# Electric flux density between the 2 plates of a capacitor

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?

• Puzzled why (at the time of writing this) two people flagged to close as "homework", yet nobody tagged it as such. I think this question is clearly asking about a principle and falls in the scope of "on topic" questions. – Floris Jan 15 '16 at 13:53
• @Floris I can't see the close votes but I can't understand why anyone would vote this as a homework question when I really don't ask anything about homework. – Adam Jan 15 '16 at 14:06
• @Floris: Note that it is not required that a VTC as HW on a post have the HW tag in order to do so. – Kyle Kanos Jan 15 '16 at 23:56
• @KyleKanos I appreciate that - but wouldn't it be a reasonable thing to do? – Floris Jan 16 '16 at 0:01
• @Floris: probably, but sometimes we don't have time beyond a few mouse clicks...or are lazy. – Kyle Kanos Jan 16 '16 at 0:11

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