The capacitor shown in Fig. 4-36 (P4.54) consists of two parallel dielectric layers. We wish to use energy considerations to show that the equivalent capacitance of the overall capacitor, C, is equal to the series combination of the capacitances of the individual layers, C1 and C2, namely



v (a) Let V1 and V2 be the electric potentials across the upper and lower dielectrics, respectively. What are the corresponding electric fields E1 and E2? By applying the appropriate boundary condition at the interface between the two dielectrics, obtain explicit expressions for E1 and E2 in terms of 1, 2, V, and the indicated dimensions of the capacitor.
(b) Calculate the energy stored in each of the dielectric layers and then use the sum to obtain an expression for C.
(c) Show that C is given by Eq. (4.136).

C;C2 (4.136) Ci+C2 where C1 = 81 di C2 = 2- Ip [d2 I3 73 (a) (b) Figure P4.54: (a) Capacitor with parallel dielectric layers, and (b) equivalent circuit (Problem 4.54). E, Vi E1 E, Vi Figure P4.54: (c) Electric fields inside of capacitor.