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Electrodynamics (I): Homework 4 Due: October 30, 2014 Exercises in Griffiths 3.3 (hand-in 5%), 3.7, 3.8, 3.10(hand in 10%), 3.11(hand-in 5%), 3.16, 3.39(hand-in 10%), 3.41(hand in 10%), 3.42(hand in 10%), 3.47 Ex.1 hand-in Consider an array of infinitely Ph501 long wires that3,lieProblem on the xz 1 plane. inceton University 1999 Set Each wire carries a charg 1 density λ. The direction of wires are all in parallel to the z axis and are equally spaced in x direction with locations being y = 0 and x = na, n = 0, ±1, ±2, · · ·, as shown in the 1. A grid of inﬁnitely long wires is located in the (x, y) plane at y = 0, x = ±na, n = 0, 1, 2, . . ..following Each figure. line carries charge λ per unit length. Obtain a series expansion for the potential φ(x, y). Show that for large y the ﬁeld is just ∞ n z that due to a plane of charge density λ/a. By noting that = − ln(1 − z) n n=1 (a) 5 % Using method of separation of variables, find fundamental solutions of the potentail for z complex or real, sum the series to show V (x, y) for y > 0 and y < 0. 2πx (b)10 % Consider general2πy solutions constructed by fundamental solutions −2πy/a −4πy/a obtained in φ(x, y) = −λ + ln 1 − 2e cos +e a a (a). Using appropriate boundary condition at y = 0, find V (x, y) as the summation of a 2πy 2πx mathematical series. Resum series to find−the = −λ ln the 2 cosh . of V (x, y). cosfunction form a a (1) Show that the equipotentials are circles for small x and y, as if each wire were alone. 1