Question
Three point charges , , and are placed at the corners of an equilateral triangle of side . Find the net electrostatic potential energy of the system.
Solution — Step by Step
For each pair of charges, the potential energy is:
For three charges, total PE is the sum of all three pairs: .
All three pairs have the same separation . The charge products:
- Pair 1-2:
- Pair 1-3:
- Pair 2-3:
Sum: .
.
Total electrostatic PE: .
The negative sign means the system is bound — we’d need to do positive work to separate all three charges to infinity.
Why This Works
Potential energy is a scalar, so we just add the contributions. No vector book-keeping, no angle calculations — that is what makes PE problems faster than force or field problems.
Each pair contributes independently. The factor of that some students remember from does NOT appear when we sum pairwise — because the sum of pairs already counts each bond once.
Speed shortcut: When all separations are equal (equilateral triangle, square with diagonal asked separately, etc.), factor out and just sum the charge products. We get the answer in three lines.
Alternative Method — Using Potentials
We could compute the potential at each corner due to the other two charges, multiply by that corner’s charge, and divide by 2:
This works but doubles the arithmetic. Use it only when the question gives potentials directly.
Common Mistake
Students often forget the signs of charges and treat all products as positive — getting instead of . The sign is half the answer.
Another classic: counting each pair twice. If we sum , we get double the right answer. Always count each pair once.
JEE Advanced 2022 had a four-charge configuration (square corners). Six pairs to count, with two pairs at separation and four at . Same template, just more bookkeeping. Top scorers solved it in under 90 seconds by tabulating cleanly.