Effect of Hall effect on resistance

[kneeslider]

(The topic was previously posted in Electrical Engineering forum)
https://www.physicsforums.com/showthread.php?t=510793
The charge carriers in a current carrying wire subjected to a magnetic field will move to a side due to the Hall effect. But doesn't that also decrease the effective cross section area through which the charge carriers are moving? Does the resistance increase? If so, how significant is it in electric machines like motors?

 

[jambaugh]

Keep in mind the capacitance of the conductor is very small so, although a magnetic field does slightly "compress" the electron gas, the density will not change appreciably and thus neither will the effective conducting cross section.

(Note though that magnetic fields can affect conductivity through other mechanisms.)