If q_A=0 in Figure, under what conditions will there be no net Coulomb force on q?

Figure Four point chargesq_a, q_b,q_c, and lie on the corners of a square andq is located at its center.

The Coulomb force between two charges Q and q separated by a distance r is given as,

$\mathit{F}\mathbf{=}\frac{\mathbf{K}\mathbf{Q}\mathbf{q}}{{\mathbf{r}}^{\mathbf{2}}}$

Here, K is the electrostatic force constant.

The force at the charge q located at the center of the square is,

Force acting on the charge q located at the center of the square.

The force of q due to q_a is directed along OD is given as,

$F_a=\frac{KqQ}{r^2}$

The force of q due to q_b is directed along OC is given as,

role="math" localid="1653629756645" $F_b=\frac{KqQ}{r^2}$

The force of q due to q_c is directed along OB is given as,

$F_c=\frac{KqQ}{r^2}$

The force of q due to q_d is directed along OA is given as,

$F_d=\frac{KqQ}{r^2}$

The net Coulomb force on charge q at the center of square will be zero when all the forces at the charge q adds to zero.

The forces directed along OB and OC can cancel each other only if q_b = q_c .

If q_a = 0 , the charge q will experience no force only if q_d = 0 .

Hence, the charge q is in equilibrium if q_a = q_d = 0 , and q_b = q_c .