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I am not seeing the "big picture" here. If I have two conducting spheres separated by a long conducting wire, why would the spheres share the same electric potential?

I think of the spheres as point charges, what does the conducting wire do? The $E$ field inside the conducting wire is 0, so what is it really doing?

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The sphere should be conductible, or you may face 1/0. – puresky Jan 6 '12 at 3:53
Hi @puresky - it looks like you may have intended to comment on the question, so I converted your post into a comment. – David Z Jan 6 '12 at 4:39

Because electrons flow when there is potential difference. So only when every point has the same potential will the system reach electrostatic equilibrium.

The above holds only if everything is conductor.

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But what does the electron flowing have to do with the potential differences? – lam Jan 6 '12 at 3:59
potentials are caused because of excess or deficiency of electrons. Now when there exist a pot. difference, a current flows which is flow of electrons. Now, this will happen till there is a pot. difference. So, when current flow stops, you have two equipotential spheres. – Vineet Menon Jan 6 '12 at 5:11
you mean "till there is NO potential difference" – ThePopMachine Jan 6 '12 at 5:21

The E field inside the conducting wire is 0, so what is it really doing?

The potential difference between two points is related to the electric field along the path between them:

$$V_{ba}=\int_a^b \vec{E}\cdot{}\vec{dl}$$

So the fact that the high-conductivity material forces a (near-)zero electric field is exactly why the two ends of the conductor are at the same potential.

As other answers have said, the reason why the E field is zero inside a perfect conductor is because if it wasn't, current would flow until it became zero; and electro-statics only considers situations where all such flows have reached steady state.

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Think of a gas. We will ignore gravitational potential, and we'll consider the situation to be at steady state. Compressing the gas takes energy, so we conclude that the entire container of the gas will be at a constant level of compression - in this case, constant density as well. You may intuitively understand that this is true no matter how oblong the shape of the container is. This container could be a compressed air pipe running a mile. The two ends still have the same densities because that's just how gases work.

Electrons in a conductor are like a gas which occupies the same space, and is confined by, the conducting material. The only critical difference between the electron gas and a classical gas is the existence of long-range forces. The electron gas is charged so one end affects the other by the 1/r^2 electrostatic repulsion. Otherwise, the gas still takes some energy to compress (on the Debye scale), but its distribution is mostly dictated by minimizing the electric potential.

Hope that analogy helps your understanding some.

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Opposite charges attract, electrons are moving charges. A conducting wire is the perfect way for electrons to move until the charge of the two ends is equal, and this happens very quickly.

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See. If there is some potential difference between the 2 ends of a wire the electrons will flow in a direction where there is less potential difference. Hence the potential difference will be uniform throughout the conductor.

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The two spheres share the same potential if and only if they are in equilibrium, which in general, as you pointed out, does not have to always hold.

In fact, if you take any two spheres and connect them with a wire, they will not share the same potential. They eventually will when all the charge carriers have flown from one to another according to the direction of the electric field therein.

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The electric potential in a place is the potential energy per unit charge that would be placed in that position. So, for a particular charge, a difference in electric potential between two places, means a difference in potential energy in those two places.

Now, as everything else in this universe, a charged particle tends to minimize it's potential energy when you lift the restrictions. To make this clear, stand up and take something not to valuable in your hands. Now watch what happens with that object when you remove the restrictions (your hands): it lowers its potential energy as much as possible.

Now back to the conducting spheres. Let's assume they start at a different potential. That means free electrons on one of the spheres can lower their potential energy by moving to the other sphere. However, there are restrictions: the electrons can't travel through the air between the spheres. This restriction is lifted when they are connected with a conducting wire. Free electrons move from one sphere to the other. As this happens, the potential in both spheres changes as well. Free electrons will continue to flow from one to the other as long as the potentials are different, but eventually, they become the same. Now, there is no longer a difference in potential and the charges stop moving. When this equilibrium is reached, and only then, the potentials are the same.

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