Hall Effect If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor. This is most evident in a thin flat conductor as illustrated. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. The presence of this measurable transverse voltage is called the Hall effect after E. H. Hall who discovered it in 1879.

Note that the direction of the current I in the diagram is that of conventional current, so that the motion of electrons is in the opposite direction. That further confuses all the "right-hand rule" manipulations you have to go through to get the direction of the forces.

The Hall voltage is given by

The Hall effect can be used to measure magnetic fields with a Hall probe.

Hall Voltage for Positive Charge Carriers The transverse voltage (Hall effect) measured in a Hall probe has its origin in the magnetic force on a moving charge carrier.

The magnetic force is where is the drift velocity of the charge.

The current expressed in terms of the drift velocity is

where n is the density of charge carriers. Then

At equilibrium

And substituting gives

Hall Probe The measurement of large magnetic fields on the order of a Tesla is often done by making use of the Hall effect. A thin film Hall probe is placed in the magnetic field and the transverse voltage (on the order of microvolts) is measured.

Sometimes a thin copper film of thickness d on the order of 100 micrometers is used for a Hall probe. Taking the charge carrier density to be

Charge Carriers in the Hall Effect The Hall effect is a conduction phenomenon which is different for different charge carriers. In most common electrical applications, the conventional current is used partly because it makes no difference whether you consider positive or negative charge to be moving. But the Hall voltage has a different polarity for positive and negative charge carriers, and it has been used to study the details of conduction in semiconductors and other materials which show a combination of negative and positive charge carriers.

The Hall effect can be used to measure the average drift velocity of the charge carriers by mechanically moving the Hall probe at different speeds until the Hall voltage disappears, showing that the charge carriers are now not moving with respect to the magnetic field. Other types of investigations of carrier behavior are studied in the quantum Hall effect.

Magnetic Force The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge:

The implications of this expression include:

1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B.

2. The magnitude of the force is F = qvB sin where is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero.

3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product.

From the force relationship above it can be deduced that the units of magnetic field are Newton seconds /(Coulomb meter) or Newtons per Ampere meter. This unit is named the Tesla. It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss.

Lorentz Force Law Both the electric field and magnetic field can be defined from the Lorentz force law:

The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule.

http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/hall.htmHall EffectHall Voltage for Positive Charge CarriersHall ProbeCharge Carriers in the Hall EffectMagnetic ForceLorentz Force Law