Electrodacus has agreed that there is power available to do work in the return circuit. And that this current flowed in and out of the capacitor terminals. As to whether this power went "through" the capacitor i guess we will never agree.
Am I correct ?
The energy will go in a capacitor when it is charged and out of the capacitor when it is discharged. There will never be a case where energy goes through capacitor unless the capacitor is defective in the sense that plates are shorted.
Energy flows through a resistor / wire but flows in or out of a capacitor.
As I see that problem was not solved I will provide the correct results.
12V ideal voltage power supply no internal resistance
2x 0.5Ohm wires connecting a 1000uF capacitor also with no significant DC ESR compared to wire resistance so we ignore for simplicity.
Initial connection we get:
The supply providing 144mWs in order to charge the capacitor with 72mWs worth of energy 0.5 * 0.001F * 12V
2The other 72mWs end as heat on the wires that have 1Ohm of resistance.
If now after capacitor is charged we reverse the polarity of the power supply:
The supply will provide 288mWs and all of that will end up as heat on the wires so 288mWs of heat on wires while capacitor will still have the 72mWs stored from the initial connection.
So since capacitor was already fully charged even if it will be discharged and charged with reverse polarity all energy provided by the source will end up as heat on the wires.
If you disagree with this results we can discuss in more details what happens.
This is for the questions about AC supplies that a few of you asked. So this will be the first cycle.
After this first cycle if power supply keeps switching polarity (AC) then all energy will end up as heat/doing work as capacitor remains with same charged energy as it is discharged and then charged with reverse polarity for each half cycle.
So no energy passes through the capacitor. The current in the wires is due to capacitor being discharged (energy out) and charged (energy in) on each half cycle in case of AC.