Lesson 5 of 13 · 6 min
Potential energy of a system of charges
NCERT §2.7
Meera wants to glue three small beads, each +2 nC, at the corners of a triangle with 10 cm sides. They repel each other. How much work does it take to build the triangle?
The lesson in notes
In short
The potential energy of a system of charges is the total external work needed to bring them from infinity, one by one, to their places.
Two charges: U = q₁q₂/(4πε₀r₁₂). The first charge costs nothing; the second costs q₂ times the potential of the first.
Like charges (q₁q₂ > 0) give positive U, since work must be done against repulsion. Unlike charges give negative U, since work is needed to pull them apart.
Three charges: U = (1/4πε₀)(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃), one term for every pair.
Because the force is conservative, U depends only on the final arrangement and not on the order or paths used to assemble it.
Worked value: +q, −q, +q, −q at the corners of a square of side d (alternating) have U = −(q²/4πε₀d)(4 − √2). The centre of that square is at V = 0, so a further charge can be brought there for no work.