Potential Due to a Group of Point Charges — Question 10

Question 10

A symmetric neutral triplet begins with an inverse-cube axial potential

Charges +Q are at x=$\pm a$ and -2Q is at x=0. At x=r>a use infinity reference.

1. Combine the three terms into one exact potential.

2. Find the leading far-field potential and compare its radial decay with a single charge and a dipole.

Question 10 - Solution

A symmetric neutral triplet begins with an inverse-cube axial potential

(a) Method and calculation

$$V(r)=kQ\left(\frac1{r-a}+\frac1{r+a}-\frac2r\right).$$$$V(r)=\frac{2kQa^2}{r(r^2-a^2)}.$$

(b) Interpretation and further calculation

$$r\gg a:\quad V(r)=\frac{2kQa^2}{r^3}\left[1+O(a^2/r^2)\right].$$$$V_{\mathrm{monopole}}\propto r^{-1},\quad V_{\mathrm{dipole}}\propto r^{-2},\quad V_{\mathrm{triplet}}\propto r^{-3}.$$

Physical check

The net charge and dipole moment both vanish, cancelling the first two far-field orders.