Chapter 23, Problem 40AP

Show that the maximum magnitude E_max of the electric field along the axis of a uniformly charged ring occurs at $x=a/\sqrt{2}$ (see Fig. 23.3) and has the value $Q/(6\sqrt{3}\pi {ϵ}_0{a}^2)$.

Figure 23.3 (Example 23.2) A uniformly charged ring of radius c. (a) The field at P on the x axis due to an element of charge dq. (b) The perpendicular component of the field at P due to segment 1 is canceled by the perpendicular component due to segment 2.