Question

Potential of a Finite Rod

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 .

a) What is V_A, the electric potential at point A (see the figure), located a distance above the midpoint of the rod on the y axis? Express your answer in terms of L,d,q, and k.

b) What is V_B, the electric potential at point B, located at distance d from one end of the rod (on the x-axis)?

Answer 1

Part A
dq = (q/L) dx
dV = k dq/r
r = √(x² + d²)
dV = (kq/L) dx/√(x² + d²)
integrate, limits: -L/2 to L/2
VA = (kq/L) ln[√(x² + d²) +x] = (plug limits ) = (kq/L) {ln[√((L/2)² +d²) + L/2] - ln[√((L/2)² +d²) - L/2]}


Part B
dq = (q/L) dr
dV = k dq/r = (kq/L) dr/r
integrate, limits: d to d + L
VB = (kq/L) ln r = (plug limits) = (kq/L) ln[(d +L)/d}