Distribution of Macroscopic Electric Charge within Conductor

Theorem

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

Then it is not possible for different volume elements of $B$ to have different (macroscopic) electric fields caused by intrinsic imbalance of the electric charges distributed throughout $B$.

Proof

In an electrical conductor, electric charges can flow, because of the action of conduction electrons within the substance.

Hence an electric charge in one point within $B$ will flow to another point within $B$ under a potential difference caused by differences in (macroscopic) electric fields within $B$.

Hence any imbalances caused by internal (macroscopic) electric fields will not be possible.

$\blacksquare$

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