Two charges q and -3q are fixed on x axis

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

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Now, the Coulomb force $F$ on a charge $q$ due to another charge $Q$ placed at a distance $r$ from it is given by $F=K\dfrac{Qq}{{{r}^{2}}}$ --(1)Where $K=9\times {{10}^{9}}kg.{{m}^{3}}{{s}^{-2}}{{C}^{-2}}$ Now, using (1), we get the sum of the forces on the charge $2q$ due to the charges $q$ and $-3q$ as$F=K\dfrac{q\left( 2q \right)}{{{x}^{2}}}+K\dfrac{-3q\left( 2q \right)}{{{\left( d+x \right)}^{2}}}=2K{{q}^{2}}\left[ \dfrac{1}{{{x}^{2}}}-\dfrac{3}{{{\left( d+x \right)}^{2}}} \right]$ Now, for equilibrium, $F=0$ $\Rightarrow 2K{{q}^{2}}\left[ \dfrac{1}{{{x}^{2}}}-\dfrac{3}{{{\left( d+x \right)}^{2}}} \right]=0$$\Rightarrow \left[ \dfrac{1}{{{x}^{2}}}-\dfrac{3}{{{\left( d+x \right)}^{2}}} \right]=0$$\Rightarrow \dfrac{1}{{{x}^{2}}}=\dfrac{3}{{{\left( d+x \right)}^{2}}}$ $\Rightarrow {{\left( d+x \right)}^{2}}=3{{x}^{2}}$ Square rooting both sides we get$\Rightarrow \sqrt{{{\left( d+x \right)}^{2}}}=\sqrt{3{{x}^{2}}}$$\Rightarrow d+x=\sqrt{3}x$ $\Rightarrow \sqrt{3}x-x=d$ $\Rightarrow x\left( \sqrt{3}-1 \right)=d$ $\Rightarrow x=\dfrac{d}{\sqrt{3}-1}$ Rationalizing, we get$x=\dfrac{d\left( \sqrt{3}+1 \right)}{\left( \sqrt{3}-1 \right)\left( \sqrt{3}+1 \right)}=\dfrac{d\left( \sqrt{3}+1 \right)}{{{\sqrt{3}}^{2}}-{{1}^{2}}}=\dfrac{d\left( \sqrt{3}+1 \right)}{3-1}=\dfrac{d+\sqrt{3}d}{2}$ $\left( \because \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}} \right)$ Therefore, we have got the required expression for the distance from $A$ that the third charge has to be kept.Therefore, the correct option is $B)\text{ }\dfrac{d+\sqrt{3}d}{2}$.

So, the correct answer is “Option B”.

Note: While putting the square root in the calculation, we neglected the negative root since doing that would mean that $d$ would turn out to be negative which is not possible as $d$ is the measure of a distance and it has to be positive. Students must realize that if they put the charge in between the two charges, it will never be in equilibrium upon seeing the question and hence, must not waste time by placing the charge in between the two charges and waste time in calculating an impossible answer.

Tardigrade - CET NEET JEE Exam App

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 2 q be placed such that it will not experience any force?A. d +√3 d /2B. d +3 d /22C. d 3 d /2D. d √3 d /2

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Two charges q and -3q are fixed on x axis

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