APPLICATION OF GAUSS LAW

ENERGY DENSITY STORED BETWEEN THE PLATE OF THE CAPACITOR

Suppose the conductor is charged gradually at any stage , the charge on the conductor is q

Potential difference (v) = q/c

Small amount of work done in given an additional charge dq to the conductor

                       dw = q/c × dq

Total work done in giving a charge Q to the conductor  is under limit q=Q,q=0

                        W =  ∫ q/c dq  = 1/c [q²/2]^q=Q_q=0

                         W = 1/c × Q²/2

 Energy stored in the conductor

                       U = W = 1/c × Q²/2

                    As    Q = cV

                    So   U = 1/2 × (cV) ²/c = 1/2 cV²

                                   cV = Q ,            W = ½ QV

Energy density is defined as total energy stored per unit volume of the conductor

            u = total energy / volume

               = 1/2 cV²/Ad

            c = є₀A/d  and  V = E . d

               =  ½  (є₀A/d  ) (E² d²/Ad)

       Energy density = 1/2 є₀ E²