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Why is current drawn in an RC circuit (in a circuit powered by DC voltage supply) independent of the capacitor used?

While the capacitor is charging current drawn from the battery only depends on resistor and on the value of capacitor.

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In a charging RC circuit where the capacitor is initially uncharged, the charges will move as if the capacitor is essentially absent. Therefore, the initial value of the current is just equal to $V/R$.

If the RC circuit starts with a fully charged capacitor and is discharging, then once the current starts the capacitor acts like a battery. The circuit is then essentially a resistor in series with a battery, and the initial current is once again $V/R$ (since the capacitor's initial potential difference has a magnitude of $V$).

At times when the current is changing, the value of the capacitance does effect how quickly the current changes with time constant $RC$. This is because the amount of charge stored on the capacitor changes the potential drop across the capacitor due to $V=CQ$. The more charge that is present, the more it “fights the battery” in the charging case, and the more it pushes charge of itself in the discharging case.

In either case, the magnitude of the current is given by $$I(t)=\frac VRe^{-t/RC}$$ so the current does depend on the capacitance, but the initial current does not.

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