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
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 / (4πεo |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 / (4πεo |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 / (2πεo |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 / (2πεo |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) = 0∫z 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.