Poisson and Helmholtz equations

[Sailaway]

Hello guru's,

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,

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

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

$$\nabla \sigma \nabla (V - V^{p}) = 0$$

with a voltage source $$V^{p}$$, or

$$\nabla \sigma (E - E^{p}) = 0$$

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

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

(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 $$\sigma$$ 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!

 

[Sailaway]

Sorry the Poisson's equation forms

$$\nabla \circ (\sigma \nabla (V - V^{p})) = 0$$

$$\nabla \circ (\sigma (E - E^{p})) = 0$$