Charges remain on opposite sheets after the battery is removed because they are trapped on the surface of the conductors due to charge conservation and lack of a path to dissipate. The electric field established between charged plates keeps individual charges in equilibrium. The battery's work during charging ensures that charges stay separated even after disconnection.
Why Charges Remain on Sheets After Battery is Removed
When a battery is used to charge two metallic plates, one plate is connected to the positive terminal and becomes positively charged, while the other plate is connected to the negative terminal, becoming negatively charged. This setup creates an electric field between the plates. If we imagine placing a free-body diagram for a single charge in the center of each sheet, the charge would experience an equal and opposite force from each side of the sheet, thus remaining in equilibrium.
The reason the charges remain on the plates even after the battery is removed is due to the principle of charge conservation and the nature of conductors. On a charged conductor, excess charges reside on the surface to minimize repulsive forces. Once the charging process is finished and the battery is disconnected, there is no path for these charges to leave the plates, and they remain because of this separation of charges created by the battery's chemical reaction.
The concept can be further understood through the process of charging the plates, as described in the question. Initially, it requires very little work to place the first opposite charges on each plate. As more charges are added, they repel each other, requiring more work to be done by the battery to place additional charges on the plates. The work done by the battery converts chemical potential energy into the electrical potential energy that holds the oppositely charged plates in this state even after disconnecting from the battery.
The diagram is attached below.