Macroscopic Electric Field in Conductor in Static Equilibrium is Zero

Theorem

Let $B$ be a body made out of an electrically conducting substance.

Let $B$ be in static equilibrium.

Then the macroscopic electric field within $B$ is everywhere zero.

Proof

By definition of static equilibrium there are no forces upon $B$.

That includes forces caused by an electric field.

From Distribution of Macroscopic Electric Charge within Conductor, the macroscopic electric field is constant throughout $B$.

An electric field arises through differences in electric charges.

As we have seen, there are no such differences.

Hence the result.

$\blacksquare$

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.3$ The macroscopic electric field