The Electric Field Due to an Electric Dipole — Question 8

Question 8

Zero total dipole moment can leave a nonzero higher-order field

Charges $+q,-2q,+q$ lie at $x=-a,0,+a$. Use Coulomb constant k and observe at $x>a$.

1. Find net charge and dipole moment about the origin, then derive the exact axial field.

2. Using $(1\pm u)^{-2}=1\mp2u+3u^2+O(u^3)$, find the leading far-field term and explain why zero dipole moment does not mean zero field.

Question 8 - Solution

Zero total dipole moment can leave a nonzero higher-order field

(a) Method and calculation

$$Q_{\mathrm{total}}=q-2q+q=0,\quad p_x=q(-a)+q(a)=0.$$$$E_x=kq[(x+a)^{-2}-2x^{-2}+(x-a)^{-2}].$$

(b) Interpretation and further calculation

$$x\gg a:\quad E_x=\frac{kq}{x^2}[6(a/x)^2+O((a/x)^4)].$$$$E_x\simeq6kqa^2/x^4.$$The charge and dipole terms cancel, exposing a faster-decaying spatial moment of the distribution.

Physical check

Vanishing leading moments constrain distant behavior; they do not eliminate all local fields.