Question

A charge +q is fixed to each of three corners of square. On the empty corner a charge q is placed such that there is no net electrostatic force acting on the diagonally opposite charge then q equal

Answer 1

Let ABCD is a square. we place a charge +q to each of three corners {e.g., A, B and C} of square . now on the empty corner { e.g., D} a charge Q is placed such that no net electrostatic force acting on the diagonally opposite charge.

electrostatic force between B and D, $\bf{F_1=\frac{kqQ}{(\sqrt{2}r)^2}}$

electrostatic force between A and B ,$\bf{F_2=\frac{kq^2}{r^2}}$

and electrostatic force between B and C, $\bf{F_3=\frac{kq^2}{r^2}}\hat$

resultant of $F_2\:\textbf{and},F_3$ is $F_4=\sqrt{\left(\frac{kq^2}{r^2}\right)^2+\left(\frac{kq^2}{r^2}\right)^2}$

= $\sqrt{2}\frac{kq^2}{r^2}$

a/c to question,$F_1+F_4=0$

or, $\frac{kqQ}{2r^2}+\sqrt{2}\frac{kq^2}{r^2}=0$

or, $\frac{Q}{2}+\frac{\sqrt{2}q}{r^2}=0$

or, $Q=-2\sqrt{2}q$

hence, value of Q = -2√2q