D.2.12 (HL)—Two-charge potential energy

Use the pair-energy expression

For two point charges or spherical conductors represented at their centres, keep the signs of $q_1$ and $q_2$ and use centre separation $r$. The zero reference is infinite separation.

Worked example — two negative conductors

Radii $2.5\,\mathrm{cm}$ and $1.5\,\mathrm{cm}$, separated by a $1.7\,\mathrm{cm}$ surface gap, give $r=5.7\times10^{-2}\,\mathrm{m}$. For charges $-4.7\times10^{-8}\,\mathrm{C}$ and $-6.3\times10^{-8}\,\mathrm{C}$, $E_p=(8.99\times10^9)q_1q_2/r=+4.7\times10^{-4}\,\mathrm{J}$. Positive energy matches repulsion between like charges.

Interpret the sign

Like charges give positive potential energy because work is required to bring them together. Unlike charges give negative potential energy because the electric field releases energy as they approach. Increasing separation moves the energy toward zero.

Use changes in energy

When a charge configuration changes, calculate the final minus initial potential energy. The external work and work done by the electric field have opposite signs under a quasistatic convention. Use the actual separation between charge centres, not the physical radius of either object.

Common trap

Do not use absolute values for q1q2 before deciding the sign of the energy, and do not put r² in the potential-energy formula; r² belongs to Coulomb force.