When a capacitor is connected to a battery and the distance between its plates is increased, the energy between the plates decreases by E, while the battery gains an additional energy of 2E. The extra energy, E, is attributed to the work done by an external force during the process of pulling the plates apart. This indicates that energy is conserved, as the work input results in the observed energy changes. The discussion emphasizes the relationship between work, energy, and the behavior of capacitors in electric circuits. Understanding this principle is crucial for grasping energy dynamics in capacitive systems.