Foundations — RC circuits — charging, discharging, time constant τ = RC

This page assumes nothing. Every letter the parent note leans on is built here from a picture first. Read it before the derivations, and none of them will surprise you.

1. Charge $Q$ — the "stuff" that moves

The picture. Imagine a bucket. The water level is the charge stored on the capacitor. An empty bucket has $Q=0$; a full one has some maximum $Q_{\max }$.

Why the topic needs it. The whole story of an RC circuit is "how does the water level $Q$ change over time?" $Q$ is the star of the show — the quantity our final equation solves for.

2. Current $I$ — charge in motion, and why it is a rate

Now the key move. If $Q$ is the water level and $I$ is how fast water pours in, then $I$ is literally the rate at which $Q$ changes. We write that as

$I=\frac{dQ}{dt}$

Why this tool and not just "$Q$ divided by $t$"? Because the flow is not steady — it changes every instant. Plain "$Q/t$" only works when the rate is constant (like a car at fixed speed). The moment the rate varies, we need the instantaneous rate $dQ/dt$, which is the derivative. This is exactly the tool you meet in Exponential Decay and Differential Equations.

Look at the figure: the steepness of the $Q$-versus-time curve at any moment IS the current at that moment. Where the curve is steep, current is large; where it flattens, current is small.

3. Voltage $V$ — the "push"

Two special voltages appear again and again:

• $\mathcal{E}$ (script E) — the emf of the battery, the steady push it supplies.
• $V_C$ — the voltage across the capacitor, which we'll see is not steady: it grows as charge accumulates.

Why the topic needs it. Push (voltage) is what causes flow (current). Every rule in the topic is a bookkeeping statement about how these pushes add up.

4. Resistance $R$ — the throttle on the flow

Read it as a story: the voltage used up crossing the resistor equals the current through it times its resistance. Double the resistance and you need double the push for the same flow — or you get half the flow for the same push.

Why the topic needs it. The resistor is what makes charging take time. Without it, current would be infinite and the capacitor would fill instantly. $R$ is the referee that slows the race. This is built fully in Ohm's Law and Resistance.

5. Capacitance $C$ — how big the bucket is

Why the topic needs it. $V_C=Q/C$ is the link that turns the capacitor's charge into a voltage we can plug into the loop rule. It is the reason the capacitor "fights back" more and more as it fills. Built fully in Capacitors and Capacitance.

6. Kirchhoff's Voltage Law — the accounting rule

The picture. Think of a hiking loop on a hill. You climb up at the battery (a rise, $+\mathcal{E}$), then walk downhill across the resistor (a drop, $-IR$) and downhill again across the capacitor (a drop, $-Q/C$). Because you return to the start, all the ups must exactly cancel all the downs:

$\mathcal{E}-IR-\frac{Q}{C}=0$

Why the topic needs it. This is the single equation the entire derivation springs from. Every symbol above meets here in one line. Built fully in Kirchhoff's Voltage Law.

7. The exponential $e^{-t/\tau }$ — the shape of "self-slowing"

The parent note's answers all wear the same coat: $e^{-t/\tau }$. Here is that coat, from zero.

Why THIS tool and not a straight line? Recall from §2 that current is the slope of the $Q$-curve, and from §5 that a fuller capacitor pushes back harder, shrinking the driving voltage and hence the current. So the rate of change is proportional to how far from finished we are. A quantity whose rate is proportional to itself is exactly what $e^{-t/\tau }$ describes — no polynomial, no straight line, does this. The same shape governs Newton's Law of Cooling and radioactive decay.

All the cases you must recognise (the curve never surprises you):

• At $t=0$: $e^0=1$ (full value, nothing has decayed).
• At $t=\tau$: $e^{-1}\approx 0.368$ (down to 37%).
• At $t=2\tau$: $e^{-2}\approx 0.135$.
• At $t=5\tau$: $e^{-5}\approx 0.0067$ (essentially zero — "done").
• As $t\to \infty$: $e^{-t/\tau }\to 0$ but is never exactly $0$.
• The rising version $1-e^{-t/\tau }$ is the mirror image: $0$ at the start, $0.632$ at $t=\tau$, climbing toward $1$.

8. The logarithm $\ln$ — the tool that undoes the exponential

Why the topic needs it. In §5's worked examples we know the level and want the time — e.g. "when does $V_C$ fall to 5 V?" The unknown $t$ is trapped up in the exponent of $e^{-t/\tau }$. The only way to pull it down is to apply the operation that undoes $e$: the logarithm. So $\ln$ is the key that unlocks $t$.

$e^{-t/\tau }=k⟹-\frac{t}{\tau }=\ln k⟹t=-\tau \ln k$

9. How the foundations feed the topic

Read top to bottom: charge and voltage are the raw ideas; current is charge's rate; Ohm's law and the capacitor relation turn them into voltages; Kirchhoff sums those voltages into one differential equation; the exponential solves it; the logarithm reads times back out.

Equipment checklist

Self-test: can you answer each before the reveal? If not, re-read the matching section.

Connections

• Capacitors and Capacitance — where $V_C=Q/C$ comes from.
• Ohm's Law and Resistance — where $V_R=IR$ comes from.
• Kirchhoff's Voltage Law — the loop accounting rule.
• Exponential Decay and Differential Equations — the $e^{-t/\tau }$ machinery and the derivative $dQ/dt$.
• Newton's Law of Cooling — the same self-slowing exponential in a thermal disguise.
• Energy Stored in a Capacitor · LR Circuits — where these symbols reappear.
• Back to the parent: RC circuits (main note).