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.