The Electric Field Due to a Line of Charge — Question 8

Question 8

A ring field on its axis follows from projection of every charge element

A uniformly charged thin ring of radius R and total charge Q lies in the xy plane. Find its field on the z axis for $z>0$. Use Coulomb constant k.

1. Derive the axial field by integrating charge elements at their common distance.

2. Find its small-z and large-z limits and explain why replacing the ring by a central point charge fails near the center.

Question 8 - Solution

A ring field on its axis follows from projection of every charge element

(a) Method and calculation

$$r=\sqrt{R^2+z^2},\quad dE_z=\frac{k\,dq}{r^2}\frac zr.$$$$E_x=E_y=0,\quad E_z=\frac{kQz}{(R^2+z^2)^{3/2}}.$$

(b) Interpretation and further calculation

$$z\ll R:\quad E_z\simeq(kQ/R^3)z\to0.$$$$z\gg R:\quad E_z\simeq kQ/z^2.$$A point replacement diverges as $z\to0$, while the actual ring field vanishes by symmetry.

Physical check

All ring elements are equally far from P, but the axial projection factor is still required.