Question

Two point charges +4q and +q are separated by distance r, where should a third point charge Q be placed so that the whole system remains in equilibrium?

Answer 1

Let place the charge Q on the line between +4q and q charges as shown on the figure.

Then, according to the Coulomb's low, the interaction force between +4q and Q charges is:

$F_1 = k\dfrac{4q\cdot Q}{x^2}.\\$

The interaction force between +q and Q charges is:

$F_2 = k\dfrac{q\cdot Q}{(r - x)^2}, \\$ where $r$ is the distance between +4q and +q and $k$ is the Coulomb's constant.

In the equilibrium state both forces are equal, thus obtain the equation:

$k\dfrac{4q\cdot Q}{x^2} = k\dfrac{q\cdot Q}{(r - x)^2}$ .

Solving it for $x$ , obtain:

$\dfrac{4}{x^2} = \dfrac{1}{(r - x)^2} \Rightarrow 4(r-x)^2 = x^2 \Rightarrow 2(r-x) = x \\ x = \dfrac{2}{3}r.\\$

Answer. The charge Q should be placed at the distance 2/3r from the charge +4q.