3 A Ring Revisited [15 Points]
Problem: Consider a uniform ring of charge of radius R and constant linear charge density $\lambda$; the ring is lying in the $(x, y)$-plane, and the z-axis passes through its center, as shown below.
$V(z)$
$z$
$R$
$\theta$
$d\theta$
We employ the usual convention in which the electric potential is zero at infinity.
(a) [5 Points] Compute the electric potential $V(z)$ at the indicated point $(x, y, z)=(0,0, z)$ on the z-axis.
(b) [5 Points] What amount of work $W$ must be done to a point charge $q$ to bring it from infinity to the center of the ring at the origin?
(c) [5 Points] Directly from the result of part (a), compute the electric field $\vec{E}(z)$ at the indicated point $(x, y, z)=(0,0, z)$. Does your answer agree with the result found in Recitation 2?
