Multipole Expansion: Exact vs Truncated Potential

Far from a localized charge cluster the potential can be expanded in inverse powers of distance, $V(r) = \frac{1}{4\pi\varepsilon_0}\left[\frac{q}{r} + \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2} + \frac{\text{quadrupole}}{r^3} + \ldots\right]$, the multipole expansion: a far observer sees first the net charge, then the dipole, then finer structure. The three field maps show the exact potential, the expansion truncated at a chosen order, and their difference. The error is large near the charges (where the expansion does not converge) and falls off rapidly with distance, and each extra term tightens it further out; a sweeping probe traces the relative error against distance so the $1/r$, $1/r^2$, $1/r^3$ hierarchy is explicit.