Two charge $q$ and $-3 q$ are placed fixed on $x$ -axis separated by distance $d$. Where should a third charge $2 q$ be placed such that it will not experience any force?

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Aicharya · Accepted Answer

Let a charge $2 q$ be placed at $P$, at a distance $\ell$ from $A$ where charge $q$ is placed, as shown in figure. The charge $2 q$ will not experience any force, when force of repulsion on it due to $q$ is balanced by force of attraction on it due to $-3 q$ at $B$ where $\mathrm{AB}$ $=d$
or $\frac{(2 q)(q)}{4 \pi \varepsilon_{0} \ell^{2}}=\frac{(2 q)(-3 q)}{4 \pi \varepsilon_{0}(\ell+d)^{2}}$
$(\ell+d)^{2}=3 \ell^{2} \quad$ or $\quad 2 \ell^{2}-2 \ell d-d^{2}=0$
$\therefore \quad \ell=\frac{2 d \pm \sqrt{4 d^{2}+2 d^{2}}}{4}=\frac{d}{2} \pm \frac{\sqrt{3} d}{2}$ /

Two charge $q$ and $-3 q$ are placed fixed on $x$ -axis separated by distance $d$. Where should a third…

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