The complex Poynting vector

[Noname]

Hi guys,
Consider a circular capacitor with a disk of radius a and plate separation d, as shown in the figure below. Assuming the capacitor is filled with a dielectric constant epsilon and the capacitor is fed by a time harmonic current I₀
(a) Find the magnetic field distribution inside the capacitor assuming that the electric field is constant.
(b) Compute the complex Poynting vector and prove that the capacitor does not radiate an electromagnetic field. (Hint: S=(1/2)ExH ; only Re[S ] radiates an electromagnetic field.)
(c) Compute the total stored energy density W and show that far away from the center (kr<< 1): grad(S)-iwW = 0
(d) Find the surface current density as a function of radial distance on the top plate.

I don't really know how to do the (b). I have S=(1/2)ExH. I think I need to write E and H in complex, I have E=Eo*exp(iwt) and H=Ho*exp(iwt). But I can't prove S is only an imaginary part.

 

[Cryo]

Normally, for time-harmonic fields the time-average of the real Poynting vector is:

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

So, I guess, the complex Poynting vector you want is $\mathbf{E}^{\dagger} \times \mathbf{H}$, i.e. you are missing the complex conjugation (which will remove $\exp\left(i \omega t\right)$)

 

[Charles Link]

Feynman treats this problem, (without the dielectric material), in his lectures. See Fig. 27-3 and thereabouts. http://www.feynmanlectures.caltech.edu/II_27.html $\\$ He uses a slightly different type of units, but you should find it good reading.