# Using Gauss's Law to find E for an infinite volume charge density

• Hakkinen
In summary, there is a problem with using Gauss's Law to find the electric field at a point when considering a uniform charge density that is infinite in volume. The usual symmetry arguments used do not work, and the left hand side of the differential form of Gauss's Law becomes zero. There are some suggestions for resolving this paradox, such as viewing the problem in a higher dimensional space or breaking down the infinite volume into an infinite sum of infinite plane charges. However, the problem may lie in the fact that the field is not compactly supported, meaning it does not go to zero at a finite value. This is why the examples of an infinite line or plane of charge do not have a similar paradox.
Hakkinen
My E&M professor brought up this problem of considering a uniform charge density, rho, that is infinite in volume and then using Gauss's Law to find the electric field at a point. It's resulted in a lot of head scratching and I'd appreciate some help/discussion to guide me towards a resolution.

Basically the "usual" symmetry arguments used for examples like an infinite line/plane of charge when first applied to this problem seem to suggest that the field anywhere should be 0. However then it's straightforward to see from the differential form of Gauss's Law
$\nabla\cdot \vec{E}=\frac{\rho }{\varepsilon _{0}}$
that the left hand side is now zero. So now you have zero being proportional to the constant charge density rho!

The first thing I thought about this result is that Gauss's Law is saying you simply can't have a nonzero uniform charge density in an infinite volume but it's clearly not all there is to it.

I'm almost certain that the problem lies not in Gauss's Law but in the symmetry arguments used or in some subtlety of the way the problem is described. Once it can be shown that E=/= 0 then the problem should be resolved.

1
In all of the lower dimensional cases of an infinite charge density, the object was placed in 3d space so there were dimensions where the electric field could permeate and be nonzero. So could the problem be that we are viewing a 3d object in 3d space and if viewed in a 4d space (what would that even entail? surely it wouldn't a time dimension and I can't think of reasons to simply add a 4th spatial dimension) then there would be another dimension for the electric field to have a nonzero component.

2
It's actually a problem with the divergence theorem since it requires vector fields to be compactly supported, meaning they go to zero at some finite value. So in this case with an infinite volume the field would not be compactly supported and you cannot use Gauss's Law.

If this, #2, is the case then why do the examples of an infinite line/plane charge have no similar paradox like this example?

3

You can treat the infinite volume as an infinite sum of infinite plane charges but it might be that the integral is not absolutely convergent, that is, the electric field you get will depend on the order of which you summed the contributions from each plane.

Maybe you might first consider the problem of a plane slab, or the problem of a sphere and let the dimension increase to see what happens and resolve the puzzle?

Answer # 2 is correct. Gauss law just went kabooey. Of course # 3 is also correct - That's why Gauss law went kabooey. The infinite line or plane do not cause similar problems because the integral for such cases does converge.

Another way to see that paradox is to chose any point is space and dissect the whole space as a series of spheric shells onion-like all the way to infinite. chose a second point. For that point, by Gauss' law the outer shells do not contribute to the field at that location but the inner shells do and the total field is proportional to the radius (The charge inside goes with the cube of the radius but the area goes with the square of the radius) That radius is just the distance between the two points. How can that be? The first point was arbitrary and the distance between the two points could be anything at all. Something seriously wrong just happened

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I would first suggest carefully reviewing the assumptions and constraints of the problem. It is important to clearly define the boundaries and limitations of the system, as this can greatly impact the results. In this case, the infinite volume charge density may not be physically realistic or may require additional considerations.

Secondly, I would recommend exploring alternative methods for solving this problem. While Gauss's Law is a powerful tool, it may not be the most appropriate or accurate approach for this particular scenario. Other methods, such as using Coulomb's Law or solving for the potential, may provide more insight into the behavior of the electric field in this system.

Additionally, it may be helpful to consider the implications of an infinite charge density. Is it possible for a physical system to have an infinite charge density? What would be the consequences of such a situation? These questions can help guide your thinking and potentially lead to a better understanding of the problem.

In conclusion, while Gauss's Law is a fundamental principle in electromagnetics, it is important to carefully consider the assumptions and limitations of any problem before applying it. In cases where it may not be applicable, exploring alternative methods and critically examining the physical implications can help guide towards a resolution.

## 1. What is Gauss's Law and how is it used to find electric field for an infinite volume charge density?

Gauss's Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the enclosed charge. It can be used to find the electric field for an infinite volume charge density by considering a Gaussian surface that encloses the charge distribution. The electric field at any point on the surface can be determined by integrating the electric flux over the surface.

## 2. What is an infinite volume charge density and how does it differ from a finite volume charge density?

An infinite volume charge density refers to a charge distribution that extends infinitely in all directions. This is in contrast to a finite volume charge density, which is limited to a certain region or volume. Gauss's Law can be used to find the electric field for both types of charge density, but the mathematical calculations may differ.

## 3. Can Gauss's Law be used to find the electric field at any point in space for an infinite volume charge density?

Yes, Gauss's Law can be used to find the electric field at any point in space for an infinite volume charge density. This is because the law takes into account the entire charge distribution, regardless of its size or shape, as long as it is enclosed by the Gaussian surface.

## 4. What are some practical applications of using Gauss's Law to find electric field for an infinite volume charge density?

Gauss's Law has many practical applications, including in the design of electronic devices, such as capacitors and antennas. It is also used in calculations for the electric field inside and around conductors, which is important in understanding the behavior of electric currents.

## 5. Are there any limitations or assumptions when using Gauss's Law to find electric field for an infinite volume charge density?

One limitation of using Gauss's Law is that it assumes a static charge distribution. This means that it may not accurately predict the electric field for time-varying or dynamic charge distributions. Additionally, the law assumes a vacuum or air as the medium, so it may not be applicable in other materials with different dielectric constants.

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