Formulation of the Continuous Model

The total potential at the point P(x, y) is obtained by superimposing the potentials from all segments along the wire. In the limiting case where the length of the segment shrinks to zero, the contribution to the potential at P(x, y) due to the infinitesimal segment will be:

Therefore, the total potential, obtained by integration, is given by:

A more general charge density function could be defined as , where k is a non-negative integer. We say that follows the power dependence position law. Equation (2) becomes:

Equations (2) and (3) do not express the "actual" potential at a point P(x, y). For a complete calculation of the potential, the right-hand side of the above formulae should be multiplied by a constant , where

Examining in detail the specific case where the charged wire occupies the interval [-1, 1] does not constitute a restricted case of the "1-D finite length wire" problem; one may readily convert any 1-D finite length wire problem to the [-1, 1] case. This can be demonstated by the following argument: Let be a function representing the potential dV at the point P(x, y) due to an element of a charged wire residing on the interval [a,b] on the y-axis. The potential at a given point can be expressed by the integral:

By introducing the substitution we obtain the following "mapping":

where . Note that the new integral (involving ) is of the same "difficulty" as the one given in equation (3).

In case you need to review some ideas about basic quadrature rules (left rectangle, right rectangle and midpoint rule), please see:

Overview of Three Basic Quadrature Rules

For the remainder of the discussion in this module, we will concentrate on the various methods for evaluating the integral(s) involved in obtaining the scaled potential at a given point, which is defined as:

The solution process could be explored using any of the following methods:

• Solution Using the Trapezoidal Rule
• Solution Using Simpson's Rule
• Solution Using Maple
• Solution Using Mathematica