RC
RC·dV/dt + V = 0 at every frame — the exponential goodbye, audited.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A charged capacitor, a resistor, and a switch — the smallest interesting circuit there is. Close the switch and the stored charge must leave; the only question is the schedule, and the schedule turns out to be the most famous curve in engineering.
The law
Charge conservation writes the whole equation: whatever leaves the plates must cross the resistor, so I = −dQ/dt, and with Ohm’s law the loop closes on itself — RC·dV/dt + V = 0. The capacitor’s voltage drives the very current that drains it.
Solve
V(t) = V₀e^{−t/τ} with τ = RC — one constant rules everything. The initial tangent hits zero at exactly τ; the curve itself is at 36.8% there; and after five τ an engineer calls it done, with 0.674% left to argue about.
Discharge
The discharge: charge dots drain from the plates as the current arrow fades — two views of one bookkeeping identity. The curve halves, and halves, and never quite dies; it models a camera-flash discharge through its lamp and the timing behind a debounced input.
Audit
Audited like its mechanical sibling: the differential equation’s residual is zero at every sample — the drawn curve IS the solution — charge conservation holds as an identity, the tangent lands on τ with zero error, and the energy ledger closes at every instant, not just at the end of time.