below, you can then multiply by the area of the face of the plate the wave is incident on to find the power absorbed by the plate. $\endgroup$ Commented Oct 17, 2017 at 15:22
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So I gather that you have a plane wave at normal incidence to a metal plate, and you're looking for the power absorption coefficient, $A$. The thing to do is to compute the power reflection coefficient, $R$, and then use $A=1-R$ since the metal is thick enough that you'll get no transmission (unless it is a terrible metal). $R$ can be calculated in the usual way with the Fresnel equations, using the complex refractive index of the metal: $$ R=\left|\frac{n_1-n_2}{n_1+n_2}\right|^2 $$ If instead you have the complex conductivity or permittivity, you may want to convert to refractive index. Also, if this is a real situation, you may need to make assumptions about the smoothness of the metal surface (if it is rough, there could be scattering which isn't captured in $R$). But due to the long wavelength of your source, you probably don't have to worry much about scattering.
Finally, the total absorbed power is the total incident power times $A$.
