Magnetic monopoles, Galilean invariance, and Maxwell’s equations

Maxwell’s equations have space reserved for magnetic monopoles. Whether or not they exist in our part of the universe, monopoles provide a useful didactic tool to help us recognize relations among Maxwell’s equations less easily apparent in the approach followed by many introductory textbooks, wherein Coulomb’s law, Biot and Savart’s law, Ampere’s law, Faraday’s law, Maxwell’s displacement current, etc., are introduced independently, ‘‘as demanded by experiment.’’ Instead a conceptual path that deduces all of Maxwell’s equations from the near‐minimal set of assumptions: (a) Inertial frames exist, in which Newton’s laws hold, to a first approximation; (b) the laws of electrodynamics are Galilean invariant—i.e., they have the same form in every inertial frame, to a first approximation; (c) magnetic poles (as well as the usual electric charges) exist; (d) the complete Lorentz force on an electric charge is known; (e) the force on a monopole at rest is known; (f) the Coulomb‐like field produced by a resting electric charge and by a resting monopole are known. Everything else is deduced. History is followed in the assumption that Newtonian mechanics have been discovered, but not special relativity. (Only particle velocities v≪c are considered.) This ends up with Maxwell’s equations (Maxwell did not need special relativity, so why should we,) but facing Einstein’s paradox, the solution of which is encapsulated in the Einstein velocity‐addition formula.