Appendix B - A digression on electromagnetic force, divergences of Maxwell equations, and lengths in atomic physics

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In this second appendix we show how the electromagnetic force and the magnetic field can be seen a side effect electrostatics and special relativity, namely the contraction of lengths seen by a particle with a given speed.

We then discuss the two divergences in Maxwell equations: firstly, the well known divergence of the energy contained in the electrostatic field of a point charge, leading to the definition of the classical radius of the electron. The second divergence is for a current loop with infinite current density. We show how the inductance of a loop depends on the wire diameter, and diverges for small wire thickness. We show how to treat this divergence for the self-field contribution in codes like ROXIE, when computing the peak field in the coil.

The 20th century physics is characterized by two limits: an upper limit to speed given by special relativity and a lower limit to angular momentum given by quantum mechanics. Using these two constants one canbuild a fundamental length (Compton radius) and a adimensionless electromagnetic coupling constant (alpha). The ratio between the classical electron radius and the Compton radius is given by alpha.

We then briefly discuss the quantization of the angular momentum to compute the Bohr radius, as an estimate of the atom size. The simple formalism also allows to compute the average speed of electron in the Bohr model, and estimating the associated magnetic field of this current loop. The average speed of the elecron in this semiclassical model is alpha times the speed of light, i.e. the motion is non relativistic.

Finally we discuss the enigma of the intrinsic magnetic momentum of the electron, the anomaly in the gyromagnetic ratio.

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