<p>The beam envelope is not an equipotential surface in the traditional Kapchinskij–Vladimirskij (KV) potential, it would not be a good model in the case where the outer edge of the beam is close to the wall of the beam transport tube. Here, the “conductive potential” is introduced, where the beam envelope is the equipotential surface. The KV potential is a good model when there is a large distance from the beam to the inner surface of the beam duct, and the conductive potential is a good approximation when the beam is distributed close to the beam-duct wall. The actual beam is assumed to be somewhere in between these states. This hypothesis is confirmed by simulations, and simple notation for the combination of the two types of potentials has been obtained. It gives an appropriate space-charge force for any geometry between the beam and the beam channel wall.</p> Graphical abstract <p></p>