Discharging and the Time Constant

Disconnect a charged capacitor from its source and let it drive current through a resistor. The current depends on the voltage across the capacitor, which depends on the charge still sitting on its plates, so as charge leaves, the driving voltage drops, and the rate of leaving drops too. That feedback loop is what produces exponential decay: q(t) = q0 e^−t/RC. Every discharge curve, no matter the starting voltage, has this same shape because the equation only cares about the ratio q/q0, not the absolute value. Double the starting voltage and you just scale the whole curve up; the fraction remaining at any given time is identical.

The time constant, tau = RC, is the single number that sets the clock for that decay. After one tau, the charge has fallen to e⁻¹ of its starting value, about 37 percent, regardless of R, C, or the starting voltage. After two tau it is at 37% of 37%, and so on — the capacitor never hits exactly zero, it just keeps shedding the same fraction of whatever remains. That's why a capacitor is never 'empty' after one time constant: 37 percent left is still charge, and the decay only approaches zero as t goes to infinity.

This sits right after charging a capacitor through a resistor, where the same RC shows up but charge builds toward a maximum instead of decaying from one. Both processes share the same time constant idea because the governing differential equation is the same, just with opposite sign. Once you can read tau off a discharge curve or compute it from R and C, you can predict how long any RC circuit takes to respond, which matters for timing circuits, sensors, and anything that needs a controlled delay.