Magnetic hyperfine interaction made easier

We present two derivations of the hyperfine interaction in the ground state of hydrogen using classical electrodynamics. We calculate, at the site of the proton moment m → p, the magnetic field B → e due to the magnetization source M → e ( r → ) of the relatively extended 1 s electron state. This gives the magnetic interaction via − m → p · B → e. One derivation applies the Biot–Savart law to the bound 1 s electric current J → b = ∇ → × M → to directly find B → e; the other derivation applies the magnetic version of the Coulomb Law to the bound 1 s magnetic charge density ρ b = − ∇ → · M → to first obtain μ 0 H → e and then adds μ 0 M → to find B → e. We show, for any source M →, that these two approaches give the same B → ( r → ), as is expected within classical electrodynamics. This article describes two classical derivations of the hyperfine interaction in hydrogen. In contrast to familiar derivations, these involve no singularities in the magnetic field of the electron. Instead, the interaction energy of a localized proton magnetic moment in the smooth field of an electron in a 1 s orbital is calculated using both bound magnetic charges and bound currents, and the general equivalence of these two approaches is proved. The analysis is accessible in a junior-level electrodynamics course and connects classical electrodynamics to quantum mechanics, as well as the 21 cm line used in astronomy and astrophysics.