Hong Life CalculatorsUnderstanding the Hall coefficient calculator — concept and principle
This calculator relates the Hall coefficient, the charge-carrier density and the Hall voltage of a conducting strip in a magnetic field. Give it the carrier density to predict the Hall voltage, or a measured Hall voltage to find the density. It is used in solid-state physics labs and to characterise semiconductors and metals.
How it is calculated
R_H = 1 ÷ (n q), negative for electrons and positive for holes
V_H = I B ÷ (n e t)
From a measured voltage: |R_H| = V_H t ÷ (I B) and n = 1 ÷ (|R_H| e)
- n is entered in units of 10²⁸ per m³, V_H in µV, I in A, B in T and the strip thickness t (along B) in mm.
- e = 1.602176634 × 10⁻¹⁹ C.
The principle
Carriers drifting along a strip in a magnetic field feel a sideways magnetic force. They pile up on one edge until the electric field from that charge balances the magnetic force, and the voltage across the strip at that point is the Hall voltage. Fewer carriers must drift faster to carry the same current, so they feel a stronger force and produce a larger voltage; that is why V_H is inversely proportional to n. The sign of the voltage reveals whether the carriers are negative electrons or positive holes.
Worked example
Default inputs: from carrier density, electrons, n = 8.34 × 10²⁸ m⁻³, I = 10 A, B = 1 T, t = 0.1 mm.
- R_H = −1 ÷ (8.34 × 10²⁸ × 1.602 × 10⁻¹⁹) = −7.484 × 10⁻¹¹ m³/C.
- V_H = 10 × 1 ÷ (8.34 × 10²⁸ × 1.602 × 10⁻¹⁹ × 1 × 10⁻⁴) = 7.484 µV.
- Carrier density = 8.34 × 10²⁸ per m³; sign of the carriers: negative (electrons).
How to read the result
Metals have huge carrier densities, so their Hall voltages are only microvolts even with large currents and strong fields. Lightly doped semiconductors have far fewer carriers and give much larger voltages, which is why Hall sensors are made from semiconductors. A positive R_H shows that holes dominate.
Common mistakes
- Using the strip width instead of the thickness along B.
- Forgetting unit prefixes: µV, mm and the 10²⁸ scale.
- Applying the single-carrier formula to materials where electrons and holes both conduct.
- Ignoring an offset voltage from misaligned contacts; reverse B and average to remove it.
Frequently asked questions
Why is the Hall voltage so small in copper?
Copper has about 10²⁹ free electrons per cubic metre, so they drift slowly and the sideways force is small.
What does a positive Hall coefficient mean?
The current is carried mainly by positive holes, as in p-type semiconductors.
How is the Hall effect used in practice?
Hall sensors measure magnetic fields and detect position, speed and current without contact.
Sources: OpenStax, University Physics Volume 2 (2016), section 11.6 "The Hall Effect"