Potential Due to a Group of Point Charges — Question 4

Question 4

A square with alternating charges cancels both center potential and field

Four fixed charges occupy square vertices $(\pm a,\pm a)$. Charges +Q are at (a,a) and (-a,-a); charges -Q occupy the other two vertices. Use infinity reference.

1. Find V and the electric field at the center.

2. Find V on the entire z axis and explain whether these cancellations imply that the field is zero everywhere.

Question 4 - Solution

A square with alternating charges cancels both center potential and field

(a) Method and calculation

$$r_i=\sqrt2a,\quad V(0)=\frac{k}{\sqrt2a}(Q+Q-Q-Q)=0.$$$$\mathbf E(0)=-\frac{k}{(\sqrt2a)^3}\sum_i q_i\mathbf r_i=\mathbf0.$$

(b) Interpretation and further calculation

$$V(0,0,z)=\frac{k\sum_iq_i}{\sqrt{2a^2+z^2}}=0.$$$$V(x,y,0)\ne0\ \text{in general};\qquad V\to\pm\infty\ \text{near the respective sources}.$$

Physical check

Symmetry creates special cancellation sets, not an identically vanishing potential throughout space.