2.2.1 The Model of a Free Electron Gas

Basically, a metal is a region containing a large number of electrons which can move freely but are not allowed to leave the region (model of a free electron gas). In a state of neutrality, the free electrons are distributed all across the metal, thus guaranteeing not only global but also local neutrality. The macroscopic electric field is zero inside and outside the metal. If an excess charge is supplied to the metal, the situation is close to the previous simulation (where the positive ions forming a lattice are not shown, as well as the free electron gas). The effect of the core ions is cancelled by that of the free electrons. Only the effect of the excess charge remains. It is his charge what is visualized in the simulations.

The excess charges can also move freely inside the metal, but cannot escape. The Coulomb interaction between these charges makes them to distribute over the surface, producing a zero field inside the metal. The force upon the charges distributed over the surface must be normal to this surface (cancelled by the fact that they are constrained to stay in he metal): any tangent component of the force would result in a redistribution of the charges. Thus, the tangent component must vanish at the steady state.

The vanishing of the field inside the metal is a consequence of the value 2 of the exponent in Coulomb's Law. There is very accurate experimental confirmation of his value, which is known to be exact up to 16 significant digits, through measurements of the vanishing field inside a charged metal.

In an imaginary 2-D world Coulomb's Law would be 1/r, as explained above. This is the reason of the singular behaviour of a 2-D system simulated with a 1/r2 law. To see this, let us go back to a previous simulation.
 

Click "animation" and look at the steady state. A test charge (with value 1/1000 of the rest) is introduced to visualize the field.

It is now observed that the charge does not move to the surface and, though the force upon the charges vanishes as it should be in a state of equilibrium, the field inside the region is not zero everywhere.