There is an interaction between the the electron's orbital angular momentum and the magnetic moment of the nucleus. It is called the "hyperfine structure" or "hyperfine splitting".
In the Wikipedia article you will notice that some of the development of the interaction is done using velocity and position, but that the actual computation is made using the orbital angular momentum. This is important because the latter quantity is a good quantum number of the bound states and the former are not.
Observing the hyperfine structure requires a high resolution spectrometer, but it can be done in a bench-top experiment. I believe the we used a Fabry-Pérot interferometer as the core of the instrument we used in my graduate spectroscopy lab.