Discharging a Capacitor and the Time Constant

A resistor in series with a capacitor sets the pace of charging and discharging. The time constant tau equals R times C, in seconds when R is in ohms and C in farads. After one time constant the discharging voltage has fallen to about 37 percent of its start, after two to about 14 percent, and after five it is effectively zero. These fixed fractions let you predict the voltage after any stated number of time constants without touching exponentials. Read tau straight off a voltage against time curve as the time taken to fall to 37 percent.

Discharge follows an exponential: voltage equals the starting voltage times e to the minus t over RC. Taking the natural log of voltage turns the curve into a straight line against time, with gradient minus one over RC. That trick lets you pull RC from messy data with a ruler. Rearranging the same form finds either the remaining charge or the elapsed time. Keep units strict: ohms times farads gives seconds, and milliseconds must be converted before they meet microfarads.

Hit an RC circuit with a sudden voltage step and the capacitor voltage climbs toward the new level on the same exponential shape, rising instead of falling. One time constant takes it about 63 percent of the way there, and five take it essentially all the way. The resistor current starts large and dies away as the capacitor fills. Speed the response by cutting R or C, and slow it by raising either. Smoothing circuits and camera flashes both exploit this: one tames ripple, the other dumps stored charge fast through a small resistance.