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A problem set for phy 712 students, focusing on solving the poisson equation for a one-dimensional charge distribution using given boundary conditions. Students are required to find the electrostatic potential φ(x) and the corresponding electrostatic field e(x), as well as plot both functions. The problem is discussed in terms of gauss's law arguments.
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January 25, 2008
PHY 712 – Problem Set #
ρ(x) =
0 for x ≤ −a/ 2 ρ 0 x/a for −a/ 2 ≤ x ≤ a/ 2 0 for x ≥ a/ 2 , where ρ 0 and a are constants. (a) Solve the Poisson equation for the electrostatic potential Φ(x) with the boundary conditions Φ(−a/2) = 0 and dΦ(−a/2)/dx = 0. (b) Find the corresponding electrostatic field E(x). (c) Plot Φ(x) and E(x). (d) Discuss your results in terms of elementary Gauss’s Law arguments.