Resistor-Capacitor (RC) Circuits

Capacitors combine the opposite way from resistors: capacitances add in parallel and their reciprocals add in series, so a series combination is smaller than its smallest member, and series capacitors carry equal magnitudes of charge. At the instant a switch closes, an uncharged capacitor has no potential difference across it and behaves like a plain wire, so the current in its branch is at its largest and is found by replacing the capacitor with a wire. After a long time no current flows in that branch, so it is treated as open and the capacitor's potential difference is whatever appears across the two points it bridges. The crossing has time constant $\tau = RC$.

Four errors dominate. Reusing the resistor combining rules for capacitors, which inverts every equivalent capacitance. Treating a capacitor as an open switch from the moment the circuit closes, which puts the initial current at zero when it is at its maximum. Charging every capacitor up to the source emf regardless of where it sits, when it reads only the potential difference at its own terminals. And reading $\tau$ as a finishing time, or scaling the speed backward so a larger resistor is expected to fill the capacitor faster.