# Electrostatics and Magnetostatics

# Hall Effect

## Fondamental : Action of a magnetic field on the movement of charge carriers

An electrical conductor under a voltage difference sees its carrier electrons set in motion.

The lesson about Ohm's local law specified the notations.

The electrical conductor is additionally placed in a magnetic field .

Newton's second law applied to a charge carrier is expressed by :

The following picture outlines the geometry of the conductor used : it is a parallelepiped rectangle of length .

The exterior magnetic field is .

The electric field created by the generator which sets charge carriers in motion is called .

In steady-state, electrons move with the velocity .

In steady-state ( ) :

The magnetic term is oriented as Ox axis. Thus an electric field appears, called the Hall field such as :

With :

Thus, the speed of the carriers becomes :

This Hall field is induced by the motion of electrons during a brief moment (length of transitory regime) towards side (2).

This creates a dissymmetry of charges, thus an electric field oriented from side (1) to side (2).

## Fondamental : Calculation of the Hall Voltage

The Hall electric field is :

The Hall voltage is obtained by calculating the line integral around a loop of the field :

Thus :

The intensity of the current is :

Consequently :

If (carrier electrons case, with on the picture) :

Hence we can see the Hall voltage is proportional to the magnetic field that is applied.

Hall probes can measure this voltage and deduce the value of the magnetic field.

## Complément : Laplace force

See the lesson about Laplace force.

## Simulation : Animation JAVA de JJ.Rousseau (Université du Mans)

Hall probe : click HERE