According to this video, and multiple other sources, we can calculate the energy stored by a capacitor using the idea integrating the work done by moving a charge from one plate to the other at instantaneous charge levels $dq$. But I do not understand how we can talk about the work done moving each charge from one plate to the other, when in reality, charges are not jumping from a plate to the next. They have to go through the battery, where they are given a potential difference $\epsilon$ and then move from that higher voltage to the instantaneous lower voltage of the capacitor, $V$. So I see how there's an electric field caused by the battery having a higher voltage than the capacitor, but I don't understand how the voltage difference between the plates has to do with what's happening if considering moving one charge to the other plate.
I'll give you an example of what I'm thinking to maybe clear it up. At the beginning, when both plates are neutral in charge, and a $10V$ battery is connected, (in conventional current), protons will flow through the battery and gain a potential of 10V, so the work done by the battery is $10 * dq$, and that should happen for every charge, as each charge is given a potential of $10V$ as it moves through the battery, no matter where you are in the capacitance charging process. But it is said that for the first charge no work is done to move charge from one plate to the other. But the battery had to do work to move the charge through the battery! So where have I gone wrong in thought?
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