A charge Q is placed at each of two opposite corners of a square. A charge q is placed at each of the other two corners. If the net electric force on each charge Q is zero, then Q/q equals
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( -2\sqrt{2} \)
By Coulomb’s law and the superposition principle, each charge \( Q \) feels three forces. Their sum is zero only when the two forces due to the charges \( q \) together balance the force due to the other \( Q \).
The two charges \( q \) sit at distance \( a \) (the side) from \( Q \). Their forces are equal in magnitude, \( kQ|q|/a^{2} \), and at right angles. So their resultant \( \sqrt{2}\,kQ|q|/a^{2} \) lies along the diagonal.
The other \( Q \), at distance \( a\sqrt{2} \) along the same diagonal, repels \( Q \) outward. So \( q \) must attract \( Q \). Hence \( q \) and \( Q \) have opposite signs.
Balancing the forces on \( Q \) along the diagonal:
\( \displaystyle \frac{kQ^{2}}{(a\sqrt{2})^{2}}=\sqrt{2}\,\frac{k\,|Q|\,|q|}{a^{2}} \)
\( \displaystyle \frac{|Q|}{2}=\sqrt{2}\,|q| \)
\( \displaystyle \Rightarrow\; \frac{|Q|}{|q|}=2\sqrt{2} \)
Since \( Q \) and \( q \) have opposite signs, \( Q/q=-2\sqrt{2} \).