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.

 

[PJC01]

Hello,

It seems that there may be a misunderstanding in your understanding of the Poisson equation. The Poisson equation relates the electric potential, \psi, to the charge density, \rho, through the permittivity of the medium, \epsilon_m, and the permittivity of free space, \epsilon_o. This equation is derived from the fundamental laws of electromagnetism, specifically Gauss's law and the relationship between electric field and potential, which you have correctly stated as E=-\nabla\psi. However, the Poisson equation does not relate the potential to the electric field directly, as you have done in your calculation. Instead, it relates the potential to the charge density and the properties of the medium. Therefore, it is not correct to compare the Laplacian of the potential, which is a function of the electric field, to the Poisson equation, which is a function of the charge density.

To further clarify, the Poisson equation states that the Laplacian of the electric potential is equal to the negative of the charge density divided by the permittivity of the medium and the permittivity of free space. This means that the Laplacian of the potential is not zero, as you have calculated, unless the charge density is zero. In the case of a uniform field, the charge density is indeed zero, which is why your calculation yields a Laplacian of zero. However, in a more general case with a non-zero charge density, the Laplacian of the potential would not be zero and would match the Poisson equation.

In summary, there is no mismatch between the uniform field and the Poisson equation. The two are related through the charge density and the properties of the medium, not directly through the electric field. I hope this helps clarify any confusion and allows you to proceed with your research. Good luck!