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Introduction to Numerical Electrostatics Using MATLAB by Lawrence N. Dworsky

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8The Finite Difference Method

8.1 Introduction and a Simple Example

We begin this chapter by examining a two-dimensional (2d) rectangular structure. Extending the analysis to three-dimensional (3d) structures in rectangular, cylindrical, or spherical coordinates is straightforward and will be discussed later. Also, for now consider only prescribed voltage boundary conditions.

Laplace’s equation in rectangular coordinates is

(8.1)images

For a uniform structure infinitely long in the Z direction, all partial derivatives with respect to Z vanish, leaving us with the 2d equation

(8.2)images

Figure 8.1 shows a rectangular grid of points in 2d space.

c8-fig-0001

FIGURE 8.1 A rectangular grid of points.

The lines in the figure are just for our eyes to follow easily. Of interest are the intersections of these lines, which we’ll call nodes. The X-axis separation of nodes is hx [equation (8.3)]; the Y-axis separation of nodes is hy [equation (8.4)]:

For now, assume that hx  = hy  = h.

Voltages at the nodes are ...

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