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 (see the figure). The central perpendicular axis through the ring is a z axis, with the origin at the center of the ring. What is the magnitude of the electric field due to the rod at (a) z = 0 and (b) z = ?? (c) In terms of R, at what positive value of z is that magnitude maximum? (d) If R = 2.07 cm and Q = 4.00 ?C, what is the maximum magnitude?

Solution

Electric field for a ring is given by

E = kqz/(z^2 + R^2)^1.5

A. at z = 0

E = 0

B. at z = infinite

as z tends to infinite 1/z^2 tends to zero so E = 0 in this case also.

C. value is maximum at

dE/dz = [d(kqz/(z^2+R^2)^1.5)]/dz = 0

k*q*(R^2 - 2*z^2)/(z^2+R^2)^2.5 = 0

z = R/sqrt 2 = 0.707*R

z = 0.707*R

D. electric field is given by

E = 9*10^9*4*10^-6*0.707*2.07*10^-2/((2.07^2*10^-4 + 0.707^2*2.07^2*10^-4)^1.5) = 3.23*10^7 N/C