RC and RL circuit transients

Close a switch on a battery, resistor and uncharged capacitor and the capacitor does not jump to the battery voltage. Charge has to flow onto the plates through the resistor, and as the plates fill, the voltage they push back with grows, which shrinks the current, which slows the filling further. The result is an exponential approach: fast at first, then ever slower, never quite arriving.

The pace is set by the product RC, the time constant, and only by it — the supply voltage scales how much charge ends up on the plates but not how long it takes to get there. After one time constant the capacitor has reached about 63% of its final voltage, after two about 86%, after five about 99%. "One time constant" is not "fully charged", and the exponential in principle never finishes; five RC is the usual engineering convention for close enough.

The same shape governs discharge, running the other way, and an inductor with a resistor behaves analogously with L/R in place of RC, delaying the current rather than the voltage. This deliberate sluggishness is useful rather than a nuisance: it sets the flash interval on a camera, debounces a mechanical switch, smooths the ripple in a power supply, and defines the cutoff of a simple filter. It is also the limit on how fast a digital circuit can switch, since every wire and gate carries some unavoidable resistance and capacitance.