The Electric Field Due to a Point Charge — Question 5

Question 5

A weak uniform background shifts a point-charge field null

Two equal positive charges Q are fixed at $x=\pm a$. A weak uniform background field $E_b\hat{\mathbf x}$ is added. Seek a null near $x=0$, with $|x|\ll a$. Use $(1+u)^{-2}=1-2u+O(u^2)$.

1. Expand the source field to first order and find the shifted null.

2. Express a self-consistency condition on $E_b$, and determine the shift if $E_b=.0400\,kQ/a^2$.

Question 5 - Solution

A weak uniform background shifts a point-charge field null

(a) Method and calculation

$$E_x=kQ[(a+x)^{-2}-(a-x)^{-2}]+E_b.$$$$E_x\simeq-\frac{4kQ}{a^3}x+E_b\Rightarrow x_*\simeq\frac{E_ba^3}{4kQ}.$$

(b) Interpretation and further calculation

$$|x_*|/a=\frac{|E_b|a^2}{4kQ}\ll1.$$$$E_b=.0400\,kQ/a^2\Rightarrow x_*/a\simeq.0100.$$The null shifts right so the stronger leftward contribution of the right-hand charge cancels the positive background.

Physical check

This is a local expansion, not an assertion that the approximate root is exact for large backgrounds.