The Method of Moments in Electromagnetics

34 The Method of Moments in Electromagnetics2aLx^Figure 3.1: Thin wire dimensions. xr mr mn^xr nx 1x 2x 3x N-1x NFigure 3.2: Thin wire segmentation.3.1.1 Charged WireConsider a thin, conducting wire of length Ä and radius oriented along the Ü axis,as shown in Figure 3.1. If the radius of the wire is very small compared to the length( Ä), the electric potential on the wire can be expressed via the integral Ä ´rµ ¼Õ ´Ü ¼ µ¯r r ¼ ܼ (3.2)whereÔr r ¼ ´Ü Ü ¼ µ ¾ ·´Ý Ý ¼ µ ¾ (3.3)With the intent of converting (3.2) into a linear system of equations, let us subdividethe wire into Æ subsegments, each of length ¡ Ü , as shown in Figure 3.2. Within eachsubsegment, we assume that the charge density has a constant value so that Õ ´Ü ¼ µ ispiecewise constant over the length of the wire. Mathematically, we write this asÕ ´Ü ¼ µÆÒ½ Ò Ò´Ü ¼ µ (3.4)where Ò are unknown weighting coefficients, and Ò´Ü ¼ µ is a set of pulse functionsthat are constant on one segment but zero on all other segments, i.e. Ò´Ü ¼ µ¼ Ü ¼ ´Ò ½µ¡ ܽ ´Ò ½µ¡ Ü Ü ¼ Ò¡ Ü(3.5)¼ Ü ¼ Ò¡ ÜLet us now assign the potential on the wire a value of ½V. Substituting (3.4)into (3.2) then yields½ ļÆÒ½ Ò Ò´Ü ¼ ½µ¯r r ¼ ܼ (3.6)