I have an exercise in which I have the following situation: there are two conductors spheres (far apart) both of a given radius (R1, R2=2R1). The lesser sphere has a positive charge q and the other shpere is chargeless. The two spheres are connected with a thin cable and then disconnected.

Now I have to establish the final charges q1 and q2 and the final potentials V1 and V2.

That's what I thought: when they are connected they stay at the same V so I can write: $$k*Q1/R1 = k*Q2/R2 \ \ \ \ where \ Q1+Q2=q$$ and using this I can establish Q1 and Q2. But what exactly happen when I disconnect them? Do they just stay at the same V, with the charges they were having while connected, or do they have a different behavior?

Thank you in advance for the help.


When you connect them, charge can flow, and it will do so until there is no net electric field it can move along (the surfaces are the same potential). Imagine I just suddenly destroyed the wire; now the charges are stuck on the spheres. They are still under the condition that they want to equilibrate and put a constant voltage across the surface, but they were already in that situation (and removing the wire doesn't change that) so nothing will change when you remove the wire.

An interesting extension to think about conceptually, if you want: you're essentially imagining these spheres to be very far apart, so that the field from one sphere doesn't affect the other. What happens if they're very close? Each sphere is generating a field that will push the surface charge on the other away. So I end up with more charge on the outside-facing sides of each sphere, and less on the inside-facing sides. (Calculating the exact the distribution is a lot more work, though, so don't worry about that! It's just a good thing to keep in mind in these problems.)


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