Question

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

• A. $\frac{d-\sqrt{3}d}{2}$
• B. $\frac{d+\sqrt{3}d}{2}$
• C. $\frac{d+3d}{2}$
• D. $\frac{d-3d}{2}$

Answer 1

The correct option is B $\frac{d+\sqrt{3}d}{2}$

Let a charge 2q be placed at P, at a distance l from A (where charge q is placed), as shown in figure.
The charge 2q will not experience any force, when force of repulsion on it due to q is balanced by force of attraction on it due to -3q at B where AB = d
or $\frac{\left(2q\right)\left(q\right)}{4\pi {ϵ}_0{l}^2}=\frac{\left(2q\right)\left(3q\right)}{4\pi {ϵ}_0(l+d{)}^2}$
$(l+d{)}^2=3l^2$
or $2l^2-2ld-d^2=0$
$\therefore l=\frac{2d\pm \sqrt{4d^2+8d^2}}{4}=\frac{d}{2}\pm \frac{\sqrt{3}d}{2}$
$l=\frac{d+\sqrt{3}d}{2}$