If the plate separation of a parallel plate capacitor is doubled while the capacitor remains connected to the battery, the electric field between the plates decreases, and the capacitance of the capacitor increases. However, the energy stored in the capacitor does not change instantaneously.
The energy stored in a capacitor is given by the formula:
E = (1/2) * C * V^2
where E is the energy stored, C is the capacitance, and V is the voltage across the capacitor.
When the plate separation is doubled, the capacitance (C) of the capacitor increases by a factor of two. Assuming the voltage (V) across the capacitor remains constant (since it is connected to a battery), the energy stored in the capacitor can be expressed as:
E' = (1/2) * (2C) * V^2 = 2 * (1/2) * C * V^2 = 2E
So, when the plate separation is doubled, the energy stored in the capacitor doubles. This implies that the additional energy required to store the increased charge due to the larger capacitance comes from the battery. The battery will supply the extra energy needed to maintain the voltage across the capacitor.
It's important to note that the process of changing the plate separation may involve the charging or discharging of the capacitor until it reaches a new equilibrium. Once the new equilibrium is reached, the energy stored in the capacitor will be twice the initial value.