RC CIRCUITS AND THE CHARGING CURVE

Plug a battery into a capacitor through a resistor. Close the switch. What would you expect the capacitor's voltage to do? If the wires were perfect and the cap were a magic tank, you would expect V_c to snap from zero to whatever the battery supplies, the way a lightbulb's brightness snaps on. That is not what happens. The voltage eases up, tentative at first, then slower, then slower still, and it never quite arrives.

The reason is one line of accounting: the voltage across a capacitor is V_c = Q/C. To raise V_c by any amount you have to push that same amount of Q/C of extra charge onto the plate. Charge has to flow through the resistor to get there, and the resistor — obeying Ohm's law — only lets charge past at a finite rate i = (V₀ − V_c)/R. At t = 0 the capacitor is empty, the full battery voltage drops across R, and current is at its maximum. One microsecond later the cap has a tiny voltage on it, the resistor has a tiny bit less to drop, and the current has shrunk by the same tiny bit. The system is eating its own appetite. That is the whole phenomenology.

A capacitor cannot change its voltage discontinuously because its charge cannot change discontinuously — and charge, like water in a tank, only arrives through pipes that have flow limits. This is the rule you lean on every time you reason about a filter, a debouncer, or a flash unit: V_c is never ahead of its history.

Take the series loop: battery V₀, resistor R, capacitor C. Apply Kirchhoff's voltage law around it. The voltage the battery supplies has to equal the sum of what the resistor drops and what the capacitor holds:

Now use the two bridge facts. The current i is the rate at which charge piles onto the cap — in plain words, i is the derivative dQ/dt, how fast Q is changing per second. Substitute:

One line of calculus finishes the job. Rearrange to dQ/dt = (V₀ − Q/C)/R, separate variables, integrate with Q(0) = 0, and collect:

That is the whole chapter in one equation. The current i(t) = (V₀/R)·e^(−t/τ) falls along a mirror exponential — maximum at t = 0, zero at infinity. Notice what τ means: a big capacitor (bigger tank) or a big resistor (narrower hose) both slow the process down, and they slow it by exactly their product. The units work: ohms × farads = seconds, a numerical surprise the first time you check it.

The time constant is how long it takes V_c to cover roughly two-thirds of the distance to V₀. At t = τ, 1 − 1/e ≈ 63.2 %. At 2τ, 86.5 %. At 3τ, 95 %. At 5τ, 99.3 % — close enough that every datasheet lists "5 time constants" as the rule-of-thumb settling time.

Close the switch and watch. The lilac trace is V_c(t) climbing toward V₀. The amber trace is i(t) falling from V₀/R toward zero. They share a time constant — τ controls both rhythms because they are the same first-order system observed from two angles. Drag the sliders: increase R and everything slows down together; increase C and the same. The 63 % line is a universal marker that every RC-filter, debouncer, and sample-and-hold circuit designer recognises in their sleep.

The never-arrives is not a bug. In every real circuit some tiny leakage will pin V_c asymptotically close to V₀ after 5–7 τ, but the formal "infinity" in the exponential is the mathematics being honest: first-order linear systems are patient.

Disconnect the battery and short the capacitor through the same resistor. The KVL loop now reads V_c = i·R with i = −dQ/dt (the cap is losing charge), giving R·C · dV_c/dt = −V_c. Integrate with V_c(0) = V₀:

At t = τ the cap is at 1/e ≈ 36.8 % of its starting voltage. At 5τ it is below 1 %. The current magnitude mirrors it — same envelope, reversed direction.

A pre-charged cap drains through R. The amber arrows on the loop now sweep the other way — this is the cap acting as a source, pushing current the direction that will empty it. The discharge is the same mathematics with a sign flipped, which is how first-order linear systems always work.

Here is the twist that surprises every undergraduate who meets it. Charge a cap from 0 to V₀ through any resistor R. The energy stored on the capacitor at the end is ½·C·V₀². The energy the battery put out over the whole charge is C·V₀². Where did the other half go?

Look at that cancellation. R sits in the numerator of i(t)² · R, and the factor of τ = R·C from the integral brings in another R on the other side. They cancel. The total heat dumped in the resistor is ½CV₀² — independent of R. A bigger resistor dissipates at a lower rate for longer; a smaller resistor dissipates at a higher rate for less time. The product integrates to the same number either way. Joule would have raised an eyebrow and then nodded: i²R is his heating law, and here it delivers the same total regardless of how you spread it over time.

Half the battery's joules become heat. You cannot charge a capacitor from zero through a resistor with better than 50 % efficiency — not with a bigger R, not with a smaller R, not at any speed. The other half — exactly the other half — is the ½CV² now stored in the capacitor's electric field (see FIG.19, §02). Switched-mode power supplies escape this tax by charging capacitors through inductors instead; that is their whole reason to exist.

What engineers actually measure on a scope is τ, not R or C separately. A blurry resistor and a tolerancey capacitor still pin down τ beautifully: close the switch, hit the cursor on the scope at 63 % of final value, read the time. That is your τ, and from it you back out either parameter given the other.

Three curves, same V₀, same C, three resistors — three different rhythms. The 63 % guide crosses each one at its own τ, with τ varying linearly with R. This is the picture every EE keeps in their head. "Make the filter 10× slower" means "make R or C 10× bigger." "Make the circuit settle in 5 ms" means "τ ≈ 1 ms so R·C ≈ 1 ms — choose the capacitor you have, divide into it, done."

RC is the most reached-for filter in electronics. A resistor followed by a capacitor to ground is an RC low-pass — high-frequency noise sees a cap that looks like a short to ground; low-frequency signal sees a cap that looks like an open. The corner frequency is f = 1/(2π·τ), the one formula every audio and RF engineer can recite in their sleep.

Debouncing is pure RC: press a mechanical switch, the contacts chatter for 5 ms, you feed the signal through an RC with τ ≈ 10 ms, the chatter averages out. PWM smoothing — turning a square wave into a DC voltage — is an RC whose time constant is much longer than the pulse period. Camera flash capacitors charge slowly through a small trickle resistor then dump their ½CV² through a xenon tube in microseconds when you press the shutter. Sample-and-hold front-ends on every ADC hold their voltage between conversions using the fact that a capacitor has τ = ∞ when disconnected — it holds its state with patience.

Every one of these is a line of V_c(t) = V₀(1 − e^(−t/τ)) or V₀·e^(−t/τ) turned sideways into product design. FIG.28 meets the coil — the same mathematics with the sign of hesitation reversed.