Develop postprocessing for the 1-D FEM algorithm of Problem 8.7 to compute the capacitance of an…

Develop postprocessing for the 1-D FEM algorithm of Problem 8.7 to compute the capacitance of an inhomogeneous dielectric-filled parallel plate capacitor, with one plate at x = a and the other at x = b (see Figure 8.9). Neglecting fringing fields at the plate edges effectively makes the problem one-dimensional. (a) Convert the potential to electric field intensity, using a finite difference approximation for the derivative in (4.65). (b) Convert the electric field intensity to electric flux density using the constitutive relation (2.2a). (c) By Gauss’s law, the electric charge stored on one

»Develop postprocessing for the 1-D FEM algorithm of Problem 8.7 to compute the capacitance of an inhomogeneous dielectric-filled parallel plate capacitor, with one plate at x = a and the other at x = b (see Figure 8.9). Neglecting fringing fields at the plate edges effectively makes the problem one-dimensional. (a) Convert the potential to electric field intensity, using a finite difference approximation for the derivative in (4.65). (b) Convert the electric field intensity to electric flux density using the constitutive relation (2.2a). (c) By Gauss’s law, the electric charge stored on one plate is equal to the integral of the electric flux density over a closed surface enclosing the plate. Since the electric field is small outside the region between the plates, this integral can be evaluated by multiplying the flux density by an arbitrary plate area A. From the stored charge, use the relationship C = Q/V to find the capacitance. (d) Check the code by verifying the capacitance computed for a parallel plate capacitor filled with a homogeneous dielectric (e = constant). (e) Compare the capacitance obtained using this method for the two-layer dielectricfilled parallel plate capacitor of Problem 8.6(b) with the exact result obtained from the series combination of two parallel plate capacitors with homogeneous dielectrics, one for each dielectric layer.

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