Energy stored in a Charged capacitor

Let q’ be the charge and V’ the potential difference established between the plates of the capacitor at any instant during the process of charging. If an additional charge dq’ is given to the plates, the work done by the battery is given by

    dW = V’ dq’ = (q’ / C) dq’            (. : V’ = q’ / C)

Total work done to charge a capacitor to a charge q is

    W = ∫ dW = ∫^q q’ / C dq’ = 1 q²
                _____C                  2 C

The work done is stored as electrostatic potential energy in the capacitor.

. :     U = 1 q₂     = 1 CV²                 (. : q = CV)
       ___2 C           2

This energy can be recovered if the capacitor is allowed to discharge.

Energy Density. Consider a parallel plate capacitor of area A and plate separation d.

Energy of the capacitor = U = 1 CV² = 1 (ε₀A/d) V²
                  ______________   2            2

Volume of the space between the plates = Ad.
Energy density u is the potential energy per unit volume.

. :    u = U = 1 (ε₀A V²) x 1 = 1 ε₀ (V / d) ²
        __ Ad  2    d            Ad   2

. :    u = 1 ε₀ E²                 (. : V/d E)
              2

Thus we can associate in electrostatic energy density u = 1/2 ε₀ E² with every point in space where an electric field E exists.

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