An interesting electrostatics puzzle!(adsbygoogle = window.adsbygoogle || []).push({});

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.

1. The problem statement, all variables and given/known data

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 thewholeobject is Q1.

Next, we change the potentials such that all six faces areallat 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?

2. Relevant equations

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

3. 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 havenegativecharge 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 whereallthe faces are at 1V. To raise potential on those other 5 faces, you had to add charge to the faces andsubtractcharge 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 thejustificationfor 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 lets 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!

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

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