6 Electric field of a point - like charge in a strong magnetic field

We describe the potential produced by a point electric charge placed into a constant magnetic field, so strong that the electron Larmour length is much shorter than its Compton length. The standard Coulomb law is modified due to the vacuum polarization by the external magnetic field. Only mode-2 photons mediate the static interaction. The corresponding vacuum polarization component, taken in the one-loop approximation, grows linearly with the magnetic field. Thanks to this fact a scaling regime occurs in the limit of infinite magnetic field, where the potential is determined by a universal function, independent the magnetic field. The scaling regime implies a short-range character of interaction in the Larmour scale, expressed as a Yukawa law. On the contrary, the electromagnetic interaction regains its long-range character in a larger scale, characterized by the Compton length. In this scale the tail of the Yukawa potential follows an anisotropic Coulomb law: it decreases as the distance from the charge increases, slower along the magnetic field and faster across. The equipotential surface is an ellipsoid stretched along the magnetic field. As a whole, the modified Coulomb potential is a narrower function than the standard Coulomb function, the narrower the stronger the field. The singular behavior in the vicinity of the charge remains unsuppressed by the magnetic field. These results may be useful for studying atomic spectra in super-strong magnetic fields of several Schwinger's characteristic values.