Electric Field Near a Straight Line Charge

In this example, our goal is to calculate the electric field, , at any point produced by a charged wire whose length is and has a charge density (charge/length), . We assume that the charge is spread uniformly along the wire, e.g. is a constant. The geometry is sketched in Figure given right.

The wire can be divided into small segments each has length and carries the charge as shown in figure given left. The electric field, , produced by this segment, at a distance from it, is calculated from: (1) where is a constant and has the value: (2)

and is the unit vector in the direction of the vector . Putting , Equation (1) becomes:

(3)

By using vector a operation, the vector can be written in terms of position vector such that:

(4)

and the length of can be expressed as a function of geometric parameters:

(5)

The charge of the segment, , can be written as:

(6)

Subsitution of Equations (4), (5) and (6) into Equation (3) yields:

(7)

The total electric field, , is the integral of electric field over the wire. So, the componets of are found by evaluating the following integrals:

(8)

(9)

In our program, we will evaluate these integrals numerially. Integrating a function can be calculated by Simpson's method. The approximation to the integral is:

(10)

where is a finite length of . Note that numerical errors can be reduced by making smaller. The Simpson method in Equation (10) is used to solve the integrals in Equation (8) and (9).