26.12: Electrical Conductivity
In perfect conductors, the electric field inside is always zero due to the abundance of free electrons, which nullify any field by flowing. As a result, any residual charge resides on the surface.
In a practical conductor, an applied electric field may be sustained, causing a flow of electrons, which produce a current. The differential form of the current, the current density, is related to the electric field.
More generally, it is related to the force per unit charge, which involves the magnetic component of the Lorentz force. However, the electrons in conductors move so slowly that this term is negligible. In plasmas, this component is significant.
The current produced and the applied electric field can be related by the definition of current density. Moreover, the definition of the potential difference across the conductor can then be used to relate the current with the potential difference.
The proportionality constant is called the resistance. Other than geometric factors like the length of the conducting wire and the cross-sectional area, it depends inversely on the conductivity, which is called resistivity and is an intrinsic property of the material.
This is called Ohm's law. However, it is not a fundamental law. It is a model of current flow that is experimentally verified.
Since Ohm's law follows from the relationship between the current density and the electric field, the latter is sometimes called the differential form of Ohm's law. Since the former relates the integral forms, the current and the potential difference, it is called the integral form of Ohm's law.
The dependence of resistance on geometric factors can be easily explained. Since a practical conductor has positive ions bound to each other and does not move, electrons must drift past them. Hence, the longer the conductor, the greater the number of bound positive ions and the greater the resistance. On the other hand, increasing the cross-sectional area reduces the resistance simply because that leaves more space for an electron to pass through the positive ions.