Assembling charges, one at a time.

Charges brought from infinity one at a time, the work for each step added pair by pair into the system's potential energy.

Assemble +q, −q, +q, −q on a square, confirm U < 0 and the same in any order, then bring a fifth charge to the centre for zero work.

Giving unlike charges a positive potential energy.

• 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.

Teaching this idea? Put the simulation on your own page or LMS with this code.