nLab Hall effect

Idea

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.

Details

Write

• $B$ for the absolute value of the ambient magnetic field;

• $E_{Hall}$ for the absolute value of the electric field induced by charge separation;

• $e$ for the electron‘s electric charge;

• $v$ for the absolute value of the drift velocity of the electrons in the material;

• $\rho$ for the number density of electrons;

• $j = \rho e v$ for the absolute value of the electric current density.

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:

• the absolute value of the Lorentz force on the electrons is

  $$F_L = e v B$$

• the absolute value of the electrostatic force on the electrons is

  $$F_S = e E_{Hall}$$

• and hence in equilibrium $F_S = F_L$ we have

  $$E_{Hall} \;=\; v B \,.$$

  This is called the Hall electric field.

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

$$V_{Hall} \;=\; d E_{Hall} \;=\; d v B \,,$$

called the Hall voltage.

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

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

Here

• 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).

Related concepts

• quantum Hall effect

• spin Hall effect, quantum spin Hall effect

References

The original article:

• Edwin Hall, On a New Action of the Magnet on Electric Currents, American Journal of Mathematics 2 3 (1879) 287-292 [doi:10.2307/2369245]

See also:

• Wikipedia: Hall effect