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pdes for electrostatics & heat flow 4491000V(x, y)100806040200-20-40-60-80w100 Vd–100 V (X, Y)LL–10002040y6080100 02040x6080100Figure 17.4 Left: A simple model of a parallel-plate capacitor within a box. A realistic modelwould have the plates close together, in order to condense the field, and the enclosinggrounded box so large that it has no effect on the field near the capacitor. Right: A numericalsolution for the electric potential for this geometry. The projection on the xy plane gives theequipotential lines.+100 V++ + + + + + + + + + + + + + + + + + + + + + + +− − − − − − − − − − − − − − − − − − − − − −−-100V−Figure 17.5 A guess as to how charge may rearrange itself on finite conducting plates.given program to solve Laplace’s equation such that the plates have fixedvoltages.2. For the next version of this problem, assume that the plates are composed of aline of dielectric material with uniform charge densities ρ on the top and −ρ onthe bottom. Solve Poisson’s equation (17.3) in the region including the plates,and Laplace’s equation elsewhere. Experiment until you find a numericalvalue for ρ that gives a potential similar to that shown in Figure 17.6 forplates with fixed voltages.3. For the final version of this problem investigate how the charges on a capacitorwith finite-thickness conducting plates (Figure 17.5) distribute themselves.Since the plates are conductors, they are still equipotential surfaces at 100and −100 V, only now they have a thickness of at least 2∆ (so we can see−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 449