Effect of Hall effect on resistance

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SUMMARY

The Hall effect causes charge carriers in a current-carrying wire to deflect when subjected to a magnetic field, potentially reducing the effective cross-sectional area available for conduction. This deflection can lead to an increase in resistance, although the impact is minimal in electric machines like motors due to the small capacitance of the conductor. The density of charge carriers remains largely unchanged, meaning that while there is a slight compression of the electron gas, the overall effect on resistance is not significant. Other mechanisms may also influence conductivity in the presence of magnetic fields.

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(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?
 
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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.)
 

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