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I have a few doubts about this problem. So we have two charged spheres of radius $r_1$ and $r_2$, one is initially charged with a charge $Q$, while the other one is initially without charge. The problem has a lot of questions that I already solved and I don't have any doubts about that. However, I still have one question that may be very obvious to some but I still cannot figure it out.

When the spheres are both at the same potential, they both generate an electric potential throughout space that depends on the distance to the spheres. So, if we take two separate points in the wire (and let's assume these two are really far from one of the spheres so that the electric potential of that sphere can be neglected for this particular case) there should be a potential difference between these two points because they are at different distances from the center of the closer sphere right? Then there should be current. If there's not please explain to me why. Thanks for your help whoever reads this!

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It is true that the potential of sphere is space-dependent but the outside potentials do not affect the potential, electric fields and motion of electrons inside the conducting wire. You must be knowing that outside electric fields fo not have any influence on the inside of a conductor. Probably the confusion is because you forgot this fact that inside the metallic wire, at static situation, electric field is zero.

Just as you connect the wire, one end of wire is at higher potential than the other. A current does flow from the sphere at higher. potential to the one at lower potential. But eventually, as the electrons redistribute, the electric field within the wire becomes zero and the whole wire reaches a constant potential. This happens even though outside the wire, the electric field and potential are still space-dependent.

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  • $\begingroup$ I forgot to take that into account. Thank you! $\endgroup$ May 16, 2020 at 22:42
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Your problem suggests that the spheres are both conductors. If we assume that one starts with a negative charge, when they are connected by a wire, electrons will flow until all parts of the system (and just outside) are at the same potential. The charge will not be uniformly distributed on the spheres, and they cannot be treated as point charges.

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