and conductivity 
=
1/
were introduced.
This page is under construction.
Background: views of resistance
In the early nineteen century, resistance R (and conductance
G
= 1/R) were thought to be sufficient to describe the electric behavior
of materials. R was can be directly obtained from two-terminal
measurement of current I and voltage V. However different shapes of the
same material had different resistance, so a geometry-free quantities of
resistivity
and conductivity =
1/
were introduced.
Different materials can have the same resistivity, while on the other hand the samples of the same material might have different values of resistivity depending on how they were synthesized. Furthermore, in the case of semiconductors, the resistivity alone was not sufficient to explan all phenomena.
Over the years many theories of conduction were developed with varied
success, until the era of quantum mechanics ushered new views on carriers
in crystals. Using the carrier density n and the carrier
mobility 
is
sufficient to describe all observables of measurements performed today.
Current density is equal to electric charge e times the number of carriers that traverses unit area in unit time. Expressed quantitatively:
where I and J are the current and the current
density respectively, A is area, e is the charge
of the electron (e < 0), n is the density
of electrons, p is the density of holes, and 
and 
are the mean velocities of electrons and holes respectively.
The Hall Effect
In 1879 E. H. Hall was investigating a metal strip with a current immersed in a strong magnetic field. He discovered that there was a potential difference accross the strip, in a direction perpendicular to both the current and the magnetic field. The setup is shown in the following sketch:
The Hall effect can in fact be easily explained by a Lorentz force that
acts upon the charge carriers. The carriers consistenly move
to one side of the crystal, creating a potential difference 
.
Defining the x axis along the positive direction of the
current (and thus the direction of driving elctric field E),
z axis along the magnetic field B, and the
y axis perpendicular to the two (as on the sketch), the Lorentz
force 
is compensated by the Hall field, 
so combining this two equations we get 
.
If we replace the average drift velocity with 
,
finally get that 
.
We now define the Hall coefficient:
So to measure 
we need all three: the Hall voltage, the current and the magnetic
field.
Instructions for performing the measurement
The local setup for investigating the Hall Effect has been manufactured
by Klinger (Leybold didactic). The summary of the instructions for
performing this measurement have been written by Mariko Ninomiya and can
be dowloaded from here
(or here in .pdf).
Notes
Given the direction of the current and the orientation of the magnetic field, the side with the higher potential is determined by the charge of the carrier. Hall's work demonstrated that in metals the charge carriers are negative. (We know now that the carrier is the electron.)
