First slide

Electrostatic force

Question

It is required to hold equal charges q in equilibrium at the corners of a square. What charge when placed at the center of the square will do this?

Moderate

A

$- \frac{q}{2} (1 + 2 \sqrt{2})$
B

$\frac{q}{2} (1 + 2 \sqrt{2})$
C

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

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

Solution

Let place a charge Q at center of square such that Q and q are of opposite sign.
$A C = \sqrt{2} l = B D or B O = \frac{l}{\sqrt{2}}$
For equilibrium of charge at B
$F_{B O} = F_{B D} + (F_{B A} + F_{B C}) \cos 45^{\circ}$
$\Rightarrow \frac{k Q q}{{(\frac{l}{\sqrt{2}})}^{2}} = \frac{k q q}{{(\sqrt{2} l)}^{2}} + (2 \frac{k q q}{{(l)}^{2}}) (\frac{1}{\sqrt{2}})$
$\Rightarrow Q = \frac{q}{4} (1 + 2 \sqrt{2}) and Q should be negative of q .$

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