Finite Length Wire

Assume that a charged wire resides on the y-axis in the interval [-1, 1]. The problem is to calculate the electrostatic potential at a given (x,y) point. We will examine the formulation of a discrete as well as a continuous model. A Computer Algebra System (Maple/ Mathematica) is essential for the solution of the discrete model. On the other hand, various methods of attack will be used to present a solution to the continuous model. Such methods use: a) Fortran/C code to implement elementary quadrature rules (Trapezoidal and Simpson's rule), b) The symbolic, numeric and graphical capabilities of a Computer Algebra System.

Let's take a closer look to the initial stages of the model formulation. Assume that is a function that describes the charge distribution per unit length along the wire. For example, if we assume that a constant charge is uniformly distributed along the wire, then the charge distribution function is defined as:

Our task is to compute the potential at any point (x,y) in space, not including the wire itself. Let's see how we can formulate the problem and express the potential at a point P(x,y).

According to Coulomb's law of electrostatic interactions, the potential at a point P(x,y) due to a charged element of length will be:

where r is the distance between P(x,y) and the charged element . Note that the charge of the element is .

The derivation of the expression for the total potential at a point P(x,y) depends on the choice of the model. The formulation of two models is considered in this module:

Formulation of the Discrete Model and Formulation of the Continuous Model

• Formulation of the Discrete Model
  • Solution Using Maple
  • Solution Using Mathematica
• Formulation of the Continuous Model