A finite rod of length L has total charge q, distributed uniformly along its len

Question

A finite rod of length L has total charge q, distributed uniformly along its length. The rod lies on the x -axis and is centered at the origin. Thus one endpoint is located at (-L/2, 0), and the other is located at (L/2, 0). Define the electric potential to be zero at an infinite distance away from the rod. Throughout this problem, you may use the constant k in place of the expression 1(4 pi epsilon_0). http://session.masteringphysics.com/problemAsset/1003405/39/15783_a.jpg

Part A What is V_A, the electric potential at point A (see the figure), located a distance d above the midpoint of the rod on the y axis?

Explanation / Answer

dV = k dq/r
= k(q/L dx)/(x² + d²)


dV = V = [-L/2, L/2] kq dx/[L(x² + d²)]
= kq/L [-L/2, L/2] dx/(x² + d²)


[-L/2, L/2] dx/(x² + d²) = 2[0, L/2] dx/(x² + d²)

the indefinite integral of dx/(x² + d²) = ln[(x² + d²) + x]

V = 2kq/L ln[(x² + d²) + x], with x ranging from 0 to L/2
V= 2kq/L (ln[[(L/2)² + d²] + L/2] - ln[d²])
V = 2kq/L ln[((L/2)² + d²) + L/2] - ln |d|
= 2kq/L ln[(((L/2)² + d²) + L/2)/d]


[((L/2)² + d²) + L/2]/d
=[(L²/4 + 4d²/4) + L/2]/d
=[(1/2)(L² + 4d²) + L/2]/d
=[(L² + 4d²) + L]/[2d]


V = 2kq/L ln[((L² + 4d²) + L)/(2d)]

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