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Charge-voltage relation for a universal capacitor

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Abstract: Most capacitors do not satisfy the conventional assumption of a constantcapacitance. They exhibit memory which is often described by a time-varyingcapacitance. It is shown that the classical relation,$Q left(t right)=CV left(t right)$, that relates the charge, $Q$, with thecapacitance, $C$, and the voltage, $V$, is not applicable for capacitors with atime-varying capacitance. The expression for the current, $dQ dt$, that issubsequently obtained following the substitution of $C$ by $C left(t right)$ inthe classical relation corresponds to an inconsistent circuit. In order toaddress the inconsistency, I propose a charge-voltage relation according towhich the charge on a capacitor is expressed by the convolution of itstime-varying capacitance with the first-order time-derivative of the appliedvoltage, i.e., $Q left(t right)=C left(t right) ast dV dt$. This relationcorresponds to the universal capacitor which is also known as the fractionalcapacitor among the fractional calculus community. Since the fractionalcapacitor has an inherent connection with the universal dielectric responsethat is expressed by the century old Curie-von Schweidler law, the findingextends to the study of dielectrics as well.

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