An air-filled parallel-plate capacitor with initial capacitance \(C_0\) is connected to a battery that maintains a constant potential difference \(V_0\). The initial plate separation is \(d\). Two experimental procedures are conducted:

Case 1: The capacitor remains connected to the battery while an external agent slowly pulls the plates apart to a final separation of \(2d\).

Case 2: The capacitor is disconnected from the battery, and then an external agent slowly pulls the plates apart to a final separation of \(2d\).

If \(W_1\) and \(W_2\) represent the work done by the external agent in Case 1 and Case 2, respectively, what is the ratio \(\dfrac{W_2}{W_1}\)?