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Physics Question on electrostatic potential and capacitance

A charge qq is placed at the centre of line joining two equal point charges each equal to +Q+Q. The system of 33 charges will be in equilibrium if qq is equal to

A

−Q4- \frac{Q}{4}

B

Q4\frac{Q}{4}

C

−Q2- \frac{Q}{2}

D

Q2\frac{Q}{2}

Answer

−Q4- \frac{Q}{4}

Explanation

Solution

The situation is as shown in the figure.

Let two equal charges Q each placed at points A and B at a distance r apart. C is the centre of AB where charge q is placed.
For equilibrium, net force on charge Q = 0
∴1πε014πε0Qqr/22=0\therefore \frac{1}{\pi \varepsilon _0} \frac{1}{4\pi \varepsilon _0} \frac{Qq}{r/2}^2 = 0
∴1πε0QQr2+14πε0Qq(r/2)2=0\therefore \frac{1}{\pi \varepsilon _0} \frac{QQ}{r^2} + \frac{1}{4\pi \varepsilon _0} \frac{Qq}{(r/2)^2} = 0
14πε0Q2r2=−14πε04Qqr2  \frac{1}{4\pi \varepsilon _0} \frac{Q^2}{r^2} = - \frac{1}{4\pi \varepsilon _0} \frac{4Qq}{r^2}\, \, or   Q=−4q  \, \, Q = -4q\, \, or   q=−Q4\, \, q = -\frac{Q}{4}