nLab Hall effect




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In electromagnetism the Hall effect (Hall 1879) is essentially the manifestation of the Lorentz force in conductors:

It is the phenomenon that an electric current flowing perpendicular to a magnetic field induces an electric field perpendicular to both of these, sourced from the separation of positive and magnetic charge carriers in the current, which are driven apart by the Lorentz force.



Now, assuming, for simplicity and as usual, that the magnetic field and the electric current are both (as vector fields) constant in space and time as well as perpendicular to each other:

In a conductor of extension dd perpendicular to both the current and the magnetic field (hence: along the electric field), this produces a voltage?

V Hall=dE Hall=dvB, V_{Hall} \;=\; d E_{Hall} \;=\; d v B \,,

called the Hall voltage.

In practice one typically refers to the Hall resistance R HallR_{Hall}, defined to be the ratio of Hall voltage over the electron current J=AjJ = A j through the cross-section area AA of the conductor:

R HallV HallJ=dAvρBe. R_{Hall} \;\coloneqq\; \frac{ V_{Hall} }{ J } \;=\; \frac{ d }{ A } \cdot \frac{ v }{ \rho } \cdot \frac{ B }{ e } \,.


  • the first factor on the right encodes the geometry of the conductor;

  • the second factor encodes the charge transport properties of its material;

  • the third factor encodes the external magnetic field.

Since the first term is typicall known by construction, knowledge of either of the remaining two terms makes the Hall effect applicable for Hall sensors measuring the remaining term:

  • when the material properties (second term) are known, the Hall effect serves in magnetometers (measuring the last term);

  • when the external magentic field is known (the third term) the Hall effect provides information about the current and/or its conducting material (the second term).


The original article:

  • Edwin Hall, On a New Action of the Magnet on Electric Currents, American Journal of Mathematics Vol. 2, No. 3 (Sep., 1879), pp. 287-292 (doi:10.2307/2369245)

See also:

Last revised on February 18, 2021 at 07:37:08. See the history of this page for a list of all contributions to it.