Question

Three point charges $+Q, +2Q$ and $q$ are placed at the vertices of an equilateral triangle. The value of charge $q$ in terms of $Q$, so that electrical potential energy of the system is zero, is given by

• A. $\text{q} = \frac{1}{3}\text{Q}$
• B. $\text{q} = \frac{2}{3}\text{Q}$
• C. $\text{q} = -\frac{2}{3}\text{Q}$
• D. $\text{q} = -\frac{3}{2}\text{Q}$

Hint:

For potential energy to be zero in a system of point charges, at least one charge must be of opposite sign to the others.

Answer 1

The Correct Option is C

Step 1: Concept
The total potential energy ($U$) of a system of three charges at distance $r$ is $U = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1q_2}{r} + \frac{q_2q_3}{r} + \frac{q_3q_1}{r} \right)$.

Step 2: Meaning
Since it's an equilateral triangle, all distances $r$ are equal. For $U=0$, the sum of the charge products must be zero: $Q(2Q) + (2Q)q + qQ = 0$.

Step 3: Analysis
$2Q^2 + 2Qq + qQ = 0$
$2Q^2 + 3Qq = 0$
$Q(2Q + 3q) = 0 \implies 3q = -2Q \implies q = -\frac{2}{3}Q$.

Step 4: Conclusion
The value of $q$ is $-\frac{2}{3}Q$. Final Answer: (C)