Electrostatics and Magnetostatics
Local Ohm's law
Fondamental : Drude model presentation (1900)
In a metallic (“ohmic”) conductor under an electric voltage, the carrier electrons begin to move.
Intensity of electric current and current density vector are defined by :
With :
: density of mobile charges ( , where is the density of mobile charges).
: the velocity of charge carriers.
Let be the electric field responsible for the motion of mobile charges.

A mobile charge undergoes an electric force :

It also undergoes a force due to stationary charges which compose the crystal lattice of the metallic conductor.
This force can be modeled.
We model it by a fluid friction :
is a phenomenological constant which depends on the metallic conductor studied.
Fondamental : Fundamentals : ohm's local law and conductivity of a metallic conductor
Newton's second law applied to a mobile charge in the laboratory referential gives : ( is the mass of a charge carrier)
Let be defined by :
is the relaxation time of the ohmic environment.
The differential equation becomes :
Suppose the electric field does not depend on time.
The solution to this equation is :
In a steadystate (for ) :
The current density vector can be deduced :
Let , the electric conductivity of the environment be defined by :
Attention : Ohm's local law
For a conductor whose conductivity is (For a conductor whose conductivity is ) :
The limitation of the carriers migration speed explains Ohm's local law.
The interactions with the matter of the medium (the stationary cations of the metallic grid) slow the charge carriers down.
Order of magnitude :
The charge carriers are electrons.
The following table shows the conductivity of several usual metals at room temperature (300 K) :
Métaux 
Ag 
Cu 
Au 
Al 
Hg 

6,21 
5,88 
4,55 
3,65 
0,10 
We can evaluate the relaxation time for copper :
The steadystate is rapidly reached.
At least as long as the characteristic duration of field variations are greatly superior to .
Thus, Ohm's local law remains true as long as the frequencies of the electric field are not too high (it is called lowfrequency approximation).
Complément : Electric resistance and macroscopic Ohm's Law
Let's consider a cylindrical metallic conductor which has a transverse section and a length (for instance an electric copper cable).
The field inside the wire is (in a steadystate regime) :
According to Ohm's local law :
However :
Thus :
Let :
be the electrical resistance of the wire.
Then (in receiver convention) :
It is the macroscopic Ohm's Law.