Macroscopic Electric Field in Conductor in Static Equilibrium is Zero
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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