2 A macroscopic approach to Ohm’s laws

The usual macroscopic approach to electrical conduction is based on the following experimental observations on metallic conductors:

  1. The application of a steady voltage difference \(V\) to a metallic wire produces a steady electric current \(I\) proportional to \(V\). This holds true at least for small values of \(I\) when the temperature of the wire does not increase appreciably.

This allows the definition of the electric resistance as follows, according to the first Ohm’s law:

\[\begin{equation} R=\frac{ V }{ I } \tag{2.1} \end{equation}\]

  1. For high currents the wire temperature increases and the power \(W=VI\) supplied by the generator to the moving charges, instead of accelerating more and more the circulating charge, is converted into heat (Joule effect).

It could appear that the moving charges are subjected to some kind of force, like a body falling in a viscous medium, so that they reach a steady motion and give up part of their kinetic energy to the “body” of the wire (i.e. to the crystal lattice).

  1. The resistance \(R\) increases with increasing temperature.

  2. Using wires of different length \(l\) and sections \(S\) the second Ohm’s law determines the resistance:

\[\begin{equation} R= \frac{ \rho l}{S} \tag{2.2} \end{equation}\]

where the constant \(\rho\) (the electrical resistivity) has a characteristic value for any material and increases with temperature.

The inverse quantity, the electrical conductivity \(\sigma\), can be expressed using the two Ohm’s laws as:

\[\begin{equation} \sigma = \frac{1}{ \rho }=\frac{J}{E} \tag{2.3} \end{equation}\]

where \(J\) is the current density and \(E\) the electric field intensity. This is the starting relation needed to pass to a microscopic picture, that will allow a better understanding of the phenomena.