A parallel plate capacitor, when connected to a battery, develops a potential difference across its plates. This potential difference is key to the operation of the capacitor, as it determines how much electrical energy the capacitor can store.

By integrating the equation that relates voltage and current in a capacitor, one can derive an equation for the voltage across the capacitor at any given time. This equation is crucial in understanding and predicting the behavior of capacitors in electronic circuits.

One interesting property of capacitors is that they possess a kind of memory. The voltage across a capacitor at any moment depends on the past flow of current through it. This means that capacitors can "remember" their charging and discharging history, which can be useful in various applications such as memory storage in computers.

The instantaneous power delivered to a capacitor can be used to determine the amount of energy stored in the capacitor. If we consider an uncharged capacitor at time equals minus infinity, it has zero voltage. This means that the energy stored in the capacitor can be determined in terms of charge and capacitance. This represents the energy present in the electric field between the plates.

This stored energy can be retrieved in terms of power since an ideal capacitor does not dissipate energy. However, real-world capacitors are not ideal. A non-ideal capacitor has a parallel-model leakage resistance, but this is usually high enough to be neglected in most practical applications.

A unique characteristic of capacitors is that they act as an open circuit to direct current (DC) voltage but can get charged when connected to a battery. This property allows capacitors to block DC while letting alternating current (AC) pass.

Another important feature of capacitors is that the voltage across them is always continuous and cannot change abruptly. This behavior is essential in many applications, such as smoothing out voltage in power supplies and filtering out noise in signal processing.

In conclusion, understanding capacitors, their properties, and their behavior in circuits is a fundamental aspect of electronics. It enables engineers to design and construct complex electronic systems that are integral to modern life.