Uniform Field & Poisson equation Mismatch?

[Apteronotus]

Hi,

I'm getting some confusing results and can't figure out what is wrong
Suppose we have a uniform field

E=[0,0,E_z] in a dielectric media.

By E=-\nabla\psi we can deduce that \psi(x,y,z)=-z E_z

But, taking the Laplacian
\nabla^2\psi=\frac{\partial^2 (-zE_z)}{\partial z^2}=0
does not match the results of the Poisson equation
\nabla^2\psi=-\frac{\rho}{\epsilon_m \epsilon_o}

what am I missing?

 

[gabbagabbahey]

In the region where the field is uniform, the charge density is zero by Gauss' Law:

\mathbf{ \nabla } \cdot \mathbf{E} = \frac{ \partial }{ \partial z }E_z = \frac{ \rho }{ \epsilon_0 }

 

[Apteronotus]

Ahh I see. So \rho=0 is an implicit condition for us to have the uniform field in the first place?

 

[vanhees71]

Sure, you have one of Maxwell's (microscopic) equations, saying that
\vec{\nabla} \cdot \vec{E}=\rho
(Heaviside-Lorentz units). This means that a homogeneous electric field necessarily leads to 0 charge density.

Of course, in nature there is no such thing as a global homogeneous field. You get such a field only in a quite unphysical situation. To that end consider the electrostatic potential of a point charge Q at rest at the position \vec{a}. That's of course the corresponding Coulomb potential,
\Phi(\vec{x})=\frac{Q}{4 \pi |\vec{x}-\vec{a}|}.
Now let |\vec{a}| \gg \vec{x}. Then you can expand the potential around \vec{x}=0. You find
\Phi(\vec{x})=\frac{Q}{4 \pi |\vec{a}|}+\frac{Q \vec{x} \cdot \vec{a}}{4 \pi |\vec{a}|^3}.
Now you obtain the potential for a homogeneous field, by letting |\vec{a}| \rightarrow \infty in such a way that Q \vec{a}/|\vec{a}|^3=-\vec{E}=\text{const}. The constant first term you can subtract beforehand. Then you get
\Phi(\vec{x}) \rightarrow -\vec{x} \cdot \vec{E}.
As you see you have to use an infinite charge at infinite distance to make a homogeneous electric field everywhere in space. That's a rather unphysical situation.

In practice you get a quite good approximation of a homogenous electric field between two large charged plates at small distance, in the region in the middle between the plates.