Hall effect

Every Hall effect formula looks alike, and every one hangs on a single choice: which side of the sample you call the thickness. Pick the wrong side and the answer is wrong, even when every substitution is clean. Use the wide side instead of the thin one and the answer shrinks by the ratio of the two sides, so a ribbon five times wider than it is thick comes out five times too small.

That choice matters to first-year engineers in Pune. The 2024 pattern Engineering Physics syllabus of Savitribai Phule Pune University (SPPU) asks for the derivation of Hall voltage and Hall coefficient, plus numericals. Its lab asks you to find a semiconductor's Hall coefficient and charge carrier density.

Below you get the meaning in plain words, a four-step card for numericals, worked numbers you can redo, and the sign rule that separates n-type from p-type. Padhna aasaan hai once you sketch the sample properly.

That voltage is the Hall voltage. NIST explains the cause as the Lorentz force, the magnetic push on moving charges.

Carriers are shoved to one face, charge piles up, and a sideways electric field grows until it cancels the magnetic push. After that the voltage stays constant. Edwin Hall discovered the effect in 1879.

The thickness t is the dimension of the sample along the magnetic field. The Hall voltage is read across the width w, at right angles to both current and field. Here n is carrier density (electrons per m³), p is hole density (per m³), e is the electron charge and R_H is the Hall coefficient:

Which side is the thickness t?

• V_H = IB / (n e t)
• R_H = V_H t / (IB), which equals −1/(ne) for electrons and +1/(pe) for holes
• n = 1/(e |R_H|)
• If drift speed v is given instead of current: V_H = B w v

The four-step Hall numerical card

1. Sketch. Draw I, B and the voltage direction, then mark the side B passes through as t. Pass: the voltmeter spans w, never t.
2. Convert. Change mm and cm to m, and cm⁻³ to m⁻³ (multiply by 10⁶). Pass: every length is in metres.
3. Pick the formula. Use the table below. Pass: the quantity you are asked for appears alone on the left of the formula you chose, and every other symbol is given in the question.
4. Check units. R_H must come out in m³/C. Fail: m²/C means t was dropped.

The thickness sits in the denominator of V_H and in the numerator of R_H, and any sheet that flips either one fails a unit check. Run it yourself. A tesla is V·s/m², so I·B has units A·V·s/m². Then V_H/(IB) comes to m²/C, while V_H·t/(IB) gives the m³/C a Hall coefficient needs.

A t in the numerator of V_H is also wrong physically. A thicker sample spreads the same current over more carriers, so the Hall voltage drops. Test any formula sheet you copy with the same unit check before you trust it.

A silver ribbon 1.0 cm by 0.20 cm carrying 100 A in a 1.5 T field gives about 7.9 µV, as in OpenStax University Physics Volume 2, section 11.6 (6 October 2016). Rerunning it with t = 0.20 cm and n = 5.9×10²⁸ m⁻³ gives 100 × 1.5 / (5.9×10²⁸ × 1.602×10⁻¹⁹ × 0.0020) = 7.93 µV.

Metal against semiconductor. Keep I = 10 mA, B = 0.5 T and t = 1 mm. Silver gives 0.53 nV. A silicon sample with an assumed n = 1×10²¹ m⁻³ gives 31 mV, about 5.9×10⁷ times more, because V_H falls as n rises. That is why a semiconductor gives a readable signal and a metal strip barely does.

Read the polarity. A negative Hall coefficient means electrons carry the current (n-type), and a positive one means holes do (p-type), as NIST sets out for semiconductors.

The reason trips people up. For the same current direction, electrons and holes drift in opposite directions and carry opposite charge, so the magnetic force pushes both to the same face. That face collects negative charge in n-type and positive charge in p-type, so the voltage flips. OpenStax College Physics 2e, section 22.6 (13 July 2022) states the same opposite-sign result.

Which terminal counts as positive depends on your sketch, so follow your lab manual's convention. When a material has sizeable numbers of both carriers, the single-carrier formulas no longer hold.

Four neighbours cause most mix-ups:

• Hall voltage and Hall coefficient. The voltage changes with current, field and sample size. The coefficient, 1/(nq) for one carrier type, belongs to the material.
• Mobility. Hall measurements, combined with resistivity, give how freely carriers move, as NIST notes.
• Hall-effect sensors. These use the effect to sense magnetic fields and position, and Hall-effect thrusters use it for spacecraft propulsion, per Wikipedia.
• Quantum Hall effect. This is a separate high-field, low-temperature phenomenon.

It sits in Unit IV, Semiconductor Physics and Ultrasonics (8 hours), of course BSC-102-BES. The unit lists "Hall effect: derivation for Hall voltage and Hall coefficient and related numerical problems" in the SPPU First Year Engineering 2024 Pattern curriculum. Theory is split into 40 marks of continuous assessment and 60 end-semester, with 25 term-work marks. Lab options include finding the Hall coefficient and carrier density, or an Arduino Hall sensor experiment.

Our Engineering Physics course for Pune University (2026-27) is taught by Rahul Jewaani (RG Sir) in Hindi, with pre-recorded digital-board lectures and notes. As of October 2026 it costs ₹1,299 (list price ₹1,599). Browse the Pune University courses in our store.