I am reading a solved exercise about a parallel plate capacitor in which states that the electric flux density between the 2 plates is:
$$D=p_{s}$$
where $p_{s}$ is the surface current density of one plate.
My question is why is this correct? Isn't the previous relationship a boundary condition which is true only in the surface of the plate? Why is this correct for between the plates also?
In a parallel plate capacitor we assume that the electric field between the plates is uniform, i.e. it doesn't spread out. Given that it doesn't spread out, the electric flux at the surface is the same as the electric flux in between.
In general the electric field is given by $E = \frac{Q}{\epsilon A}$. For a point particle, its electric field spreads out into a sphere, so $A = 4\pi r^2$. Given that $A$ depends on $r$, then the electric flux changes with distance.
However in the case of a uniform field $A$ is constant and for a parallel plate capacitor equal to the area of the capacitor plates. So $E$ doesn't vary with distance between the plates.
$D$ is the flux on any surface $S$ intersected by one of the plates, divided by the intersected area on the plate $A$. Therefore, the actual flux $\Phi = DA$. In addition, from the Gauss Law, $\Phi = Q$, being $Q$ the net charged enclosed by the surface $S$, which is $p_s A$. Finally $D = p_s$ for any surface $S$ (of course, assuming an uniformly distributed charge density
