Potential Due to a Group of Point Charges — Question 6

Question 6

A remote cluster exposes net charge before its internal arrangement

Charges +2Q at x=a and -Q at x=-a are viewed at x=r>a. Let Q,a be positive and r much larger than a. Use infinity reference.

1. Find the exact axial potential and expand through order $a/r$.

2. Estimate the fractional correction to the net-charge approximation and find the leading criterion for it to be below one percent.

Question 6 - Solution

A remote cluster exposes net charge before its internal arrangement

(a) Method and calculation

$$V(r)=kQ\left(\frac2{r-a}-\frac1{r+a}\right)=kQ\frac{r+3a}{r^2-a^2}.$$$$V(r)=\frac{kQ}{r}\left[1+\frac{3a}{r}+O(a^2/r^2)\right].$$

(b) Interpretation and further calculation

$$V_{\mathrm{mono}}=kQ/r,\quad\frac{V-V_{\mathrm{mono}}}{V_{\mathrm{mono}}}\simeq3a/r.$$$$3a/r<.0100\Rightarrow r>300a\quad\text{to leading order}.$$

Physical check

The exact relative correction is $(3a/r+a^2/r^2)/(1-a^2/r^2)$; the leading criterion is an approximation, not a strict bound.