A sideways voltage from a magnetic field
In 1879, Edwin Hall, then a graduate student at Johns Hopkins, passed a current through a thin gold leaf placed between the poles of a magnet and measured a tiny voltage across the leaf, at right angles to both the current and the field. The moving charges are pushed sideways by the magnetic force until the charge that piles up on one edge creates an electric field strong enough to balance it. That equilibrium voltage is the Hall voltage.
The formulas
- Hall voltage: V = I × B ÷ (n × q × t)
- Hall coefficient: R_H = 1 ÷ (n × q), in m³/C
- Hall field: E = V ÷ w
- Drift velocity of the carriers: v = I ÷ (n × q × w × t)
Here I is the current, B the magnetic field, n the number of charge carriers per cubic meter, q the elementary charge (1.602 × 10⁻¹⁹ C), t the thickness of the strip along the field and w its width, across which the voltage is measured.
Example: a copper strip
A copper strip 10 mm wide and 0.1 mm thick carries 10 A in a 1 T field. Copper has about 8.49 × 10²⁸ free electrons per m³:
- Hall voltage: 10 × 1 ÷ (8.49 × 10²⁸ × 1.602 × 10⁻¹⁹ × 0.0001) = 7.35 µV
- Hall coefficient: 7.35 × 10⁻¹¹ m³/C
- Drift velocity: 0.74 mm/s, slower than a snail, even with 10 A flowing
Choose indium antimonide, with about 2 × 10²² carriers per m³, and a current of only 10 mA gives about 31 mV in the same field: four million times fewer carriers make the effect four million times larger per ampere. This is why Hall sensors are made from semiconductors, not metals.
Reading the result
- The voltage is proportional to the current and to the field, which is what makes Hall probes good magnetometers once calibrated.
- It is inversely proportional to the thickness, so sensors use very thin films.
- The sign of the voltage tells whether the carriers are negative electrons or positive holes, which showed in the 1930s that some metals and doped semiconductors conduct by holes.
Limits
The simple formula assumes a single type of carrier. In semiconductors with both electrons and holes, or in metals such as zinc or beryllium, the measured coefficient mixes the two and its sign can be surprising. At very low temperature and very high field, the quantum Hall effect appears, in which the resistance takes exact values h ÷ (ie²), now used as the standard of electrical resistance.