Hall resistance ($R_{xy}$, $R_H$)

In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus $t$ and $n$ are also fixed).

In those cases, it can be useful to talk about the "Hall resistance" defined as:So note that it is not a "regular resistance", it just has the same dimensions, and is more usefully understood as a proportionality constant for the voltage given an input $I_x$ current:

This notion can be useful because everything else being equal, if we increase the current $I_x$, then $V_y$ also increases proportionally, making this a way to talk about the voltage in a current independent manner.

And this is particularly the case for the quantum Hall effect, where $R_{xy}$ is constant for wide ranges of applied magnetic field and TODO presumably the height $t$ can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.