458 the capacitor shown in fig p458 consists of two parallel diele9456

Question

4.58 The capacitor shown in Fig. P4.58 consists of two parallel dielectric layers. 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, C¡ and C2, namely C=\frac{C_{1} C_{2}}{C_{1}+C_{2}} where C_{1}=\varepsilon_{1} \frac{A}{d_{1}}, \quad C_{2}=\varepsilon_{2} \frac{A}{d_{2}} (a) Let V1 and V½ be the electric potentials across the upper and lower dielectrics, respectively. What are the cor-responding 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 ɛ1, ɛ2, and 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. (4.149).