If I have a volume of $L$ liters of salt water at a concentration of $\approx N$ mM NaCl and I pour it into an electrophoretic apparatus (like this one: 
). Once we turn the apparatus on, and set the power level to some number of volts $V$, we would expect that there would be a Lorentz force on the individual Na+ and Cl- ions, pulling them towards the cathode and anode ends of the device, respectively. What does the physical distribution of Na+ and Cl- ions actually look like in the device as it operates?
Each ion will experience some Lorentz force, but separating the two ion types will have entropic penalties, so it's unclear to mean what the equilibrium distribution will look like with some voltage and current level.
A note on why this might be of interest: The apparatus for "gel electrophoresis" shown below is typically used to drag charged molecules (for example, nucleic acids or proteins) through a gel of some density, allowing for size sorting (where the gel leads to better size discrimination than the typical charge/mass ratio one uses to seperate molecular species for mass spectrometry). Proper molecular structure depends on having specific concentrations of specific anions or cations. Thus, I've always wondered if a gradient distributions of anions and cations over the length of distance between the anode and cathode had any undesirable effects. However, nowhere have I ever seen or heard this mentioned in the literature.
