Question

Three charges each of value q are placed at the corners of an equilateral triangle. A fourth charge Q is placed at the center of the triangle.
a. If Q= -q,will the charges at the corners move toward the center or fly away from it
b. For what value of Q at 0 will the charges remain stationary?

Answer 1

If $Q$ is a positive charge,the resultant force on $q$ at $A$ will be
$\overrightarrow{F_R}=\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}$
Here, $F_1=F_2=\frac{1}{4\pi {\epsilon }_0}\frac{q^2}{a^2}$
So, $(\overrightarrow{F_1}+\overrightarrow{F_2})=2\times \frac{1}{4\pi {\epsilon }_0}\frac{q^2}{a^2}cos{30}^{\circ }=\frac{\sqrt{3}{q}^2}{4\pi {\epsilon }_0{a}^2}along\overrightarrow{OA}$
While $\overrightarrow{F_3}=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{(a/\sqrt{3}{)}^2}=\frac{3qQ}{4\pi {\epsilon }_0{a}^2}along\overrightarrow{OA}$
So, $F_R=\frac{q}{4\pi {\epsilon }_0{a}^2}[\sqrt{3}q+3Q]along\overrightarrow{OA}$ (i)
b. For Q = -q
$F_R=\frac{q}{4\pi {\epsilon }_0{a}^2}[\sqrt{3}q+3a]along\overrightarrow{OA}$
$\frac{\sqrt{3}q}{4\pi {\epsilon }_0{a}^2}[\sqrt{3}-1]along\overrightarrow{AO}$
i.e., the charges q at corners $A,B,$ and $C$ will move toward the center $O$.
Charge will remain stationary if FR = 0, which in Eq. (i) is possible only if

$\left[\right(\sqrt{3})q+3Q]=0i.e.,Q=-\left(\frac{q}{\sqrt{3}}\right)$
d. Potential energy of the system
$U=\frac{1}{2}\times \frac{1}{4\pi {\epsilon }_0}\sum _{i=j}\frac{q_i{q}_j}{r_{ij}}=\frac{1}{4\pi {\epsilon }_0}[3\frac{q\times q}{a}+\frac{3q(-q/\sqrt{3})}{\left(a/\sqrt{3}\right)}]=0$
And for finite charge, distribution potential energy at infinity is always zero. So
$W=U_F-U_I=0-0=0$