An uncharged parallel-plate capacitor of capacitance \(C\), a resistor of resistance \(R\), an ideal battery of potential difference \(\Delta V\), and an open switch are connected in series, as shown in the circuit diagram. At time \(t = 0\), the switch is closed. Which of the following correctly compares the energy \(U_C\) stored in the capacitor long after the switch is closed (\(t \to \infty\)) to the total energy \(E_{\text{batt}}\rangle\) supplied by the battery during this process, and provides the correct physical justification?