Infinite scaling limit of an elliptic charged surface

[Vincf]

TL;DR

I consider the infinite scaling limit of an elliptic planar charge distribution and show that the horizontal component of the electric field does not tend to zero.

Hello,

This post is related to my previous discussion about the uniformly charged infinite plane. However, since the example and the approach are quite different, I thought it would be clearer to open a separate thread. I hope this is appropriate, as I am still new to this forum.

The goal is to construct an example closer to a ring decomposition, which several participants seem to prefer. The key point is that the observation point is located at a focus of the ellipse, and I consider a scaling centered at this point.

I am trying to better understand how the infinite plane result emerges as a limit of finite charge distributions.

I consider a planar surface charge distribution (uniform surface density $\sigma>0$) bounded by an ellipse whose polar equation is
$$
r=\frac{p}{1+e\cos\theta}
$$
where the origin is one focus $F$. ($p$ is the parameter and $e$ the excentricity)

I am interested in the horizontal component of the electric field at the focus, in the plane of the charge distribution. This horizontal component is continuous across the charged sheet, and no convergence issue arises.

The exact calculation is not difficult. However, symmetry considerations and dimensional analysis immediately show that the horizontal field must lie along the major axis, directed toward the other focus, and its magnitude must be of the form
$$
E=\left(\frac{\sigma}{\epsilon_0}\right)f(e)
$$
where $f(e)$ is a function which depends only on the eccentricity $e$. In particular, $f(0)=0$. The parameter $p$, which has dimensions of length, cannot influence the result.

Now consider a scaling centered at the focus. The scaling parameter is $p$. As the ellipse is scaled indefinitely, the charge distribution expands without bound but the focus $F$ where I compute the horizontal electric field does not move. Since the field does not depend on $p$, the horizontal component does not tend toward zero (Unless $e=0$)

Yet in the limit, the charge distribution covers the entire plane.

How can this be reconciled with the classical result that the electric field of an infinite plane has no horizontal component?

 

[Dale]

TL;DR: I consider the infinite scaling limit of an elliptic planar charge distribution and show that the horizontal component of the electric field does not tend to zero.

How can this be reconciled with the classical result that the electric field of an infinite plane has no horizontal component?

There is no need to reconcile them. They are both valid solutions to Maxwell’s equations. As I already told you in the other thread.

 

[Vincf]

We agree: there are infinitely many solutions for the horizontal component of the field. But then, why talk about "the" uniformly charged infinite plane if there are infinitely many?

And when I model a finite distribution with an infinite plane, which one should I choose?
If I consider a finite elliptic distribution with a parameter of several kilometers, and I am in the vicinity of the focus, a few centimeters away, the edges are very, very far away: I can use the usual model for the vertical component of the field. But I don't know the horizontal component. I cannot conclude that it is zero.

Finally, when using Gauss's theorem, should we then add : I restrict myself to an infinite plane that has symmetries such that the horizontal field is zero? (And not claim that this is always the case?)

And in this case, since it's only true in the vicinity of specific points, using Gauss's theorem to recover the classical result is considerably less straightforward. (Since the horizontal field is uniform, the flux of this field is zero. But proving that it's uniform isn't straightforward, is it?)

 

[Dale]

And when I model a finite distribution with an infinite plane, which one should I choose?

The one that best approximates your boundary conditions. If you are modeling the field close to the surface of a capacitor plate, then if the capacitor is in an external E field use the one whose E field matches. Otherwise use the one without the extra E field.

I can use the usual model for the vertical component of the field. But I don't know the horizontal component. I cannot conclude that it is zero.

Actually, you don’t know the vertical component any more than the horizontal. The constant E field can be oriented in any arbitrary direction.

 

[Vincf]

I don't believe it's necessary to introduce an external field. This is the field created by the distribution itself.

We can look at the problem another way. Since the "infinite charged plane" is part of our students' curriculum, it must be useful for modeling something. Could we discuss an example of a situation where we modeled a charge distribution using an "infinite plane" to see how we use it?

 

[Dale]

I don't believe it's necessary to introduce an external field. This is the field created by the distribution itself.

It is a boundary condition. It isn’t about introducing an E field, it is about choosing the boundary condition that best matches the scenario you are modeling.

Could we discuss an example of a situation where we modeled a charge distribution using an "infinite plane" to see how we use it?

Like modeling the field close to the surface of a capacitor plate.