Question

Four charges each equal to $Q$ are placed at the four corners of a square and a charge $q$ is placed at the centre of the square. If the system is in equilibrium then the value of $q$ is

• A. $\frac{Q}{2}(1+2 \sqrt{2})$
• B. $\frac{-Q}{4}(1+2 \sqrt{2})$
• C. $\frac{Q}{4}(1+2 \sqrt{2})$
• D. $\frac{-Q}{2}(1+2 \sqrt{2})$

Answer 1

Answer: (B)

Net force on $Q$ due to other corner charge is
$F_{123} =F_{3}+\sqrt{F_{1}^{2}+F_{2}^{2}} $
$=F_{3}+\sqrt{2} F_{1}$
$=\frac{K Q^{2}}{2 l^{2}}+\frac{\sqrt{2} K Q^{2}}{l^{2}}$
Force on $Q_{1}$ due to centre charge $-q$
$F_{4}=\frac{K Q q}{I^{2}} .2$
If net force on corner charge $Q$ is zero Then
$F_{123}+F_{4}=0$
So $q=-\frac{Q}{4}[1+2 \sqrt{2}]$