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Charges $Q$ are placed at the ends of a diagonal of a square and charges $q$ are placed at the other two corners. The condition for the net electric force on $Q$ to be zero is
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Verified Answer
The correct answer is:
$Q=-2 \sqrt{2} q, q$ being negative
Net force on $D$ due to $A$ and $B=\frac{K \sqrt{2} Q q}{a^2}$
Force on $D$ due to $B=\frac{K Q^2}{2 a^2}$
For net electric field on $D$ to be zero
$$
\begin{aligned}
K\left[\frac{Q^2}{2 a^2}+\frac{\sqrt{2} Q q}{a^2}\right] & =0 \\
\Rightarrow \quad \frac{Q}{2}+\sqrt{2} q & =0 \\
Q & =-2 \sqrt{2} q
\end{aligned}
$$
$\therefore q$ should be negative.
Force on $D$ due to $B=\frac{K Q^2}{2 a^2}$
For net electric field on $D$ to be zero
$$
\begin{aligned}
K\left[\frac{Q^2}{2 a^2}+\frac{\sqrt{2} Q q}{a^2}\right] & =0 \\
\Rightarrow \quad \frac{Q}{2}+\sqrt{2} q & =0 \\
Q & =-2 \sqrt{2} q
\end{aligned}
$$
$\therefore q$ should be negative.
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