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 8c286c60 7348 4e9e b9d9 8789898d82fd and 56c490d3 d2f7 44cf 8d45 e4b8b2baf5cb, as shown in Fig. P4.5.9a. The segment over which this charge (of line charge density 601ee513 10f2 477a 82dd c48980e5730fl) 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 559ae19e 9032 4a6e af4f d7fc80191d90 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.

Fig. P4.5.9a,b

da7702aa 0b73 463b 9b7f d4b409a37ac8

(a) Argue that in terms of 559ae19e 9032 4a6e af4f d7fc80191d90, the base and tip of the a vector are as designated in Fig. P4.5.9b along the 559ae19e 9032 4a6e af4f d7fc80191d90 axis.

(b) Show that the superposition integral for the potential due to the segment of line charge at 03ecf5ef 14ee 496d 8444 bbf9812ee071’ is

dc09ee55 cb12 458d b6d9 55207809329e

(c) Finally, show that the potential is

db19a1db 2468 426e 9362 030e275aabeb

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