A system of three charges is placed as shown in the figure [+q and -q separated by a distance d, and a charge Q at a distance D from them, all three lying on one straight line in the order +q, -q, Q]. If D d, the potential energy of the system is best given by

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(1) 1/(4πε₀)[-(q²)/d-(qQd)/(D²)]. The energy is the sum over pairs. The +q and -q pair, separated by d, contributes U₁=(k(q)(-q))/d=-(kq²)/d. For the interaction with Q, treat +q and -q as a dipole of moment p=qd pointing from -q towards +q, i.e. away from Q. On the dipole axis the potential at distance D is V=(kp cos θ)/(D²), and with Q on the negative end cos θ=-1, so V=-(kqd)/(D²). U₂=QV=-(kqQd)/(D²). U=U₁+U₂=1/(4πε₀)[-(q²)/d-(qQd)/(D²)].

A system of three charges is placed as shown in the figure [+q and -q separated by a distance d, and a charge Q at a distance D from them, all three lying on one straight line in the order +q, -q, Q]. If D d, the potential energy of the system is best given by

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