Physics /Electromagnetism /Module 1.3

The Hall Effect

Explore the historic experiment that proved electrons are the moving charge carriers in metals, and how crossed electric and magnetic fields create electrostatic equilibrium.

In 1879, a young American physicist named Edwin Hall discovered the Hall Effect , solving one of the deepest mysteries of early electromagnetism. While physicists knew that electric currents responded to magnetic fields, they did not know whether the flowing charges inside a metal were positive or negative. Hall’s elegant crossed-field experiment provided the first definitive proof.

The Mystery of the Charge Carrier

Before the discovery of the electron, electric current was defined by Benjamin Franklin’s convention as the flow of positive charge. However, in solid metals, it was impossible to see whether positive charges were drifting forward or negative charges were drifting backward. Both scenarios result in identical macroscopic currents $I$ and generate the same magnetic fields.

To resolve this, Edwin Hall placed a thin, flat gold leaf strip carrying a current into a powerful perpendicular magnetic field. According to the Lorentz force law, moving charges feel a transverse force:

\vec{F}_B = q(\vec{v}_d \times \vec{B})

If the carriers were positive, their drift velocity $\vec{v}_d$ would align with the current, deflecting them toward one edge of the strip. If they were negative, they would drift in the opposite direction, but the double sign flip (negative charge and opposite velocity direction) would deflect them toward the exact same edge. By measuring the sign of the resulting potential difference—the Hall Voltage ( $V_H$ )—across the width of the strip, Hall could finally determine the sign of the charge carriers.

Crossing the Fields: Electrostatic Equilibrium

As charge carriers accumulate on one edge of the conductor, they leave the opposite edge with a net opposite charge. This separation of charge creates a transverse electric field—the Hall Electric Field ( $\vec{E}_H$ )—pointing across the width of the slab. This electric field exerts an opposing electric force $\vec{F}_E = q\vec{E}_H$ on subsequent drifting charges.

Hover canvas to speed up simulation

At the moment the magnetic field is turned on, the Lorentz force (FB) deflects moving electrons toward the front edge. There is no electric field yet (EH = 0).

Eventually, the accumulation of charge reaches a point where the electric force perfectly balances the magnetic Lorentz force. Once this electrostatic equilibrium is established, charge carriers pass through the conductor completely undeflected. This condition forms the basis of the velocity selector :

F_E = F_B \quad \implies \quad q E_H = q v_d B \quad \implies \quad E_H = v_d B

Deriving the Hall Voltage

How do we determine the exact mathematical strength of the Hall Voltage? Since the electric field inside the slab is completely uniform, we can derive the formula step-by-step.

Click through the visual steps below to watch the algebraic cancellation unfold and discover the final macroscopic equation:

V_H = \textcolor{#10b981}{v_d} B \textcolor{#f59e0b}{w}

1. The Voltage Bridge

Connecting voltage to physical motion

The Hall Voltage (V_H) across the slab depends on three things: how fast the charges are moving (drift velocity, v_d), the strength of the background magnetic field (B), and how wide the conductor path is (width, w).

Why the Sign Matters: Proving Electrons Move

If positive charges were moving rightward to carry current, the magnetic force would push them to the front edge, making it positive. Conversely, if negative charges (electrons) were drifting leftward, the magnetic force would also push them to the front edge, accumulating negative charge there and making the front edge negative.

When Edwin Hall measured the polarity on the gold foil, he found the side where charges accumulated was negative relative to the back. This was the final proof that the charge carriers in metals are indeed negative.

Today, this effect is widely used in Hall sensors to measure magnetic fields, detect mechanical positioning in automotive systems, and determine carrier concentrations and types in semiconductors.

Concept Checks

Crossed-Field Velocity Selector

A velocity selector uses a uniform magnetic field

B = 0.40\text{ T}

crossed perpendicularly with an electric field. What magnitude of electric field

E

is required to allow electrons with a speed of

3.0 \times 10^5\text{ m/s}

to pass through completely undeflected?

Calculating Hall Voltage

A copper strip of width

w = 2.0\text{ cm}

and thickness

t = 0.10\text{ mm}

carries a current of

8.0\text{ A}

in a uniform magnetic field

B = 1.2\text{ T}

perpendicular to the strip. If the charge carrier density of copper is

n = 8.47 \times 10^{28}\text{ electrons/m}^3

and

e = 1.60 \times 10^{-19}\text{ C}

, calculate the magnitude of the measured Hall Voltage.