Question:
In the figure, find the value of \(Q\) so that the electrostatic potential energy of the system becomes zero.
In the figure, find the value of \(Q\) so that the electrostatic potential energy of the system becomes zero.
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For zero potential energy in a triangular system, sum all pairwise potential energies and solve for unknown charge.
Updated On: Jul 18, 2026
- \(\frac{q}{\sqrt{2}}\)
- \(\frac{-2q}{2+\sqrt{2}}\)
- \(\frac{2q}{2-\sqrt{2}}\)
- \(\sqrt{2} q\)
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The Correct Option is C
Solution and Explanation
Step 1: Recall electrostatic potential energy formula.
Potential energy of system of point charges:
\[ U = k \sum_{i\lt j} \frac{q_i q_j}{r_{ij}} \]
Step 2: Identify charges and distances.
Two fixed charges: \(-q\) at top, \(+q\) at bottom left; unknown charge \(Q\) at bottom right. Distance between charges \(x\) for sides, diagonal \(\sqrt{2} x\).
Step 3: Write total potential energy.
\[ U = k \left[ \frac{(-q)(+q)}{x} + \frac{(-q) Q}{x} + \frac{(+q) Q}{x} \right] + \text{diagonal terms} \]
Step 4: Substitute distances.
For diagonal distance \(\sqrt{2} x\), interaction terms: \(-q\) with \(Q\), \(+q\) with \(Q\), etc. Simplify for zero energy condition:
\[ U = 0 \]
Step 5: Solve for \(Q\).
After simplification using geometry of right triangle:
\[ Q = \frac{2q}{2 - \sqrt{2}} \]
Step 6: Final conclusion.
Hence, the value of \(Q\) is:
\[ \boxed{\frac{2q}{2-\sqrt{2}}} \]
Potential energy of system of point charges:
\[ U = k \sum_{i\lt j} \frac{q_i q_j}{r_{ij}} \]
Step 2: Identify charges and distances.
Two fixed charges: \(-q\) at top, \(+q\) at bottom left; unknown charge \(Q\) at bottom right. Distance between charges \(x\) for sides, diagonal \(\sqrt{2} x\).
Step 3: Write total potential energy.
\[ U = k \left[ \frac{(-q)(+q)}{x} + \frac{(-q) Q}{x} + \frac{(+q) Q}{x} \right] + \text{diagonal terms} \]
Step 4: Substitute distances.
For diagonal distance \(\sqrt{2} x\), interaction terms: \(-q\) with \(Q\), \(+q\) with \(Q\), etc. Simplify for zero energy condition:
\[ U = 0 \]
Step 5: Solve for \(Q\).
After simplification using geometry of right triangle:
\[ Q = \frac{2q}{2 - \sqrt{2}} \]
Step 6: Final conclusion.
Hence, the value of \(Q\) is:
\[ \boxed{\frac{2q}{2-\sqrt{2}}} \]
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