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This paradox occured to me today as I was cogitating what fiendish problem to assign to my poor, poor students in an intro to electrodynamics course. I don't claim to be the first one to have thought of this paradox, but I've never seen it treated anywhere. So I was actually not trying to be fiendish; I was trying to be nice, and have the students analyze some simple situation. What's the simplest situation I can think of? Well, what about calculating the electric field if the electric charge density, ρ(r), is the same everywhere in space (and non zero)? That's simple enough! It will prove to not be so.

If you have a background in physics, you have been trained to always utilize the symmetries of the problem. Sophisticated physicists never passes up an opportunity for an elegant solution using symmetries. Thinking myself a sophisticated physicist, I instinctively tried to solve the problem by using its symmetries:

  • No point in space is different than any other point (the problem has translational symmetry). Therefore, symmetry dictates that the electric field should be the same everywhere.
  • The problem is symmetric under arbitrary rotations of space. Therefore, the value of the electric field everywhere should be a vector that is invariant under rotations. There is only one such vector, namely zero.

It follows then that the solution is that E(r) = 0 for all r. There is one problem with this -- it violates Gauss's law. I get ∇.E = 0, whereas Gauss and Maxwell say I should get ∇.E = ρ/εo.

All right, I said, let's not try to be this clever. This is an easy problem, I bet I can solve it easily with conventional methods. I can think of three ways of solving this problem by the method of superposition. Unfortunately each of these gives a completely different answer!:

1. A superposition of concentric spherical shells

Consider the uniform charge density filling space to be composed of concentric spherical shells, each with its center at the origin, each with radius r' and width dr'. OK, if you've taken introductory electromagnetics, you know that the electric field at r due to each spherical shell is

E (r) = (4π ρr'2dr') r / (εo |r|3)

if r' < |r|, and E (r) = 0 otherwise. So to superpose all of these fields, we integrate and get

E (r) = 0|r| (4π ρr'2dr') r / (εo |r|3) = (1/3) ρr/εo

So the field points radially outward away from the origin with a magnitude that grows linearly with the distance from the origin.

2. A superposition of concentric cylindrical shells

Now consider the uniform charge density to be composed of concentric cylindrical shells of infinite length, each with its axis along the z-axis, each with radius r' and width dr'. Again, the electric field at r=rxy+zk (rxy is the component of r parallel to the xy-plane, k is the unit vector in the z-direction) due to each infinite cylinder is a familiar problem we've solved before. It is

E (r) = (2π ρr'dr') rxy / (εo |rxy|2)

if r' < |rxy|, and E (r) = 0 otherwise. So again to superpose all of these fields, we integrate and get that

E (r) = 0|rxy| (2π ρr'dr') rxy / (εo |rxy|2) = (1/2) ρrxy/εo

That is, the field points radially outward away from the z-axis, with a magnitude that grows linearly with the distance from the z-axis.

3. A superposition of parallel plate pairs

Lastly, consider the uniform charge density to be composed of pairs of parallel plates, each pair consists of two plates parallel to the xy-plane, one at z=z', the other at z=-z', and each of width dz'. The electric field at a point r, with z-coordinate z,  from each pair of plates is easily gotten to be

E(r) = ρdz'/εo k if z >z'

E(r) = - ρdz'/εo k if z < -z'

E(r) = 0 if -z' < z < z'

And we superpose by integrating, we get

E (r) = 0z  ρdz'/εo k = ρzk/εo

So the field points outward away from the xy-plane, with a magnituide that grows linearly with the distance from the xy-plane.

All right, so we have three different methods for solving for the electric field, and three different solutions. Of course, each of these solutions violates the symmetries pointed out earlier. It's worse than that even: I could have chosen any point (axis, plane) to be the center around which I built up my spherical shells (cylindrical shells, parallel plate pairs). I didn't have to pick the origin (the z-axis, the xy-plane) and each choice would have given a different solution. So it seems we have a whole slew of solutions. They can't all be right. So, what gives?

Let me put it this way: depending on how I build up to infinity, I get a different answer. In fact, it seems that by choosing carefully how I build up to infinity I can get any answer I desire. Sounds familiar? Those of you who said "alternating harmonic series" or "Riemann's rearrangement theorem", give yourselves 10 points. If you don't know what Riemann's rearrangement theorem is, it's pretty simple. It says that if you judiciously rearrange the terms of a conditionally convergent series (such as the alternating harmonic series), you can make it sum up to any number you want. Indeed, the integrals we are implicitly doing here are all rearrangements of each other, and they are all convergent, but they are not absolutely convergent. The fact that by building up our infinite charge distribution in different schemes of superposition gave us different answers is a direct consequence. So what's the real answer to the problem? It seems it has no satisfactory solution -- or horrendously many, depending the what "satisfactory" means. But it doesn't matter, because we never have infinitely extended charge distributions, and any way we make the extent finite solves our paradox.

What is the moral of this story? I guess it is Beware infinity, my son! The jaws that bite, the claws that catch!

P.S. I'm taking suggestions for a sexier name for this paradox. "The Uniform Charge Density Paradox" is functional, but it is not sexy.

When one thinks in a less-mathematical, more physical way about this problem, it becomes obvious that there are significant problems immediately. I consider the problem as set up by colonelmustard - constant charge density rho in 3 dimensions, and we desire to find the potential (or the electric field which obeys Maxwell; either will do).

Scaling Arguments

Consider an origin O again, and let points we consider be at distance away from this point R (in spherical co-ordinates). Now, consider the spherical shells again. They will be distance R away from the origin, with charge per metre rho*C*R*R total, and be subject to the inverse square law so that the potential phi at O will see a contribution of D*rho*(R*R/R*R) from this shell, where C and D are some (probably dimensioned) constants I don't care about.

Uh-oh. This is a finite contribution from a shell at distance R away - a contribution of rho*D (it doesn't matter what it is - it matters what it isn't - it isn't a function of R). I see immediately that if I add up the value of an infinite number of shells that I'm going to get an infinite potential at every point. I conclude I oughtn't proceed until I've thought a bit harder.


See also: Olbers' Paradox. This is identical, mathematically, until the resolution of the paradox.

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