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Consider the simple capacitor-battery system in the image. Assuming the battery above has a potential difference of 9V, where the positive side has potential of 9V and the negative side has potential of zero. My understanding of how charge flows into a capacitor is as follows:

Focusing only on the positive terminal, when we first connecte the plate to the terminal, a difference in potential between the plate and the positive terminal exists, assuming the plate has a potential of 3V with respect to the negative terminal, positive charges will flow from positive terminal to the plate, until the potential in the wire between the plate and the positive terminal is zero (i. e. The plate now has potential of 9V). And if we connect the negative terminal to another plate, the same will happen ( the plate would have a potential of zero with respect to the positive terminal)

The part that really threw me off is how would potential between the plates effect the potential in the wires between the plates and the terminals. For instance, a book said that when the distance between the plates increase, the potential between the plates will also increase (which is understandable considering that potential is equal to electric field times distance), and that this difference will in turn cause a difference in the wire connecting the plate to the terminal, which will cause positive charges to flow from the plate to the positive terminal. But i fail to see how would the difference between plates could effect the difference in the wire. My believe is that the wire has no idea whats going on outside of it, all the wire cares about is maintaining a zero potential difference between the plate and the terminal, which is established through keeping both plate and terminal at the same potential difference with respect to the negative terminal (i. e. Both plate and terminal should have a potential 9V). Can someone please explain how the distance between plates will effect the potential difference between the plate and the positive terminal in the wire? Thanks

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You have already stated correctly that there is no potential difference between the plates and the battery terminals due to the (ideal) wire connections. Moving the plates doesn't change this observation. However, there are transient currents in the wires during the movement. E.g., when you move the plates further apart, the capacitance diminishes in time and thus the charge $Q$ on the plates is reduced as a function of distance. This means, that a reverse current has to flow into the battery. If your wire has a finite resistance, this causes a voltage drop between the plates and the terminals during the movement.

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Distance affects the potential drop across the plates and thus the capacitance and also the charge stored since you need to maintain that potential across the plates.

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Actually there is no flow of charge inside the capacitor.What happens actually is only field lines are developed as soon as we give potential difference .In other words there is polarized di-electric medium which induces charge on the plates when we give bias.We can also explain it in terms of displacement vector(maxwell's equations)

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Q=CV C, the capacitance is inversely proportional to the distance. Since the plates are still attached to the battery, V, the potential difference will remain unchanged. However since the capacitance drops as a result of the increasing distance between plates, Q, the charge on the plates should be changed. So charges will flow back to the battery.

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