Answer to Question #299189 in Electricity and Magnetism for Sim

Question

Assume a point charge Q1 is at x=0 and a charge Q2 is at x= d. What is the relationship between these charges if the resulting electric field is zero at the following points:

a) x= d/2; b) x= 2d; c) x= -d/2

Answer 1

First let us remind the expression of the total field at the point $x$ :

$E= \frac{1}{4\pi\epsilon_0} (\frac{Q_1 x}{|x|^3}+\frac{Q_2 (x-d)}{|x-d|^3})$

1. The first case gives us $\frac{Q_1}{d^2/4}-\frac{Q_2}{d^2/4}=0$ which yields $Q_1=Q_2$
2. The second case gives $\frac{Q_1}{4d^2}+\frac{Q_2}{d^2}=0$ which yields $Q_1=-4Q_2$
3. The third case gives $\frac{-Q_1}{d^2/4}+\frac{-Q_2}{9d^2/4}=0$ which yields $Q_1=-Q_2/9$