Apparent Poynting vector contradiction

In summary, the conversation discusses the calculation of the Poynting vector for an electromagnetic wave and the importance of properly computing the cross product of complex-harmonic fields in order to get an accurate result. It also mentions the time-averaged Poynting vector and the use of complex conjugation in its calculation. The overall focus is on understanding the correct procedure for calculating the Poynting vector in order to avoid contradictions and obtain accurate results.
  • #1
frustrationboltzmann
5
2
Hello all,

Im trying to do a simulation of a poynting vector of an electromagnetic wave and I assume the following: At t=0 the E-field vector is (0,0,e^(-ikx)) and the H-field vector (0,e^(-ikx),0), hence orthogonal to it in vaccum, which is assumed, also the amplitudes are simplified both to 1 since only the direction is of interest.

When I calculate the cross product I get now: (-e^(-i2kx),0,0) and when I take the real part I get (-cos(2kx),0,0) which means the poynting vector oscillates in x-direction but this is wrong because it shouldn't oscillate.

My procedure was summarized: take the COMPLEX E-and H-vector, calculate the cross product and after get the real part of it...but this is obviously wrong.

if I take the real parts immediately of the 2 amplitudes and THEN take the cross product of the real amplitudes it works...I get a vector (cos(kx)^2,0,0) which also oscillates but not in 2 different directions but only in intensity which is correct.

between those 2 procedures is a contradiction but at the moment I am afraid I can't see why.
I appreciate any help.
Thank you in advance very much.
 
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  • #2
You have to be careful in computing products of fields when working with complex-harmonic fields. First of all, there is nothing weird about Poynting vector oscillating, it is a bit like if you multiply two sine-functions: ##\sin\left(\omega t\right)^2=\frac{1-\cos\left(2\omega t\right)}{2}## - you get a constant + oscillating part.

Now what you were probably after is time-averaged Poynting vector. Given complex harmonic electric field (##\mathbf{E}##) and magntic field (##\mathbf{H}##), the time-averaged Poynting vector is:

##\langle \mathbf{S} \rangle = \frac{1}{2} \Re \left(\mathbf{E} \times \mathbf{H}^{\dagger}\right)##

i.e. note the complex-conjugation
 
  • #3
This is a AC problem complex Poynting theorem should be applied, in which, The Poynting vector is defined as
$$\boldsymbol{S}=\frac{1}{2}\boldsymbol{E}\times\boldsymbol{H}^{*}$$
The power is,
$$ P=\frac{1}{2} \iint_{\Gamma}\boldsymbol{E}\times\boldsymbol{H}^{*}\cdot\hat{n}d\Gamma $$

##*## is complex conjugate that is important, that cannot be taken away.
 

1. What is the Apparent Poynting vector contradiction?

The Apparent Poynting vector contradiction is a phenomenon observed in electromagnetic theory where there appears to be a discrepancy between the direction of the Poynting vector, which represents the direction of energy flow, and the actual direction of the electromagnetic field.

2. How does this contradiction occur?

The contradiction occurs due to the complex nature of electromagnetic waves. In some cases, the direction of the Poynting vector may not align with the direction of the electric and magnetic fields, leading to the apparent contradiction.

3. Can you provide an example of the Apparent Poynting vector contradiction?

One example is the case of a plane electromagnetic wave reflecting off a perfectly conducting surface. In this scenario, the Poynting vector is perpendicular to both the electric and magnetic fields, creating the contradiction.

4. How is this contradiction resolved?

The Apparent Poynting vector contradiction is resolved by taking into account the boundary conditions at the interface of different materials. By considering the refractive index and polarization of the wave, the apparent contradiction can be explained.

5. Why is the Apparent Poynting vector contradiction important?

The Apparent Poynting vector contradiction is important because it highlights the complexity of electromagnetic waves and the need for a deeper understanding of their behavior. It also has practical applications in fields such as optics and telecommunications, where accurate predictions of the direction of energy flow are crucial.

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