An ideal parallel plate capacitor consists of two parallel plates of area a separated by a very small distance d. When this capacitor is connected to a battery that maintains a constant potential difference (V) between the plates, it stores energy.
The energy (U) stored in a capacitor can be calculated using the formula:
U = (1/2) * C * V²
In this equation, C represents the capacitance of the capacitor. For a parallel plate capacitor, the capacitance can be expressed as:
C = (ɛ₀ * a) / d
Here, ɛ₀ is the permittivity of free space. By substituting the expression for C into the energy formula, you can derive:
U = (1/2) * (ɛ₀ * a / d) * V²
This indicates that the energy stored in the capacitor is directly proportional to the area of the plates, inversely proportional to the separation of the plates, and increases with the square of the potential difference.
In summary, when the capacitor is maintained at a constant potential difference by a battery, it stores energy in an electric field created between the plates. The design of the capacitor (area and separation) plays a critical role in determining how much energy can be stored.