# A question about the Second Uniqueness Theorem in electrostatics

• I
• Ahmed1029
In summary, this conversation discusses how charge can be distributed on a conductor, and how two conductors that have the same charge, but are placed in different configurations, will have different electric fields.
Ahmed1029

in this example in Griffiths' electrodynamics, he says the following :(Figure 3.7 shows
a simple electrostatic configuration, consisting of four conductors with charges
±Q, situated so that the plusses are near the minuses. It all looks very comfort-
able. Now, what happens if we join them in pairs, by tiny wires, as indicated in
Fig. 3.8? Since the positive charges are very near negative charges (which is where
they like to be) you might well guess that nothing will happen—the configuration
looks stable.
Well, that sounds reasonable, but it’s wrong. The configuration in Fig. 3.8 is
impossible. For there are now effectively two conductors, and the total charge
on each is zero. One possible way to distribute zero charge over these conductors is to have no accumulation of charge anywhere, and hence zero field everywhere)

I feel like there are gaps in this explanation, that is, don't know if he implicitly invoked the first uniqueness theorem here

Delta2
This is like connecting two oppositely charged capacitors. You bet the current will flow!

Delta2
Well he seems to imply that since one way to redistribute zero total charge over a conductor, is to have zero charge everywhere in the conductor, then this is the only way. Is this the first uniqueness theorem btw?

BvU said:
Delta2 said:
Well he seems to imply that since one way to redistribute zero total charge over a conductor, is to have zero charge everywhere in the conductor, then this is the only way. Is this the first uniqueness theorem btw?
Well there are infinite ways to distribute charge on conductors, each with its own electric field. What conditions will tell me which of them is the true one? This is the second uniqueness theorem

A conductor has no potential difference. The charge may be distributed unevenly, but the surface potential is the same everywhere.

##\ ##

vanhees71
Ahmed1029 said:
I feel like there are gaps in this explanation, that is, don't know if he implicitly invoked the first uniqueness theorem here
The situation shown in Figure 3.8 is a fictitious assumption and it is likely not a stable state.

If we think of these two conductors as an isolated capacitor, there is no potential difference and no electric field between them since the net charge in the two conductors is zero. That is, there is also no charge accumulation on the surfaces of the two conductors.

Hmm, given a conductor (that is given its shape) and a total charge Q, isn't the way that this charge Q going to redistribute itself along the surface of the conductor, unique? And that depends only on the shape of the conductor?

vanhees71
Delta2 said:
Hmm, given a conductor (that is given its shape) and a total charge Q, isn't the way that this charge Q going to redistribute itself along the surface of the conductor, unique? And that depends only on the shape of the conductor?
I think so too, which seems to be what the uniqueness theorem says.

vanhees71 and Delta2
Delta2 said:
Hmm, given a conductor (that is given its shape) and a total charge Q, isn't the way that this charge Q going to redistribute itself along the surface of the conductor, unique? And that depends only on the shape of the conductor?

There is the small matter of an external field (if present)

##\ ##

vanhees71 and Delta2
Yes that's very right, I should 've said in the presence of no external field.

In fact, the situation of Figure 3.8 is impossible because it violates the conservation of energy around closed circuit paths and circuit theory.

Delta2
Yes we can't process this with circuit theory, unless the conductors have some ohmic resistance.

alan123hk

## 1. What is the Second Uniqueness Theorem in electrostatics?

The Second Uniqueness Theorem in electrostatics states that the potential at any point in a region of space is uniquely determined by the charge distribution on the boundaries of that region.

## 2. What does the Second Uniqueness Theorem imply about the behavior of electric fields?

The Second Uniqueness Theorem implies that the electric field at any point in a region of space is also uniquely determined by the charge distribution on the boundaries of that region.

## 3. Can the Second Uniqueness Theorem be applied to any charge distribution?

Yes, the Second Uniqueness Theorem can be applied to any charge distribution, as long as the boundaries of the region of space are well-defined.

## 4. How is the Second Uniqueness Theorem related to the First Uniqueness Theorem?

The First Uniqueness Theorem states that the potential at any point in a region of space is uniquely determined by the charge distribution within that region. The Second Uniqueness Theorem extends this concept to include the charge distribution on the boundaries of the region.

## 5. Are there any limitations to the Second Uniqueness Theorem?

The Second Uniqueness Theorem only applies to static electric fields, meaning that the charge distribution and boundary conditions must remain constant. It also assumes that the region of space is free of any external electric fields.

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