Let's say I have two spherical conductors with different radii and different amount of positive charge on them. The spheres are far enough from each other. I connect them with a conducting wire. I'm told that the charge is going to flow until both of the conductors gain the same potential (on top/inside of them). Now, intuitively it seems fine but I do not quite understand a few things.
1) The intuition is that the electric field should be zero everywhere when they reach the same potential, because then the charge is not going to flow. But it is not always the case.
2) Shouldn't the wire also contain some charge after the rearrangement? Because if no, then its potential will be zero, and so there will be a difference in potentials between one of the conductors and the wire, so the charge should still flow into the wire.
I'll try to rephrase the question, because it seems people keep getting my question wrong – in general, I can't understand why charges should always flow until both of the conductors reach the same potential. I mean, the fact that the conductors have same potential doesn't really tell us that the field which drives charges is zero. What's the proof that connected conductors reach the same potential?
Best Answer
1) At the conductor surfaces, the tangential electric field is zero in equilibrium (otherwise charge would flow). The normal field need not be zero.
Zero tangential field is equivalent to the surface being an equipotential. (Potential is, after all, the line integral of the field (times -1).) The fact that the conductors have the same potential really does tell us that the field which drives charges is zero, and vice versa. If the tangential field is not zero, charges will flow until it is, or equivalently until the surface is an equipotential.
2) Note that a smaller sphere requires less charge than a larger one to achieve the same potential (you can see this by integrating the field from infinity for the two cases). The wire is really small (in radius), and needs correspondingly even less charge. In the limit of a 0 radius wire, the required charge goes to 0.
Differing charges do not necessarily imply different potentials. The capacitance, which depends on the geometry, gives the constant of proportionality between charge and potential. It's different for different shapes, so, at equal potentials, different shapes will contain different charges.