A Simple Numerical Model for Exploring Electrostatic Principles: A Multipole Simulation

Abstract

The multipole expansion is a foundational yet conceptually challenging topic in electrostatics. To bridge this pedagogical gap, we present a computational model designed to visualize the fundamental behavior of electric fields originating from various multipole configurations. Based on the superposition principle for discrete charges, the model simulates dipoles, tripoles, quadrupoles, and pentapoles, and calculates the resulting electric field magnitude as a function of distance (𝑟). The results confirm the core theoretical prediction: the power-law decay of the field, given by 𝐸 ∝ 1/𝑟𝑛+2, becomes progressively steeper for higher-order multipoles. This work serves as an effective pedagogical tool, demonstrating how computational physics can render abstract theories tangible and thereby enhance the intuitive understanding of fundamental physical principles. Besides this also provides a platform for further exploration, enabling users to extend the model to more complex configurations and compare computational results with analytical predictions in a systematic manner.

Keywords

• Physics education
• computational physics
• electrostatics
• electric multipoles
• numerical simulation
• visualisation

References

1. Atkins P., de Paula J., Keeler J., 2018, Atkins’ Physical Chemistry, 11th edn. Oxford University Press google scholar
2. Byczuk K., Jakubczyk P., 2023, American Journal of Physics, 91, 840 google scholar
3. Chabay R. W., Sherwood B. A., 2008, American Journal of Physics, 76, 327 google scholar
4. Chen P., Yang Z. Q., Shi Z. Z., Hou Q. Y., Jin G. R., 2024, American Journal of Physics, 92, 78 google scholar
5. Cross R., 2025, European Journal of Physics google scholar
6. Griffiths D. J., 2017, Introduction to Electrodynamics, 4th edn. Cambridge University Press google scholar
7. Harris C. R., et al., 2020, Nature, 585, 357 google scholar
8. Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 google scholar

9. Jackson J. D., 1999, Classical Electrodynamics, 3rd edn. Wiley google scholar
10. Jacobs R., Morgan D., et al., 2025, Current Opinion in Solid State and Materials Science google scholar
11. Jones M. R., 2001, Idealization in Contemporary Physics, 63, 173 google scholar
12. Koulouridis S., 2025, Materials, 18, 2344 google scholar
13. Krasnova L., et al., 2025, ResearchGate Preprint google scholar
14. Lineweaver C. H., Patel V. M., 2024, American Journal of Physics, 92 google scholar
15. Mazibe E. N., Gaigher E., Coetzee C., 2023, Eurasia Journal of Mathematics, Science and Technology Education, 19, em2241 google scholar
16. Moreira V. C., Gavião W. D., 2020, Revista Brasileira de Ensino de Física, 42, e20190306 google scholar
17. Murray A. J., Hickman C., 2023, American Journal of Physics, 91, 847 google scholar
18. Odden T. O. B., et al., 2025, in 2025 Physics Education Research Conference Proceedings. AAPT google scholar
19. Puntel M., Gonçalves G. S., Kern D. P., 2021, Revista Brasileira de Ensino de Física, 43, e20200424 google scholar
20. Reed B. C., 2024, American Journal of Physics, 92 google scholar
21. Sullivan K. D., Sen A., Sullivan M. C., 2023, American Journal of Physics, 91 google scholar
22. Tartero G., Krauth W., 2024, American Journal of Physics, 92, 65 google scholar
23. Taylor J. R., 1997, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd edn. University Science Books google scholar
24. Thorne K. S., 1980, Reviews of Modern Physics, 52, 299 google scholar
25. Umrigar C. J., Anderson T. A., 2024, American Journal of Physics, 92 google scholar
26. Van Rossum G., Drake Jr. F. L., 2009, Python 3 Reference Manual. CreateSpace, Scotts Valley, CA google scholar
27. Wieman C. E., Adams W. K., Perkins K. K., 2008, Science, 322, 682 google scholar
28. Zhuang Y., et al., 2025, AIAA Journal google scholar
29. van Dijk W., 2023, American Journal of Physics, 91, 826 google scholar