(a) Write down the integral and differential forms of Gauss' law in a dielectric, defining all quantities used. (b) A parallel plate capacitor is completely filled with a nonconducting dielectric. Show that the electric displacement, D, is uniform between the plates and calculate its value. (You may assume that the plates each have area A and are separated by a small distance d . Each plate carries a surface charge density s C/m2 .) (c) The dielectric has a nonuniform relative permittivity K(x) = ax + b where a and b are constants and x is the perpendicular distance from one plate. Using Gauss' law, show that the electric field between the plates satisfies E(x)=E0/K(x) where E0 is a constant. Find the value of E0. (d) Show that the voltage across the capacitor is given by V=E0/e0aln(1+Qd/b) and calculate the capacitance. (e) Find the volume polarization charge density in the diele
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