Multipole Expansion: Exact vs Truncated Potential

Far enough away, any clump of charge stops looking complicated. Its potential can be written as a sum of simpler pieces: a monopole that falls off as one over distance, a dipole that falls as one over distance squared, a quadrupole as distance cubed, and so on, each weaker and shorter-ranged than the last. This is the multipole expansion, and it is why a distant galaxy is a point mass, a molecule is a dipole, and an antenna is designed term by term. The trick is that far from the cloud only the first non-zero term matters, so a neutral blob looks like a pure dipole and you can throw the rest away. Up close it is the opposite: you need many terms and the series barely helps. The scene shows the true potential of a small charge cluster with a probe you can move; the diagnostic plots how wrong each truncation is against distance, on a log-log scale where every extra term buys a steeper drop in the error.