A charge q is placed at the center of a line joining two equal positive charges Q. The system of the three charges will be in equilibrium, if q is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( -\displaystyle\frac{Q}{4} \)
The charge q at the centre is already in equilibrium, because the two equal charges Q pull or push it equally in opposite directions. The system is in equilibrium only if each end charge Q also has zero net force.
Let the two charges Q be a distance 2a apart, so q is at a distance a from each. For the charge Q at one end, the other Q repels it and q must attract it with an equal force (so q is negative):
\( \displaystyle \frac{kQ\cdot Q}{(2a)^{2}} + \frac{kQ\,q}{a^{2}} = 0 \)
\( \displaystyle \frac{Q}{4a^{2}} = -\frac{q}{a^{2}} \)
\( \displaystyle q = -\frac{Q}{4} \)