Let’s apply a magnetic field on a conductor in the form of a flat strip.
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- The magnetic field deviates the trajectory of electrons causing accumulation at one of the edges of the strip, leaving an excess positive charge at the opposite edge.
- Accumulation continues until the resulting transverse electrostatic field is large enough to counteract the external magnetic field.
- The final electric field implies there exists a potential difference between the opposite edges of the strip (distance $d$), called the Hall voltage or Hall emf.
Consider figure b), a positive charge $e$ moving along the positive x-axis ($v_{\mathrm{d}}>0$). The magnetic field is along positive y ($B_{y}>0$), while the electrostatic field is along negative x ($E_{z}<0$).
$$ e E_{z}+e v_{\mathrm{d}} B_{y}=0 $$ $$ J_{x}=n e v_{\mathrm{d}} $$,
Consider figure a), a negative charge $-e$ moving along the negative x-axis ($v_{\mathrm{d}}<0$). The magnetic field is along negative y ($B_{y}<0$), while the electrostatic field is along positive x ($E_{z}<0$).
$$ -e E_{z}-e v_{\mathrm{d}} B_{y}=0 $$ $$ J_{x}=-n e v_{\mathrm{d}} $$,
Both situations can be summarized as follows
$$ q E_{z}+q v_{\mathrm{d}} B_{y}=0 $$ $$ J_{x}=n q v_{\mathrm{d}} $$ Then $$ n q=\frac{-J_{x} B_{y}}{E_{z}}=\frac{-J_{x} B_{y} d}{V_{ab}} $$Remarks
- The sign of the Hall voltage crucially depends on the sign of the charge carriers: $V_{ab}>0$ for electrons.
- Discovered by the American physicist Edwin Hall in 1879
https://youtu.be/wpAA3qeOYiI - Hall Effect https://youtu.be/42qNfPOYlR8 - Hall Effect sensors for Arduino
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