The Electric Field Due to a Point Charge — Question 6

Question 6

Removing one member of a symmetric charge ring exposes its missing contribution

Eight equal positive point charges q occupy the vertices of a regular octagon of circumradius R. One vertex lies on the positive x axis. Coulomb constant is k.

1. Find the center field before and after removing the charge at that vertex.

2. Instead of removing it, replace that charge by $-2q$. Find the new center field using a change in source charge rather than resumming all eight contributions.

Question 6 - Solution

Removing one member of a symmetric charge ring exposes its missing contribution

(a) Method and calculation

$$\mathbf E_{\mathrm{initial}}=\mathbf0,\quad\mathbf E_{\mathrm{removed\ source}}=-\frac{kq}{R^2}\hat{\mathbf x}.$$$$\mathbf E_{\mathrm{after\ removal}}=+\frac{kq}{R^2}\hat{\mathbf x}.$$

(b) Interpretation and further calculation

$$\Delta q=-2q-q=-3q.$$$$\Delta\mathbf E=-\frac{k\Delta q}{R^2}\hat{\mathbf x}=+\frac{3kq}{R^2}\hat{\mathbf x}.$$The original symmetric sum cancels, leaving only the field of the change.

Physical check

The missing contribution points opposite the field left by its removal.