RC Circuits Right After Switching and Long After

A capacitor's voltage can't jump instantly, because that would require infinite current to dump charge onto its plates in zero time. At the moment a switch closes, an initially uncharged capacitor has zero charge, so it has zero volts across it. Zero volts across a circuit element is exactly what a plain wire gives you, so for that single instant you can replace the capacitor with a wire and solve the rest as a plain resistor circuit.

Charge keeps flowing onto the plates until the capacitor's voltage grows to match whatever voltage the rest of the circuit forces across it. Once that happens, nothing drives any more charge across the gap between the plates, so the current through that branch drops to zero. A branch carrying zero current behaves like an open switch, so long after closing, you redraw the circuit with that capacitor erased and solve again with plain resistors. Both pictures are snapshots: the messy exponential charging in between is where the actual time constant RC matters, but you don't need it to find the starting and ending currents.

This topic sits between the exponential charge and discharge equations and full RC network analysis. The two limiting pictures, wire at t = 0 and open gap at t = infinity, are the fast checks you run before or instead of grinding through the exponential formulas, and they also let you sanity check the extreme values predicted by those equations.