Hall Effect Formula
Hall effect: V_H = (I * B) / (n * e * t).
Calculate the transverse voltage produced when a magnetic field deflects charge carriers in a conductor.
The Formula
The Hall effect formula gives the transverse voltage — called the Hall voltage — that develops across a current-carrying conductor when it is placed in a perpendicular magnetic field. The effect was discovered by American physicist Edwin Hall in 1879 at Johns Hopkins University. When a magnetic field is applied perpendicular to the current flow, the Lorentz force deflects the moving charge carriers to one side of the conductor. This creates a charge imbalance — and therefore a measurable voltage — across the width of the material.
In the formula, I is the current flowing through the conductor, B is the magnetic field strength perpendicular to the current, n is the number density of charge carriers (electrons or holes per cubic meter), e is the elementary charge (1.602 × 10⁻¹⁹ C), and t is the thickness of the conductor in the direction of the magnetic field. The Hall voltage VH appears across the width of the conductor, perpendicular to both the current and the magnetic field.
The sign of the Hall voltage reveals the type of charge carriers. If electrons (negative charges) carry the current, they deflect in a specific direction and the Hall voltage is negative. If positive holes carry the current — as in p-type semiconductors — the voltage has the opposite sign. This makes the Hall effect one of the most direct methods for determining the type and density of charge carriers in semiconductor materials, which is fundamental to semiconductor physics and device engineering.
The Hall effect is also the operating principle behind Hall sensors, which are widely used to measure magnetic field strength, detect the position of magnets, and sense current without electrical contact. Modern Hall sensors appear in automotive systems, hard disk drives, brushless motors, and smartphones.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| VH | Hall voltage — transverse voltage across the conductor | V |
| I | Current flowing through the conductor | A |
| B | Magnetic field strength perpendicular to current flow | T (Tesla) |
| n | Charge carrier density — number of carriers per unit volume | m⁻³ |
| e | Elementary charge = 1.602 × 10⁻¹⁹ | C |
| t | Thickness of conductor in the direction of the magnetic field | m |
Example 1
A copper strip carries a current of 5 A in a magnetic field of 0.3 T. The strip is 2 mm thick. Copper has a charge carrier density of 8.5 × 10²⁸ m⁻³. What is the Hall voltage?
VH = (5 × 0.3) / (8.5 × 10²⁸ × 1.602 × 10⁻¹⁹ × 0.002)
VH = 1.5 / (8.5 × 10²⁸ × 3.204 × 10⁻²²)
VH = 1.5 / (2.723 × 10⁷) ≈ 5.51 × 10⁻⁸ V
Hall voltage ≈ 55 nV (very small for metals due to high carrier density)
Example 2
A semiconductor has carrier density n = 1 × 10²² m⁻³, current I = 10 mA, B = 0.5 T, thickness t = 1 mm. What is the Hall voltage?
VH = (0.01 × 0.5) / (1 × 10²² × 1.602 × 10⁻¹⁹ × 0.001)
VH = 0.005 / (1 × 10²² × 1.602 × 10⁻²²) = 0.005 / 0.1602
Hall voltage ≈ 31.2 mV (much larger than metals because semiconductors have far fewer carriers)
When to Use It
Apply the Hall effect formula when:
- Measuring the charge carrier density and carrier type in semiconductor samples
- Designing Hall sensors for non-contact measurement of magnetic field strength
- Sensing electrical current without breaking a circuit (Hall current sensors)
- Determining rotor position in brushless DC motors and automotive applications
- Characterizing new semiconductor materials or thin-film devices in research
- Studying the quantum Hall effect in two-dimensional electron systems at cryogenic temperatures