The voltage changes perpendicular to the current when magnetic field is applied.

An intuitive video is:

The key formula for it is:
where:

$V_{H}=nteI_{x}B_{z} $

- $I_{x}$: current on x direction, which we can control by changing the voltage $V_{x}$
- $B_{z}$: strength of transversal magnetic field applied
- $n$: charge carrier density, a property of the material used
- $t$: height of the plate
- $e$: electron charge

Applications:

- the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
- measure magnetic fields, TODO vs other methods

Other more precise non-classical versions:

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:

$R_{xy}=I_{x}V_{y} $

$V_{y}=R_{xy}I_{x}$

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.