Electrostatics: Computing net charge on a specific surface using multiple tests

[Fernbauer]

An interesting electrostatics puzzle!

This is a simplification of a real problem I've been hurting my brain on. Luckily the simplification is a fun challenge in its own right.

Homework Statement

There is a 3D rectangular solid object in space. Each face of the solid is a perfect conductor but insulated at the edges from the other faces so each face can have a different potential.

We set up one experiment. The top face is given a potential of 1V, and the other five faces are held at 0V. A magic oracle tells us the total charge on the whole object is Q1.

Next, we change the potentials such that all six faces are all at 1V. The magic oracle tells us the total charge on the whole object is Q2.

The question: In the second case, where all faces are at 1V, what is the total charge on just the top face alone? Is it even computable from just the Q1 and Q2 measurements?

Homework Equations

Gauss's Law! And/or classic electrostatic linearity.

The Attempt at a Solution

My initial thought went something like "Hey, electrostatics is linear. The first case, where the top face has potential 1V and everything else is grounded gives you the charge Q1. So that's still the charge on the top face when the other faces have their potential changed."
But I think this explanation is wrong (even if the answer is right). Q1 is the total charge on the object in the first case. The other 5 faces must have negative charge on them to keep their potential at 0V. If they had net 0 charge, they'd float at a potential somewhere between 0 and 1 V depending on the geometry. Thus the charge on the top face is in fact higher than Q1, and there's negative charge on the other faces, and their sum is Q1.

But while that may be true, this is also true that there's compensation for the second case where all the faces are at 1V. To raise potential on those other 5 faces, you had to add charge to the faces and subtract charge from the top face. So maybe this compensates exactly for our previous underestimate. Q1 is indeed smaller than the net charge on the top face in the first measurement, but is correct for the charge on the top face (only) in the second configuration. But the justification for this eludes me. It does have a good "feeling" it should be true, but I can't convince myself.

I also suspect the answer does not depend on the object being a 3D rectangular box. I expect that it would hold for a generalized conductor as long as my oracle let's me "freeze" potentials on the surface into the 1V/0V partitions of the region I'm interested in.

This problem isn't coming from a book, it's actually in a design problem where I want to figure out how the capacitance of an object changes if a face is perturbed slightly. I have canned routines for computing total charge on the object for any configuration of surface voltages. If I know the average charge density on the face for when the whole object is at 1V, I'll know the average electric field on that face, so I know the potential derivative over space along the face, and I can compute the change in voltage if I move the face a small distance epsilon. This tells me the change in capacitance, and I have the dC/dEPS sensitivity I'm trying to compute.

Thanks for brainstorming this with me!

 

[diazona]

Interesting question... it's been a while since I did anything with Poisson's equation (the governing equation for charge distributions and potential fields) so I don't remember the full bag of tricks that might be useful, but my first thought would be to try to exploit symmetry and superposition, much like you did at first. Of course, the shape is not fully symmetric, so that may not get you much. (It would be really easy if it were a cube, of course)

But while that may be true, this is also true that there's compensation for the second case where all the faces are at 1V. To raise potential on those other 5 faces, you had to add charge to the faces and subtract charge from the top face. So maybe this compensates exactly for our previous underestimate. Q1 is indeed smaller than the net charge on the top face in the first measurement, but is correct for the charge on the top face (only) in the second configuration. But the justification for this eludes me. It does have a good "feeling" it should be true, but I can't convince myself.

I don't share your feeling on this - that is, I'm not convinced that Q1 gives you the charge on the top face in the second situation. But I can't come up with an actual proof one way or another at the moment.

Anyway, if nothing else, I'm pretty sure the problem does have to have some solution - that is, the total charge on the top face (or on any part of the object) is computable knowing only the potentials. (You don't even need the magic oracle.) The justification is that, in the space exterior to the conducting object, you're solving a differential equation
$$\nabla^2 V = 0$$
with completely specified boundary conditions. This particular problem is known to have one unique solution (if I remember correctly). The hard part is actually finding the solution $V(x,y,z)$, but once you do, you can compute the charge density at any point on the surface using
$$\hat{n}\cdot\vec\nabla V = -\frac{\sigma}{\epsilon_0}$$
and this tells you exactly how the charge is distributed over the conductor; in particular, you can integrate over each face to find how much charge is on that face.

 

[Fernbauer]

I don't share your feeling on this - that is, I'm not convinced that Q1 gives you the charge on the top face in the second situation. But I can't come up with an actual proof one way or another at the moment.

Anyway, if nothing else, I'm pretty sure the problem does have to have some solution - that is, the total charge on the top face (or on any part of the object) is computable knowing only the potentials. (You don't even need the magic oracle.) The justification is that, in the space exterior to the conducting object, you're solving a differential equation
$$\nabla^2 V = 0$$
with completely specified boundary conditions. This particular problem is known to have one unique solution (if I remember correctly). The hard part is actually finding the solution $V(x,y,z)$

Thanks for thinking about this! You're right that it does have some solution if you can analyze all the world's geometry and solve the Laplace equation everywhere, but as you say, this is likely overkill. But that "oracle" I referred to is in fact doing some math magic that actually does do much of this: it can report the total charges, but doesn't break it down into giving the E field everywhere.

--


Two more thought experiments may help support (but certainly not prove) that the answer is Q1.

One case is when the symmetry of the problem shows that the face in question is copied in other orientations (like on a cube, 6 faces). In this case the linear superposition principle means you can raise each face to 1V in turn, each time adding Q1 charge (in an unknown distribution). At the end, you've added 6*Q1 charge, but you divide by the number of faces (6) to get average charge per face of Q1. This works for any kind of symmetry.

The other thought experiment is if the object is two distantly separated, disjoint parts. In that case, raising one object to 1V (using Q1 charge) won't really affect the other very distant object, so the charge on the distant object stays unchanged. In this case, the answer to the "how much charge is induced" question is again Q1.