Formulation of the Discrete Model

We begin our discussion by exploring a discrete model which is used to formulate the expression describing the potential at P(x,y), due to the 1-D charged wire.

Let's assume that the interval [-1, 1] is subdivided into 2n equally sized segments: (n segments below the origin and n above). Note that the width of each subinterval is , since the length of the wire is l = 2 units. For the sake of consistency with the remaining presentation of this module, we select a charge density function of the form: (power dependence position law).

Without loss of generality, we select k=1 and we define our task to be the calculation of the scaled potential at the point P(1/2,1/2). Focusing on the i^th subinterval (of width ), the distance between one of its endpoints and the origin is . Also, the distance r between the points P(1/2,1/2) and is given by:

The total scaled potential at P(1/2,1/2) is obtained by superimposing the potentials from all segments of length along the wire. Thus an approximation for V_s(1/2,1/2) can be expressed by :

It is evident that, in order to obtain a more "accurate" numeric value for V_s(1/2,1/2) we must use finer partitions for the interval [-1, 1]. Extending this process to the limiting case, we must examine the situation where (i.e., the width of each subinterval tends to zero), and try to calculate the previous sum as n tends to infinity. Such calculations are tedious to be done by hand and so a computer algebra system should be invoked. You may wish to use Maple or Mathematica.

• Solution Using Maple
• Solution Using Mathematica