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(B) Step $1$: Calculate the distances between the charges. Let the side length of the square be $a$. The diagonal length is $AC = \sqrt{a^2 + a^2} = \

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$ \frac{Q}{4}(1 + 2\sqrt{2}) $

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(B) Step $1$: Calculate the distances between the charges. Let the side length of the square be $a$. The diagonal length is $AC = \sqrt{a^2 + a^2} = \sqrt{2}a$. The distance from the center $O$ to any corner is $OC = \frac{AC}{2} = \frac{a}{\sqrt{2}}$.Step $2$: Equilibrium condition. For the system to be in equilibrium,the net force on every charge must be zero. By symmetry,the net force on the central charge $q$ is zero regardless of its value. Thus,we must ensure the net force on any one of the corner charges (e.g.,at point $C$) is zero.Step $3$: Forces acting on the charge at corner $C$. Let the charges at corners $A, B, C, D$ be $-Q$. The force due to $B$ is $F_1 = \frac{KQ^2}{a^2}$ (repulsive),the force due to $D$ is $F_2 = \frac{KQ^2}{a^2}$ (repulsive),and the force due to $A$ is $F_4 = \frac{KQ^2}{(\sqrt{2}a)^2} = \frac{KQ^2}{2a^2}$ (repulsive). The force due to central charge $q$ is $F_3 = \frac{K|q|Q}{(a/\sqrt{2})^2} = \frac{2K|q|Q}{a^2}$ (attractive,so $q$ must be positive).Step $4$: Summing the forces along the diagonal $OC$. The resultant of $F_1$ and $F_2$ is $\sqrt{F_1^2 + F_2^2} = \sqrt{2} \frac{KQ^2}{a^2}$. For equilibrium: $F_3 + F_4 + \sqrt{2}F_1 = 0$. Since $F_3$ is attractive (towards center) and others are repulsive (away from center),we set: $\frac{2KqQ}{a^2} + \frac{KQ^2}{2a^2} + \sqrt{2}\frac{KQ^2}{a^2} = 0$.Step $5$: Solving for $q$: $2q + \frac{Q}{2} + \sqrt{2}Q = 0 \Rightarrow 2q = -Q(\frac{1}{2} + \sqrt{2}) \Rightarrow q = -\frac{Q}{4}(1 + 2\sqrt{2})$. However,checking the force direction,for the corner charge to be in equilibrium,$q$ must be positive to attract the corner charge. Re-evaluating the sign convention,the correct magnitude is $q = \frac{Q}{4}(1 + 2\sqrt{2})$.

Four charges of magnitude $-Q$ are placed at the four corners of a square,and a charge $q$ is placed at the center. If the system is in equilibrium,the value of $q$ is ......

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