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Separation of Variables in Cartesian Coordinates Developed by J. D. McDonnell This set of exercises will guide the student through solving Laplace's equation for the electric potential in Cartesian coordinates via separation of variables. They will perform numerical integration and produce plots of the electric potential for situations with non-trivial boundary conditions. Exercises Exercise 1: A Two-Dimensional Case As a "warm-up", consider the following two-dimensional situation. A squared 'c'-shaped "slot" is set up, where the two parallel horizontal pieces extend from 𝑥 = 0 to 𝑥 → ∞, and the vertical connecting piece sits at 𝑥 = 0 and extends from 𝑦 = 0 to 𝑦 = 𝑎. Both horizontal pieces are grounded, and the vertical piece is held to a potential 𝑉(𝑥 = 0, 𝑦) = 𝑉0 (𝑦), where 𝑉0 (𝑦) is a function to be specified. First, set up the solution of Laplace's equation with the specified boundary conditions. Generally, it will be an infinite series. Your solution, which must be quite generic until we specify 𝑉0 (𝑦), should include an integral in terms of 𝑉0 (𝑦). For simple forms of 𝑉0 (𝑦), the integral can be done by hand. But for "interesting" forms of 𝑉0 (𝑦), it is valuable to evaluate this integral with numerical techniques. It will be impossible to evaluate every term in an infinite series - you must choose a sufficient number of terms to keep. In each example below, experiment to see how many terms are necessary to capture the solution. You might try 𝑁 = 5, 𝑁 = 10, 𝑁 = 20 …. For each of the 𝑉0 (𝑦) functions specified below, (1) evaluate your solutions for 𝑉(𝑥, 𝑦) numerically; (2) produce a contour plot of 𝑉(𝑥, 𝑦); and (3) describe your solution in physical terms - for example, does the behavior of the potential match your expectations? 3𝜋𝑦 • 𝑉0 (𝑦) = 6.0sin ( • • checking your numerical method. 𝑉0 (𝑦) = −𝑦 2 + 𝑎𝑦. Note: This example can also be evaluated by hand... 𝑎 𝑉0 (𝑦) = sinh(𝑦 − 2). 𝑎 ). Note: This example can be easily evaluated by hand, as a way of Exercise 2: A Three-Dimensional Case For a three-dimensional case, the same overall scheme allows us to solve Laplace's equation in Cartesian coordinates. Consider a semi-infinite "pipe": at 𝑥 = 0 there is a rectangular plate held at a potential (to be specified later) 𝑉(𝑥 = 0, 𝑦, 𝑧) = 𝑉0 (𝑦, 𝑧). Four infinitely long plates are joined to the four edges of the first plate, each extending from 𝑥 = 0 to 𝑥 → ∞. The four infinitely long plates are grounded. Dimensions of the semi-infinite "pipe": • • • From 𝑥 = 0 to 𝑥 → ∞. From 𝑦 = 0 to 𝑦 = 𝑎. From 𝑧 = 0 to 𝑧 = 𝑏. Again, set up the solution of Laplace's equation with the specified boundary conditions. Generally, it will be an infinite series. Similarly to the two-dimensional case, you will encounter integrals in terms of 𝑉0 (𝑦, 𝑧), but now they are double integrals over both 𝑦 and 𝑧! For each of the 𝑉0 (𝑦, 𝑧) functions specified below, (1) evaluate your solutions for 𝑉(𝑥, 𝑦, 𝑧) numerically; (2) produce a contour plot of 𝑉(𝑥, 𝑦, 𝑧), in different cross sections for constant 𝑧; and (3) describe your solution in physical terms - for example, does the behavior of the potential match your expectations? • 𝑉0 (𝑦, 𝑧) = 𝑦. • 𝑉0 (𝑦, 𝑧) = −4𝑦 2 + 4𝑎𝑦 − 𝑧 2 + 𝑏𝑧 + 4 𝑎𝑏 𝑦𝑧. 3 • 𝑎 𝑏 𝑉0 (𝑦, 𝑧) = sinh ((𝑦 − 2) (𝑧 − 2)).