Hall Effect Calculator
Compute Hall voltage V_H = IB/(nqt) from current, field, carrier density, and thickness.
Determine charge-carrier type and density from measurements.
The Hall effect is one of those rare physics results that does double duty as a measurement tool. Discovered by Edwin Hall in 1879 while he was a graduate student at Johns Hopkins (his thesis advisor Henry Rowland set him the question of whether magnets pushed currents around inside wires), it now sits inside billions of devices: every brushless DC motor, every position sensor in a car engine, every smartphone compass, every hard drive head. The same equation Hall wrote down in his notebook 145 years ago is what makes those sensors work.
The formula:
V_H = (I · B) / (n · q · t)
Where V_H is the Hall voltage, I is the current through the conductor, B is the magnetic field perpendicular to the current, n is the charge carrier density (electrons per m³ for metals, holes per m³ for p-type semiconductors), q is the carrier charge (the elementary charge e = 1.602 × 10⁻¹⁹ C in magnitude), and t is the conductor thickness in the direction of B.
Why it happens:
A current is a flow of charged particles. In a magnetic field, the Lorentz force F = qv × B pushes them sideways. They pile up on one edge of the conductor, leaving the other edge slightly depleted. The resulting transverse electric field (the Hall field) keeps growing until the electric force on each carrier exactly cancels the magnetic deflection. Steady state arrives almost instantly. Measuring the voltage across that transverse direction gives the Hall voltage.
What it tells you:
The sign of V_H reveals the type of charge carrier. Pure electron conduction gives one polarity; hole conduction (in p-type silicon, for example) gives the opposite. This is one of the most direct experimental confirmations that “current” in semiconductors really is carried by positively-charged quasiparticles, not just by electrons moving the other way. It’s how the band-theory of semiconductors got its empirical foundation.
The magnitude tells you the carrier density n. Rearranging:
n = (I · B) / (V_H · q · t)
This is why Hall measurements are the standard way to characterize new semiconductors. You apply a known current and field, measure V_H and t, and out pops the carrier density, a fundamental material property.
Worked example, copper:
Copper has n ≈ 8.5 × 10²⁸ free electrons/m³ (one per atom, with copper’s atomic density). For I = 5 A, B = 0.3 T, t = 2 mm:
V_H = (5 · 0.3) / (8.5 × 10²⁸ · 1.602 × 10⁻¹⁹ · 0.002) = 1.5 / (2.72 × 10⁷) = 5.5 × 10⁻⁸ V = 55 nV
Tiny, barely above nanovoltmeter noise. Metals have huge carrier densities, so the Hall voltage is small. This is why Hall sensors use semiconductors.
Worked example, n-type silicon:
Doped silicon at n = 10²² /m³ (about 0.001% atomic doping), I = 10 mA, B = 0.5 T, t = 1 mm:
V_H = (0.01 · 0.5) / (10²² · 1.602 × 10⁻¹⁹ · 10⁻³) = 0.005 / 1.602 = 3.12 mV
About 60,000× larger than the copper result, even though the current and field are similar. That’s why semiconductors are the practical material for Hall sensors and is why most Hall-effect chips on the market are silicon or GaAs.
Hall coefficient R_H:
The product n·q is often replaced by the Hall coefficient R_H = 1/(n·q). The formula becomes:
V_H = R_H · I · B / t
R_H is a material property, tabulated for common conductors. Positive R_H means hole conduction, negative R_H means electron conduction. Some semiconductors switch sign with temperature as different bands take over conduction — a phenomenon that was a mystery before modern band theory explained it.
Modern applications:
| Use | How |
|---|---|
| Position sensors | A magnet on a moving part triggers a Hall chip; output proportional to magnet proximity. Used in cam/crank sensors, throttle position, gear-shift detection. |
| Current sensing | Run the current through a magnetic core; the field in the core depends on current; a Hall chip in the core’s air gap reads it. No electrical contact with the high-current line. |
| Brushless motors | Three Hall chips around the rotor track its angular position so the controller can switch coils at the right time. |
| Magnetometers | Smartphone compasses use 3-axis Hall sensors to map the Earth’s field. |
| Hard drives (pre-2010) | The read head used a giant-magnetoresistance (GMR) variant; the underlying physics is the same. |
Quantum Hall effects:
At very low temperatures and high magnetic fields, the Hall voltage in 2D electron systems quantizes into precise steps — R_H = h / (q² · ν), where ν is an integer or fraction. The integer quantum Hall effect (Klitzing 1980, Nobel 1985) is now the international standard for the ohm: the kilogram of resistance. The fractional quantum Hall effect (Tsui, Störmer, Laughlin, Nobel 1998) revealed exotic quasiparticles with fractional charge — a subject still being explored in topological quantum computing research.
Practical measurement notes:
- Geometry matters: the formula assumes a rectangular sample with current along one axis and field perpendicular to a flat face. For irregular shapes, the van der Pauw technique generalizes Hall measurements to arbitrary geometries.
- Mobility from Hall: combining the Hall resistance with the sheet resistance (4-point probe) gives the carrier mobility μ = σ/(nq). This is one of the standard semiconductor characterization techniques.
- Temperature: n is roughly constant with T in metals, but rises exponentially with T in semiconductors as more carriers are thermally excited. Always note the temperature when reporting Hall results.
- Contact resistance: Hall measurements are sensitive to contact placement and geometry. Use 4-point contacts spaced symmetrically; offset and thermomagnetic effects can introduce spurious voltages.