Question

Four point charges, each –q are placed at the corners of a square and another charge Q is placed at its centre. If the system is in equilibrium, then Q is

• A. $-\frac{q}{4}[1+2\sqrt{2}]$
• B. $-\frac{q}{4}[1+2\sqrt{2}]$
• C. $-\frac{q}{2}[1+2\sqrt{2}]$
• D. $-\frac{q}{2}[1+2\sqrt{2}]$

Answer 1

The correct option is B $-\frac{q}{4}[1+2\sqrt{2}]$


Consider the equilibrium of charge on any corner, say that at A.
For equilibrium $F_{B}cos45+F_{D}cos45+F_C=F_O$
i.e., $\frac{1}{4\pi {ϵ}_0}$
i.e., $[\frac{q^2}{a^2}\frac{1}{\sqrt{2}}+\frac{q^2}{a^2}\frac{1}{\sqrt{2}}+\frac{q^2}{(a\sqrt{2}{)}^2}]=-\frac{1}{4\pi {ϵ}_0}\frac{aQ}{{\left(\frac{a\sqrt{2}}{2}\right)}^2}$
i.e., q i.e, $q[\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}+\frac{1}{2}]=-\frac{Q}{\left(\frac{1}{2}\right)}$
i.e., $|Q|=\frac{q}{4}(1+2\sqrt{2})$