# Magnetic Field Seen by Orbiting Electron

1. Apr 26, 2012

### atomicpedals

1. The problem statement, all variables and given/known data

What is the magnetic field seen by an electron in the first Bohr orbit of hydrogen due to the intrinsic spin moment of the proton?

2. The attempt at a solution

I think I've got it; but might be way off.

A charged nucleus with velocity -v constitutes a current element j,
$$\mathbf j = -Ze \mathbf v$$
Ampere's law then gives
$$\mathbf B = - \frac{Ze\mu}{4\pi} \frac{\mathbf v \times \mathbf r}{r^{3}}$$
Coulomb's law
$$\mathbf E = - \frac{Ze}{4\pi \epsilon}\frac{\mathbf r}{r^{3}}$$
and thus
$$\mathbf B = - \frac{1}{c^{2}}\mathbf v \times \mathbf E$$
The spin-orbit interaction energy is found to be
$$\Delta E = \frac{1}{2} \frac{g \mu}{\hbar} \mathbf S \cdot \mathbf B$$
Applying the electric field E and the force F acting on the electron
$$-e \mathbf E = \mathbf F$$
$$\mathbf F = - \frac{dV(r)}{dr}\frac{\mathbf r}{r}$$
and so finally
$$\mathbf B = - \frac{1}{ec^{2}}\frac{1}{r}\frac{dV(r)}{dr}\mathbf v \times \mathbf r$$
$$\mathbf B = - \frac{1}{emc^{2}}\frac{1}{r}\frac{dV(r)}{dr}\mathbf{L}$$