Potential Due to a Point Charge — Question 10

Question 10

A distant point source looks locally like a constant plus a uniform field

A positive point charge Q is at x=-R. For $|x|\ll R$, observations lie near the origin on the x axis. Use $V(x)=kQ/(R+x)$ with infinity reference.

1. Expand V through second order in x/R and identify the locally uniform field term.

2. Find the relative error of the linear approximation at x=.100R, using the exact potential as denominator.

Question 10 - Solution

A distant point source looks locally like a constant plus a uniform field

(a) Method and calculation

$$V(x)=\frac{kQ}{R}\left[1-\frac xR+\frac{x^2}{R^2}+O((x/R)^3)\right].$$$$V_{\mathrm{lin}}=\frac{kQ}{R}-\frac{kQ}{R^2}x,\quad E_{\mathrm{local}}=kQ/R^2.$$

(b) Interpretation and further calculation

$$\frac{V_{\mathrm{lin}}-V_{\mathrm{exact}}}{V_{\mathrm{exact}}}=(1-x/R)(1+x/R)-1=-x^2/R^2.$$$$x=.100R:\quad\text{relative error}=-.0100=-1.00\%.$$

Physical check

The large constant voltage is reference-dependent; the linear term supplies the leading measurable local field.