Introductory Physics Volume Two

4.5 Hall Effect 87
But this charge separation will create an electric field, that is in the
opposite direction to the magnetic force, thus the charge will continue
to build up on the face of the conductor until the electric field builds
up to the point where the electric force on the charges balances the
magnetic force on the charges.
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This will happen any time that a current carrying wire is in a magnetic
field: the wire will spontaneously generate an electric field across the
wire (not end to end but across). This effect, discovered by Edwin
Hall, is called the Hall Effect. Because there is an electric field across
the conductor there will also be an electric potential difference (called
the Hall voltage), ∆V = Ew where w is the width of the conductor.
But we know that in order for the electric force to cancel the magnetic
force, we need E = vB. So we see that
∆V = vBw −→ v = ∆V
Bw
Because it is relatively easy to measure the electric potential difference,
the width of the conductor and the magnetic field strength, this type
of device can be used to measure the velocity of the charge carriers.
Knowing the velocity allows one to determine how many charge carriers
there are per volume. In a time dt a quantity of charge dq = I dt passes
out the end of the wire, and a number of electrons dn = dq/e = I dt/e
passes out of the wire. But we can also say that the charges have moved
a distance dx = v dt in this time, so that a section of charge v dt long
has passed out of the wire. Thus a volume of charge dV = hw v dt
has passed out of the wire (where h is the thickness of the wire and hw
is the cross sectional area of the wire). So the number of conduction
electrons per volume (η) is
η = dn
dV = I dt/e
hw v dt = I
ehw v
The carrier density, η, depends on the type of material, not on the
geometry of the material or the amount of current flowing through the
material.
The Hall effect can be used to make a magnetic field sensor. By
using the equation v = ∆V
Bw
to eliminate the velocity from the equation
η =
I
ehw v
, and then solving for B we find that
B = eηh ∆V
I .
The e, η and h are all constants for a given device, while ∆V and I are
I