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Prove. The energy W stored in a charged capacitor is given by

or, equivalently, by

where Q is the charge on the capacitor, V_{AB} is the potential difference between the plates, and C
is the capacitance of the capacitor.

Proof. The energy of a charged capacitor is given by the amount of work required to charge the capacitor. We assume there is no initial charge on either plate and envision the charging process as repeatedly moving small charges against the electric field from plate B of lower potential to plate A of higher potential. At the point in the process when the total charge that has been transferred is q, the potential difference between the plates is

where C is the capacitance of the capacitor.

The work dW required to transfer a small charge dq is

The total work done in the charging process, when the amount of charge that has been transferred has increased from zero to its final value of Q, is

or

Since V_{AB} = Q/C, 1) is equivalent to

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