Distributions of line charge can be approximated by piecing together uniformly charged segments….
Distributions of line charge can be approximated by piecing together uniformly charged segments. Especially if a computer is to be used to carry out the integration by summing over the fields due to the linear elements of line charge, this provides a convenient basis for calculating the electric potential for a given line distribution of charge. In the following, you determine the potential at an arbitrary observer coordinate r due to a line charge that is uniformly distributed between the points and , as shown in Fig. P4.5.9a. The segment over which this charge (of line charge density l) is distributed is denoted by the vector a, as shown in the figure. Viewed in the plane in which the position vectors a, b, and c lie, a coordinate denoting the position along the line charge is as shown in Fig. P4.5.9b. The origin of this coordinate is at the position on the line segment collinear with a that is nearest to the observer position r.
(a) Argue that in terms of , the base and tip of the a vector are as designated in Fig. P4.5.9b along the axis.
(b) Show that the superposition integral for the potential due to the segment of line charge at ’ is
(c) Finally, show that the potential is