Phase relation between current and electromagnetic field generated

[amir11]

Dear ForumersI am having a bit problem understanding the phase relation between current source and the generated eletromagnetic field components.
Assume a very small current element( a very small current running in direction x)(essentially an electric dipole) in a non-homogenous loss periodic medium. the only knowledge about the medium is the mu and epsilon profile plus eigenvalues of the medium. The structure is like a waveguide so most of the radiation is expected to couple one of the eigenvalues. what is the phase relation between the generated electromagnetic field(dominant eigenvalue) and the driving current?(assuming phasor fields). It must be somehow related to the tangental and normal electric and magnetic fields of the dominant eigenvalue.

could it be independent of the meduim, simply 1 or -1?

Best

 

[Born2bwire]

From Maxwell's Equations, for a time-harmonic field the electric field it appears that it may lead the current by 90 degrees since the time derivative of the electric field is equal to the curl of the magnetic field and the source current. But this is only at the source point, due to the retardation of the fields, there is another phase shift that arises as the fields propagate. Let us take the z component of the electric field from a z directed point source current. The field for a unity current source is:

$$E_z = \frac{i\omega\mu}{4\pi k^2} \left[ ik - \frac{1+k^2z^2}{r^2} - \frac{3ikz^2}{r^2} + \frac{3z^2}{r^3} \right] \frac{e^{ikr}}{r^2}$$

So we find that different parts of the field are 90 degrees and 180 degrees out of phase of the current even before we take into account the spatial phase shift. As you go away from the source though, only the first term remains and you have a field that is 180 degrees out of phase plus a spatial phase shift.

So in the situation where you have a waveguide, then you have to contend with the superposition of the reflections which would make it even more difficult. But my guess is you will have a hard time determining a rule for this.