Four charges equal to – Q are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium, the value of 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
$\frac{Q}{2} (1 + 2 \sqrt{2})$

Solution:

Force on any corner charge
$F_{1} = \frac{k Q}{a^{2}}, F_{2} = \frac{k Q^{2}}{a^{2}}, F_{3} = \frac{k Q^{2}}{{(\sqrt{2} a)}^{2}}$

$\vec{F_{1}} + \vec{F_{2}} + \vec{F_{3}} = \vec{F}$ at equilibrium
$\frac{Q^{2}}{2 a^{2}} + \frac{\sqrt{2} Q^{2}}{a^{2}} = \frac{Q q}{{(a / \sqrt{2})}^{2}}$

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

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

Solution:

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