Conductor

A conductor is a material with a high conductivity, i.e., most electrons are free to move. Perfect conductors are materials with $\sigma =\infty$; this turns out to be a valid approximation for many materials (copper, silver). The opposite effect are in dielectrics.

For a finite current density $\vec{J}$ and infinite conductivity $\sigma$, this implies:

$$\vec{E}=0$$

inside a conductor, from $\vec{E}=\vec{J}/\sigma$. If we assumed $\vec{E}\ne 0$, then the charges would move internally and there would be a contradiction.

Some additional ideas:

• Perfect conductors are equipotential surfaces, i.e., $V=0$, from the below equation.
  • We saw this before with copper wires (in circuits)! We say a node has the same voltage throughout, i.e., it acts as a short circuit and is equipotential!
  • And with the parallel plates of capacitors.

$$V_{21}=-{\int }_1^2\vec{E}\cdot d\vec{l}=0$$

• An external electric field $\vec{E}$ is always perpendicular to equipotential surfaces, i.e., $\vec{E}$ will enter conductors perpendicular to their surface.
  • A nice way of rationalising this is with: $dV=-\vec{E}\cdot d\vec{l}=0$, which implies the dot product between $\vec{E}$ and $d\vec{l}$ (the surface of the conductor) is 0, i.e., those vectors are perpendicular.
  • So the field lines will always hit the surface at a $90°$ angle, implying the below.
• Since field lines always originate or enter into a charge, within a conductor there can be no volume charge density.
  • From Gauss’ law, $\nabla \cdot \vec{E}={\rho }_v/{ϵ}_0$, then ${\rho }_v=0$.
  • i.e., all free charges move to the surface.