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Let us perform an example 2d 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
.
Figure 67:
Contour plot of the electric potential generated by
a charged wire placed at the center of a grounded
rectangular channel. The wire is located at
, whereas the
channel walls are at , , , and .
Calculation performed with .

Note that when discretizing Eq. (185) the righthand side becomes

(186) 
on the gridpoint closest to the wire, with
on the remaining gridpoints.
Here, and are the grid spacings in the  and
 directions, respectively.
Figure 68:
Vector plot showing the direction of the electric field generated by a charged wire placed at the center of a grounded
rectangular channel. The wire is located at
, whereas the
channel walls are at , , , and . Calculation performed with .

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 2d Poisson solver using
.
Figure 69:
Contour plot of the electric potential generated by
a charged wire offset from the center of a grounded
rectangular channel. The wire is located at
, whereas the
channel walls are at , , , and . Calculation performed with .

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 2d Poisson solver using
.
Figure 70:
Vector plot showing the direction of the electric field generated by a charged wire
offset from the center of a grounded
rectangular channel. The wire is located at
, whereas the
channel walls are at , , , and . Calculation performed with .

Next: 3d problems
Up: Poisson's equation
Previous: An example solution of
Richard Fitzpatrick
20060329