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450 chapter 17the difference between the potential near the top and the bottom surfacesof the plates). Such being the case, we solve Laplace’s equation (17.4) muchas before to determine U(x, y). Once we have U(x, y), we substitute it intoPoisson’s equation (17.3) and determine how the charge density distributesitself along the top and bottom surfaces of the plates. Hint: Since the electricfield is no longer uniform, we know that the charge distribution alsowill no longer be uniform. In addition, since the electric field now extendsbeyond the ends of the capacitor and since field lines begin and end oncharge, some charge may end up on the edges and outer surfaces of the plates(Figure 17.4).4. The numerical solution to our PDE can be applied to arbitrary boundaryconditions. Two boundary conditions to explore are triangular andsinusoidal:⎧⎨200x/w,for x ≤ w/2,U(x)=⎩100(1 − x/w), for x ≥ w/2,U(x) = 100 sin( ) 2πx.w5. Square conductors: You have designed a piece of equipment consisting ofa small metal box at 100 V within a larger grounded one (Figure 17.8). Youfind that sparking occurs between the boxes, which means that the electricfield is too large. You need to determine where the field is greatest so that youcan change the geometry and eliminate the sparking. Modify the programto satisfy these boundary conditions and to determine the field between theboxes. Gauss’s law tells us that the field vanishes within the inner box becauseit contains no charge. Plot the potential and equipotential surfaces and sketchin the electric field lines. Deduce where the electric field is most intense andtry redesigning the equipment to reduce the field.6. Cracked cylindrical capacitor: You have designed the cylindrical capacitorcontaining a long outer cylinder surrounding a thin inner cylinder (Figure 17.8right). The cylinders have a small crack in them in order to connect them tothe battery that maintains the inner cylinder at −100 V and outer cylinder at100 V. Determine how this small crack affects the field configuration. In orderfor a unique solution to exist for this problem, place both cylinders within alarge grounded box. Note that since our algorithm is based on expansion ofthe Laplacian in rectangular coordinates, you cannot just convert it to a radialand angle grid.17.7 Implementation and Assessment1. Write or modify the CD program to find the electric potential for a capacitorwithin a grounded box. Use the labeling scheme on the left in Figure 17.4.2. To start, have your program undertake 1000 iterations and then quit. Duringdebugging, examine how the potential changes in some key locations as youiterate toward a solution.−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 450