A circuit is constructed using an ideal battery with potential difference \(\Delta V_0\), an open switch, a resistor of resistance \(R\), and an initially uncharged capacitor of capacitance \(C\), all connected in series. The switch is closed at time \(t = 0\). Which of the following correctly describes the current \(I\) through the resistor and the potential difference \(V_C\) across the capacitor as functions of time \(t\) for \(t > 0\), and provides a valid justification?