Why charging takes time
In guide 1 you met the capacitor's defining rule, I = C times dV/dt — current flows only while its voltage is changing — and the inductor's mirror, V = L times dI/dt. Those two little equations have a quiet but enormous consequence: in a real circuit a capacitor's voltage cannot jump instantly, and neither can an inductor's current. Add the one component you already know well, a resistor, into the path, and you get the whole story of this guide — things settle gradually, on a clock set by the time constant (time constant).
Keep the water picture. Charging a capacitor through a resistor is like filling a small rechargeable bucket through a narrow pipe. At the very first instant the bucket is empty, so the full pressure difference pushes water in fast — the starting current is just I0 = V/R, set by the resistor alone. But as the bucket fills, its own rising level fights back, the pressure difference shrinks, and the flow eases off. The result is a curve that rises briskly at first, then flattens as it creeps toward full.
The time constant: tau = R times C
One number names the speed of the whole curve: the time constant tau = R times C. It really is a time — check the units: ohm times farad works out to seconds. The charging voltage follows V_cap(t) = V_supply times (1 - e^(-t/tau)). After one tau the gap to full has shrunk by the same fixed fraction, 1 - e^(-1) = 63%, so tau is best read as "the time to get 63% of the way there." Wait several tau and you are essentially done.
Put numbers on it. Take R = 10 k and C = 100 uF: tau = 10000 times 100 x 10^-6 = 1 second. So this RC reaches 63% of the supply in one second and is over 99% there in about five — slow enough to watch an LED fade in. Shrink the parts to R = 1 k and C = 1 uF and tau = 1 ms, a thousand times faster. Notice it is the product R times C that sets the clock, so many different part pairs give the same tau.
TIME CHARGING (rising) DISCHARGING (falling) (in tau) fraction of final fraction of start -------- ------------------ --------------------- 1 tau 63 % 37 % 2 tau 86 % 14 % 3 tau 95 % 5 % 4 tau 98 % 2 % 5 tau 99.3 % 0.7 % <- call it "settled"
Be honest about "five tau." The exponential never mathematically reaches 100% — it forever creeps closer, an asymptote it never quite touches. "Five time constants = done" (99.3%) is just a widely used engineering convention; rough work often calls it at three tau (95%), while a precise analog-to-digital sample-and-hold might wait seven or more so the leftover error sinks below one bit. The rule is a choice about "good enough," not a law of nature.
Discharging: the mirror image
Now disconnect the supply and let the charged capacitor empty through the same resistor. The voltage falls along V_cap(t) = V_start times e^(-t/tau), with the same tau as before. One time constant now means "63% of the way down," leaving 37%. The energy you stored last guide, E = 1/2 times C times V^2, does not vanish — it flows back out through the resistor and leaves as heat, which is exactly why the curve decays.
Quick numbers, same parts: a capacitor charged to 9 V, R = 10 k, C = 100 uF, so tau = 1 s. After one second it sits at 9 times 0.37 = 3.3 V; after another it is down near 1.2 V; by five seconds it is a whisker above zero. Charging and discharging are the same curve seen from opposite ends, governed by the one product R times C.
- Find the final value the voltage (or current) is heading toward — where it would rest if you waited forever.
- Find the start value — what it was the instant just before the switch flipped (a capacitor's voltage and an inductor's current carry over; they cannot jump).
- Compute the time constant: tau = R times C for a capacitor, or tau = L/R for an inductor.
- Sketch one smooth exponential from start to final, closing 63% of the remaining gap every tau.
- After about five tau, call it settled — and the same recipe handles a rise, a fall, or any single switch event.
The inductor's clock: tau = L / R
The inductor tells the same tale with the roles swapped. It resists sudden current change — the flywheel from guide 1 — so when you switch a supply onto a coil-and-resistor, the current does not jump to its final V/R but rises along I(t) = I_final times (1 - e^(-t/tau)). The twist is the formula: here tau = L / R, divided, not multiplied. That flips the intuition — a bigger resistor makes an RL settle faster, because the resistor is what burns off the stored magnetic energy, and more resistance drains it sooner. With L = 100 mH (inductance) and R = 100 ohm, tau = 0.1 / 100 = 1 ms.
The inductor also hides a real hazard worth meeting now. Because V = L times dI/dt, if you abruptly break a coil's current — open a switch on a relay or motor — dI/dt is enormous and the inductor flings up a huge reverse voltage spike trying to keep its current going. That spike can arc across switch contacts or punch through a transistor. The standard cure is a flyback diode placed across the coil, giving the trapped current a gentle loop to die down through instead of a brick wall.
Where the time constant shows up
Once you see tau you see it everywhere. A 555 timer sets its delay by an RC charging up to a fixed threshold — pick R and C, and you pick the seconds. A bouncy switch is smoothed by deliberately slowing an RC so the rapid contact chatter blurs into one clean edge. And a decoupling capacitor is the local water tank beside a thirsty chip: when the chip suddenly gulps current, the nearby cap supplies it instantly through a tiny tau, so the long supply trace never sags.
The biggest payoff is a bridge forward. The very same RC that sets time here also rules frequency. Guide 3 brings the AC sine wave; guide 4, reactance and impedance; and guide 5 builds the RC low-pass filter, whose cutoff frequency is f_c = 1 / (2 times pi times R times C) = 1 / (2 times pi times tau) — literally this time constant turned inside out. A slow RC (long tau) only lets slow, low-frequency changes through; a fast RC passes fast ones. Time and frequency are two views of one circuit, and tau is the hinge between them.