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Specifically, let us talk about a circuit containing only an AC source and a capacitor. In this case, the current is 0 at the instant the source voltage is maximum. Same is the case when the circuit contains only the AC source and an inductor. This seems to me as an apparent violation of the Ohm's Law. Could someone please explain?

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Ohm's Law is derived only for the purely resistive part of the circuit. Suppose the resistor and the capacitor have an ammeter in series . Now as Ohm's law is valid for purely resistive part of the circuit, hence we attach a voltmeter across the resistor. The reading of this voltmeter would be different from the source voltage. When you say that the voltage is maximum, the source voltage ,which is the sum of voltages across capacitor and resistor, is maximum. Although when you plug in the voltage across the resistor only, and the current in the ammeter at any instant, you will find these values obey the Ohm's law.

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