Hello guru's,(adsbygoogle = window.adsbygoogle || []).push({});

I've been trying to figure out a way to incorporate an electric field source in the Helmholtz equation, and have been accumulating lots of question marks in my head. So in case of no static charge,

[tex]\nabla^{2} E - \mu (\epsilon\frac{d^{2}}{dt^{2}} + \sigma\frac{d}{dt}) E = 0

[/tex]

In the quasistatic case, I know many people use Poisson's,

[tex]\nabla \sigma \nabla (V - V^{p}) = 0[/tex]

with a voltage source [tex]V^{p}[/tex], or

[tex]\nabla \sigma (E - E^{p}) = 0[/tex]

Unfortunately my electric field is in hundreds of MHz range. So (1) can I use the following equation with source [tex]E^{p}[/tex], since the second term seems to be the "admittivity times electric field" term?

[tex]\nabla^{2} E - \mu (\epsilon\frac{d^{2}}{dt{2}} + \sigma\frac{d}{dt}) (E - E^{p}) = 0

[/tex]

(2) I wonder why in the low frequency limit the Helmholtz equation doesn't reduce to the Poisson's in the inhomogeneous media, since the [tex]\sigma[/tex] is in the spatial gradient in the Poisson equation. In my case the media is inhomogeneous. I did see some papers on scattering that used the Helmholtz equation in the inhomogeneous media, so that makes me wonder all the more.

Any comments would be greatly appreciated!

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# Poisson and Helmholtz equations

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