A capacitance C with a charge q0 at t=0 discharges through a...

A capacitance $C$ with a charge $q_{0}$ at $t=0$ discharges through a resistance $R .$ Use the voltage equation $q / C+I R=0$ to show that the relaxation time of this process is $R C \mathrm{~s}$; that is,
$$
q=q_{0} \mathrm{e}^{-t / R C}
$$
(Note that $t / R C$ is non-dimensional.)

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Key Concepts

RC Time Constant

Defined as the product of resistance (R) and capacitance (C), it represents the characteristic time over which the capacitor discharges (or charges) significantly. This time constant determines the exponential rate of decay or rise in voltage or charge.

Capacitance

A measure of a component's ability to store electrical charge, where the stored charge is proportional to the voltage across its terminals. In circuits, capacitors store energy that can be released when needed.

Resistance

The property of a material or component that opposes the flow of electric current, converting electrical energy into heat. It affects the rate at which charge flows in a circuit.

Differential Equation in Circuit Analysis

A mathematical equation that relates the rate of change of charge or voltage in a circuit to its instantaneous value. In the context of an RC circuit, it leads to an exponential decay solution, illustrating the time-dependent behavior of the discharge process.

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