A Better Criterion for the Discharging Time in an RC Circuit

When all parts of an electric circuit are at the same potential, no electric current flows and it is said to be in equilibrium. Otherwise, a current will flow from the higher potential parts to the lower ones, as when we make contact between the plates of a charged capacitor. The resulting discharging process towards equilibrium is a common phenomenon in electricity labs, as well as a topic covered in introductory physics, but some questions are not answered, e.g., could we establish a good criterion for the “end” of the charge transfer process? If so, typically how long does it take for electric equilibrium to be reached? The absence of these questions in textbooks and lab manuals has bothered me since my own undergraduate studies because, as the usual theory, in which charge is treated as a continuous, rather than discrete, parameter, predicts an exponential decay for the charge stored in the capacitor, rigorously speaking we should wait an infinitely long time for finding a complete equilibrium.¹ Fortunately, when teaching experimental physics last year, I was asked about these very questions, which motivated me to search for a simple answer that could be adopted universally for measuring the “lifetime” of any physical quantity that decays exponentially in time.