The Hall Effect
The Hall Effect
- Put a current-carrying slice in a magnetic field and a small voltage appears across it, proportional to . That voltage is how a Hall probe measures fields — and its formula is a derivation examiners ask for.
From sideways push to steady voltage
- Charge carriers drifting at speed through the slice feel a magnetic push towards one edge, so charge builds up there (and the opposite edge is left oppositely charged).
- The built-up charge creates an electric field across the slice. Build-up stops when the electric push balances the magnetic push: with (slice width ).
- Replace the drift speed using with cross-section (thickness ), and cancels:
Symbols
- = Hall voltage across the slice (V)
- = current through the slice (A)
- = number density of charge carriers (m⁻³)
- = thickness of the slice (m)
- = charge of one carrier (C)
- at fixed current — which is exactly what makes the probe a field meter. The 2023 paper asked what means (“number density of charge carriers”) and why probes use silicon rather than copper: a million times smaller gives a million times bigger voltage (9702/42/F/M/23 Q6(b)). Thin slices (small ) help the same way.
- Orientation matters: the probe reads maximum when the field passes perpendicular through its face and zero when the face is parallel to the field (9702/42/F/M/23 Q6(a)) — the practical-skills papers love this.
- A numerical version: , A through a slice 1.8 mm thick with gives V (9702/42/O/N/23 Q7(a)).
Common mistake
The Hall voltage tracks the field instantaneously: if holds steady, holds steady too. A 2023 question plotted the same changing field twice — the Hall voltage copies the B–t shape, while an induced e.m.f. (next chapter) is non-zero only while is changing. Confusing the two graphs throws away all the sketch marks (9702/42/O/N/23 Q7).