Hall Effect and Hall Voltage

VH
= Hall Voltage across Top and Bottom end of Strip (SI: V)
I
= Current passing through the strip. (SI: A)
= Magnetic Field perpendicular to the current I. (SI: T)
q
= Magnitude of the charge of the charge carriers in the strip. Normally q = e = 1.602x10 -19 C (SI: C)
t
= Thickness of the strip. (SI: m)
w
= Width (or height) of the strip. (SI: m)
rn
= Number of charge carriers per unit volume. (SI: #/m 3)
vd
= Drift velocity of the charge carriers. (SI: m/s)

Derivation:
As positive charges move through the strip they will experience an upward magnetic force due to their motion in the magnetic field.

Initially the positive charges will move towards the top edge of the strip. As the charges pile up, they create an electric potential between the top and bottom edges of the strip that is in the downward direction. Eventually, this potential will create a downward electric force on the charge carriers that will be equal to the upward magnetic force on them. The system will then reach a steady-state equilibrium in which there will be a constant voltage VH across the top and bottom of the strip and a constant current flow I through the strip.

Whwn the system reaches steady-state the vertical electric field created by the pile up of charges is constant in the strip, The the E-field is the gradient of the potential, the vertical E-field is equal to the voltage VH across the strip divided by the distance between the top and bottom edges w.

The current I and the drift velocity vd are related to each other by

Placing these in the force equilibrium equation we can eliminate the drift velocity and solve for the Hall voltage.


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