When a capacitor is connected to a DC circuit, what ensures that the current on both sides of the capacitor is the same? When charges arrive at one end of the capacitor they stop moving; presumably they give their kinetic energy to charges on the other side of the capacitor so they can leave. How exactly does this transfer of energy occur?
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$\begingroup$ That the current on both leads of a capacitor is the same is an approximation that characterizes lumped two-poles. In general, however, a conductive structure with the typical shape of a capacitor does not conserve current this way if it can radiate electromagnetic waves into free space, but that's very hard to calculate, so for technical applications we assume that the capacitor is a self-contained element, rather than the combination of two antennas. This works because we can often simplify the problem to a pure displacement current between the two plates and neglect the external fields. $\endgroup$– CuriousOneCommented Jul 12, 2015 at 21:44
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$\begingroup$ Sorry, what are lumped two-poles? $\endgroup$– Si ChenCommented Jul 12, 2015 at 21:48
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1$\begingroup$ A lumped two-pole is the general circuit element with two connections (like resistors, capacitors, inductors, diodes etc.), for which we can neglect the internal structure and the relative geometry of the physical component in relationship to other components, wires, etc.. We can describe them with a low frequency current/voltage relationship like Ohm's law or the capacitor or inductor formulas. For frequencies with wavelengths of the size of the physical component this is not correct, any longer, and we need refined models that take electromagnetic radiation into account. $\endgroup$– CuriousOneCommented Jul 12, 2015 at 21:53
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2 Answers
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You seem to be asking two questions: why is the current on either side the same, and what happens to the energy. To clarify where the energy goes, the kinetic energy is transferred entirely into electrical potential energy of the capacitor. This is the energy stored as a result of all the similar charges being close together on either capacitor plate. There is no actual energy transfer from one plate to the other.
For the currents, let's imagine the current on one plate was larger than the other. The capacitor would end up building a net charge, as a result of the charge on one plate being larger than the charge on the other. By conservation of charge, this extra charge must have come from somewhere else in the circuit, and there must be the opposite charge to cancel out the net charge on the capacitor. This is an energetically unfavourable situation, so doesn't happen without some additional influence. We can thus conclude that the current on either side is identical.
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It is possible for a capacitor to have a net charge. However, most components have a small capacitance with respect to infinity. Unless you're dealing with a huge metal ball, it will take a large voltage in order to get any appreciable charge there.
