Electrostatics: can a positive charge induce negative potential?

Can a charge, brought into a chargeless world filled with some geometry of conductors and dielectrics, induce a negative potential anywhere in that world?

I feel the answer is no. But I cannot think of a good way to prove it, or even attack the problem.

More explicitly, imagine a world with conductors and dielectrics in some fixed configuration.
There's no net charge anywhere, so all conductors at at 0 V, and every point in space is also at 0V. You bring some charge Q and stick it onto a conductor. The potential of that conductor will obviously go up. (The exact potential increase will depend on the world's configuration, with that conductor's capacitance relating the potential to the charge Q).
Is it possible for any other conductor in the world to reach a negative potential because of this charge?

My gut feeling is "no way", but I don't know how to prove it at all. I keep thinking of a conservation of energy explaination.. something on the lines of "if there was another conductor that had negative potential, I could bring even more charge from infinity and put it on that conductor and get free work out of it." but that's not a full argument, much less a proof.

I asked myself this question when I was learning about http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-32" I was asking myself how you might have negative entries in this matrix.. a matrix like [2, -1, 0][-1, 2, -1][0, -1, 2] is symmetric and positive definite, but is it a feasible capacitance matrix? What would such a conductor configuration be like?

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Dale
Mentor
You can always make the potential negative simply by a gauge transformation. If you define your zero potential to be any finite distance from the charge then you will have some region be negative.

Btw, charge just magically appearing violates Maxwells equations.

You can always make the potential negative simply by a gauge transformation. If you define your zero potential to be any finite distance from the charge then you will have some region be negative.

Btw, charge just magically appearing violates Maxwells equations.
OK, let's fix those loose ends.. I don't think it changes the problem. Let's say that the initial charge was brought in from infinity, and use infinity as our V=0 potential reference.

Dale
Mentor
All finite distances from the charge will then have a positive potential.

If it is the only charge in the world, how would you even know that it is positive one?

Dale
Mentor
By convention. We adopt a convention that the charge is positive and we adopt a convention that the potential at infinity is 0. Under those two conventions the potential is positive everywhere.

Of course, that really highlights the fact that there is no physical significance to the whole exercise. We could just as easily and validly adopt different conventions and get different answers.

Nature has several symmetries, and this thread has walked into two of the most important ones: EM gauge invariance and charge symmetry.

All finite distances from the charge will then have a positive potential.
This makes sense (unless you had a conductor that connected to infinity, then you'd just have to say nonnegative.) But how could we prove it, ideally with some simple explanation or counterexample?

Dale
Mentor
This makes sense (unless you had a conductor that connected to infinity, then you'd just have to say nonnegative.)
Good point.

But how could we prove it, ideally with some simple explanation or counterexample?
Conductors do not create potential, they only make two locations have the same potential. So if a conductor were grounded to infinity then it would be 0 potential and if it were grounded to some finite location then it would have a positive potential.

Drakkith
Staff Emeritus
Would Electrostatic Induction count for "induce negative potential"?

Would Electrostatic Induction count for "induce negative potential"?
Isn't induction the case where you have GROUNDED conductors, and when you put charge into the world, the new E field "induces" charge in those grounded conductors to keep their potential at 0?

That wouldn't apply to this thought experiment where we're not grounding anything (although I would think that it'd still be OK to ground whatever conductors you like... I don't know if that makes it possible to make a negative potential somewhere.

Conductors do not create potential, they only make two locations have the same potential. So if a conductor were grounded to infinity then it would be 0 potential and if it were grounded to some finite location then it would have a positive potential.

That's a good simple proof of the effect of an infinite conductor!

But I was referring to proving your statement "All finite distances from the charge will then have a positive potential." It's intuitively true, I certainly can't think of a counterexample, but I don't know how to prove it. And your phrasing is likely an elegant condensed summary of my longer original question.

A positive charge can induce negative potential during an electrostatic moment of pure potential in inertia.

Dale
Mentor
That's a good simple proof of the effect of an infinite conductor!

But I was referring to proving your statement "All finite distances from the charge will then have a positive potential." It's intuitively true, I certainly can't think of a counterexample, but I don't know how to prove it. And your phrasing is likely an elegant condensed summary of my longer original question.
In the absence of conductors Coulomb's law suffices. Then on top of that background we add conductors to get the effect described.

Dale
Mentor
A positive charge can induce negative potential during an electrostatic moment of pure potential in inertia.
Hi SandraGarcia, welcome to PF. Is English a second language? I didn't understand your post and I suspect a language barrier.

Drakkith
Staff Emeritus
Isn't induction the case where you have GROUNDED conductors, and when you put charge into the world, the new E field "induces" charge in those grounded conductors to keep their potential at 0?

That wouldn't apply to this thought experiment where we're not grounding anything (although I would think that it'd still be OK to ground whatever conductors you like... I don't know if that makes it possible to make a negative potential somewhere.
I'm not sure I understand your original question completely. Moving a positive charge near a conductor that isn't grounded will cause a negative charge to build up near that positive charge. However this is only if they don't contact each other. Is that what you meant?

i have a idea....its quite hypothetical

suppose charge +q moving in circle with very high speed. Now according to maxwell it produces emw now those emw travell amd stike a antimatter(say anti hydrogen)....due to strike positron will scape and antimatter is at negative potential

i looks funny but might be happening smwhere in the universe