 RC charging circuit

An ideal constant-voltage source $V$ charging an initially uncharged capacitor through a series resistor gives $Q(t)=CV(1-e^{-t/(RC)})$ and $I(t)=(V/R)e^{-t/(RC)}$. These follow from Kirchhoff's voltage law and $I=dQ/dt$. A smaller time constant means faster fractional settling; it does not fix the initial current or final charge unless the separate values of $R$ and $C$ are specified.