Physics of Electronics

Physicists often use the continuum form of Ohm's Law:[2]


where J is the current density (current per unit area, unlike the simpler I, units of amperes, of Ohm's law), σ is the conductivity (which can be a tensor in anisotropic materials) and E is the electric field (units of volts per meter, unlike the simpler V, units of volts, of Ohm's law). While the notation above does not explicitly depict the variables, each are vectors and each are functions of three position variables. That is, in the case of J, using cartesian coordinates, there are actually three separate equations, one for each component of the vector, each equation having three independent position variables. For example, the components of J in the x, y and z directions would be Jx(x,y,z), Jy(x,y,z) and Jz(x,y,z).


Current flowing through a uniform conductor with a uniform field appliedThe advantage of this form is that it describes electrical behaviour at a single point and does not depend on the geometry of the conductor being considered. It only depends on the material of the conductor which determines the conductivity. That this is a form of Ohm's law can be shown by considering a conductor of length l, uniform cross-sectional area a and a uniform field applied along its length.

The potential difference between two points is defined as


with the element of path along the integration of electric field vector E. For a uniform applied field and defining the voltage in the usual convention of opposite direction to the field;


Substituting current per unit area, J, for I / a (a being the cross section of the conductor), the continuum form becomes:


The electrical resistance of a uniform conductor is given, in terms of conductivity, by:


After substitution Ohm's law takes on the more familiar, yet macroscopic and averaged version:


A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,[3] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.