Resistor-Capacitor (RC) Circuits

A capacitor's voltage cannot jump, because $V_C = Q/C$ and charge takes time to arrive. That continuity turns an RC circuit into two solvable resistor networks: at the switching instant an uncharged capacitor is a wire, and after a long time it is an open circuit. The loop rule then gives $R\,dq/dt = \varepsilon - q/C$, whose solutions are $q = C\varepsilon(1 - e^{-t/\tau})$ for charging and $q = Q_0e^{-t/\tau}$ for discharging, with $\tau = R_{\text{eq}}C$ built from the resistance seen looking out of the capacitor's terminals with ideal batteries replaced by wires.

Five errors dominate. Letting $V_C$ snap to its final value the instant a switch closes, which deletes the entire transient. Swapping the two snapshots, so $t = 0$ is analysed with the branch open and steady state with it conducting. Building $\tau$ from whichever resistor is drawn beside the capacitor rather than the equivalent resistance it charges through. Reading the exponential as if it were linear, calling the capacitor full at one time constant or treating $\tau$ as a half-life. And crediting the capacitor with all the energy the battery supplied, when exactly half of $C\varepsilon^2$ is dissipated in the resistance no matter how small that resistance is.