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## Example 2-d electrostatic calculation

Let us perform an example 2-d electrostatic calculation. Consider a charged wire running parallel to the axis of a uniform, hollow, rectangular, conducting channel. Suppose that the vertices of the channel lie at , , , and . Suppose, further, that the wire carries a uniform charge per unit length of magnitude unity. The electric potential inside the channel satisfies [see Eq. (110)] (185)

where are the coordinates of the wire. Here, we have conveniently normalized our units such that the factor is absorbed into the normalization. Assuming that the box is grounded, the potential is subject to the Dirichlet boundary conditions at , , , and . We require the solution in the region and . Note that when discretizing Eq. (185) the right-hand side becomes (186)

on the grid-point closest to the wire, with on the remaining grid-points. Here, and are the grid spacings in the - and - directions, respectively. Figures 67 and 68 show the electric potential and electric field generated by a wire placed at the center of the channel: i.e., . The calculation was performed with the previously listed 2-d Poisson solver using . Figures 69 and 70 show the electric potential and electric field generated by a wire offset from the center of the channel: i.e., . The calculation was performed with the previously listed 2-d Poisson solver using .    Next: 3-d problems Up: Poisson's equation Previous: An example solution of
Richard Fitzpatrick 2006-03-29