Two identical conducting plates, each having area \(A\), are located at \(z=0\) and \(z=d\). The region between plates is filled with a material having \(z\) -dependent conductivity, \(\sigma(z)=\sigma_{0} e^{-z / d}\), where \(\sigma_{0}\) is a constant. Voltage \(V_{0}\) is applied to the plate at \(z=d ;\) the plate at \(z=0\) is at zero potential. Find, in terms of the given parameters, \((a)\) the resistance of the material; \((b)\) the total current flowing between plates; ( \(c\) ) the electric field intensity \(\mathbf{E}\) within the material.

Question: Calculate the total resistance, total current, and electric field intensity within a material with the following parameters: d = 2 cm, A = 20 cm², σ₀ = 0.04 S/m, and voltage V₀ = 50 V. Step 1: Calculate total resistance R = (d(1 - e)/σ₀A) = (2(1-2.718)/0.04*20) = (2(-1.718)/0.8) = -4.295 Ω Step 2: Calculate total current I = (σ₀AV₀)/(d(1 - e)) = (0.04 * 20 * 50)/(2(1 - 2.718)) = 40/(-3.436) = -11.642 A Step 3: Calculate electric field intensity at z E(z) = (V₀)/(d(1 - e)*e^{z/d}) = (50)/(2(1 - 2.718)*e^{z/2}) (z in cm) The total resistance within the material is -4.295 Ω, the total current flowing between the plates is -11.642 A, and the electric field intensity within the material is given by E(z) = (50)/(2(1 - 2.718)*e^{z/2}) when z is measured in cm.