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 ρb=−∇→·M→ to first obtain μ0H→e and then adds μ0M→ 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.