A thin nonconducting rod with a uniform distribution of posi

A thin nonconducting rod with a uniform distribution of positive charge Q is bent into a circle of radius R. The central perpendicular axis through the ring is a z-axis, with the origin at the canter of the ring. What is the magnitude of the electric field due to the rod at z = 0? N/C What is the magnitude of the electric field due to the rod at z = co? N/C In terms of R, at what positive value of z is that magnitude maximum? R

Solution

(a) It is clear from symmetry that the field vanishes at the center.

(b) The result (E = 0) for points infinitely far away as it goes as 1/z² as z ---> infinity or the field strength decreases as 1/r² at distant points.

c) Obtained by integration:
E = kQz/(r^2+z^2)^(3/2)
Set the first derivative of this function to zero.
Using the product rule for differentiation find:
dE/dz = kQ[ (r^2+z^2)^-(3/2) - 3z^2(r^2+z^2)^-(5/2)] = 0
multiply both sides of this equation by (r^2+z^2)^(5/2) and divide both by kQ:
dE/dz = (r^2+z^2) - 3z^2 = 0
r^2 +z^2 -3z^2 = 0
r^2 - 2z^2 = 0
r^2 = 2z^2
=>z = (+/-) r/sqrt(2)