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452 chapter 1717.8 Electric Field Visualization (Exploration)C DPlot the equipotential surfaces on a separate 2-D plot. Start with a crude, handdrawnsketch of the electric field by drawing curves orthogonal to the equipotentiallines, beginning and ending on the boundaries (where the charges lie). The regionsof high density are regions of high electric field. Physics tells us that the electricfield E is the negative gradient of the potential:∂U(x, y) ∂U(x, y)E = −∇U(x, y)=− ˆɛ x − ˆɛ y , (17.33)∂x ∂ywhere ˆɛ i is a unit vector in the i direction. While at first it may seem that some workis involved in determining these derivatives, once you have a solution for U(x, y)on a grid, it is simple to use the central-difference approximation for the derivativeto determine the field, for example:E x ≃U(x +∆,y) − U(x − ∆,y)2∆= U i+1,j − U i−1,j. (17.34)2∆Once you have a data file representing such a vector field, it can be visualized byplotting arrows of varying lengths and directions, or with just lines (Figure 17.7).This is possible in Maple and Mathematica [L 05] or with vectors style inGnuplot 2 . where N is a normalization factor.17.9 Laplace QuizYou are given a simple Laplace-type equation∂u∂x + ∂u = −ρ(x, y),∂ywhere x and y are Cartesian spatial coordinates and ρ(x, y) is the charge density inspace.1. Develop a simple algorithm that will permit you to solve for the potentialu between two square conductors kept at fixed u, with a charge density ρbetween them.2. Make a simple sketch that shows with arrows how your algorithm works.3. Make sure to specify how you start and terminate the algorithm.2 The Gnuplot command plot "Laplace_field.dat" using 1:2:3:4 with Vectors plots variablelengtharrows at (x, y) with components Dx ∝ E x and Dy ∝ E y. You determine empiricallywhat scale factor gives you the best visualization (nonoverlapping arrows). Accordingly,you output data lines of the form (x, y, Ex/N, Ey/N)−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 452