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1. Battery is a source of energy for a circuit, whereas capacitor is a passive element, which draws energy from the circuit, store and release it. 2. Usually battery is a DC component, whereas capacitor is mostly used for AC applications. It is used to block DC components in circuits.
The relationship between capacitance, charge and potential difference is linear. For example, if the potential difference across a capacitor is halved, the quantity of charge stored by that capacitor will also be halved. For most applications, the farad is an impractically large unit of capacitance.