Electric Field induces Surface Charge in Conductor/Informal Explanation

Informal Explanation
When the electric field $\mathbf E$ is first applied, there exists a macroscopic electric field $\mathbf E'$ throughout $B$.

Under the influence of $\mathbf E$, the conduction electrons move by Coulomb's Law towards the positive source of $\mathbf E$ and away from the negative source of $\mathbf E$.

Hence the conduction electrons make $P_1$ and $P_2$ charged in the opposite polarity to the electric field at $P_1$ and $P_2$.

From Distribution of Macroscopic Electric Charge within Conductor, it is not possible for there to be different (macroscopic) electric fields throughout $B$.

Hence, away from the surfaces $P_1$ and $P_2$, there is no difference in electric charge throughout the interior of $B$.