I've recently been working on relative equilibria for some systems of particles. (ie. studying equilibrium solutions in a rotating frame. Saturn's rings for example.) This has evolved into some classical notions of atomic physics and questions of stability for Coulomb interactions there. Entertain for a few moments a classical(planetary) model for the atom. Take an isolated hydrogen atom for example. It seems to me that the $q\left(v\times B\right)$ term of the Lorentz force provides an intrinsic stabilization for the two particle system. Naively play the right hand rule game with the system and you'll see. If we assume that the electron is in a classical orbit and not radiating, then could we establish the stability of atoms with the Lorentz force? The non-radiating electron is a topic of its own and appears to be far from solved. For example, if you would like to make the statement that accelerating charges radiate, therefore...... Try to go one step further and show why this must be true in general and you imediately run into difficulties. Gyro-stabilization might be a relevant topic here.
Order of Magnitude investigation:
\begin{align} &\text{Proton Charge:}\qquad&q &\approx 1.602\times 10^{-19}\: C\\[2mm] &\text{Permeability of free space:}\qquad &\mu_0 &\approx 4\pi\times10^{-7}\frac{volt\cdot s}{amp\cdot m}\\[2mm] &\text{Permitivity of free space:}\qquad &\epsilon_0 &\approx 8.854\times10^{-12}\frac{farad}{m}\\[2mm] &\text{Electron Speed: }\qquad &v &= \alpha c\approx c/137 \approx 2.19\times10^6\frac{m}{s}\\[2mm] &\text{Electron Radius:(Bohr) }\qquad &r & = \frac{\hbar}{m_ec\alpha}\approx 5.29\times10^{-11}m\\[2mm] \end{align}
Maximum Fields(Im thinking of the fields generated by the electron in a linear approximation for an infinitesimal section of the circular orbit:
\begin{align} \text{Biot - Savart}\qquad{\bf B}_{\text{max}} &= \frac{\mu_0 q v}{4\pi r^2}&\approx& 12.537\:\text{T}\\[2mm] \text{Coloumb}\qquad{\bf E}_{\text{max}} &= \frac{q}{4\pi\epsilon_0 r^2}&\approx& 5.145\times 10^{11}\frac{N}{C} \end{align}
With Force ratio:
\begin{align} \frac{q(v\times{\bf B})_{max}}{q{\bf E}_{max}} =\frac{\frac{\mu_0 q^2 v^2}{4\pi r^2}}{\frac{q^2}{4\pi\epsilon_0 r^2}} = v^2\mu_0\epsilon_0 = \frac{v^2}{c^2}\approx\frac{(2.19\times10^6)^2}{(3\times10^8)^2}\approx 5.329\times10^{-5} \end{align}
This shows the magnetic force is 5 orders of magnitude smaller than the electrostatic force for this crude model of the ground state hydrogen. Not sure what this conclusively says about the internal dynamics but at least I think I can foresee some problems with this force stabilizing the system.
A colleague of mine who does work with NMR(nuclear magnetic resonance) informed me that the magnetic fields they work with in the lab are on the order of 3$T$ and that they do not observe the large fields I am envisoning in practice. The concept he quoted as the reasoning behind this was "orbital quenching". This appears to be a sort of averaging to zero of this field. However, it was also stated that the magnetic field that I am interested in can be measured in beam type experiments.
