# Gauss's law for D

1. Aug 28, 2014

### fayled

1. The problem statement, all variables and given/known data
We have a sphere with a polarization kr. I need to show that the electric displacement D=0 everywhere.

2. Relevant equations
closed surfaceD.dS=qfree

3. The attempt at a solution
qfree=0 everywhere so the flux of D is zero everywhere. Clearly D=0 everywhere does solve this, but so could possibly many other things - how do I show D=0 is the solution? This is a very niggly and annoying to think about! Thanks for any help.

2. Aug 28, 2014

### ShayanJ

You only need to consider the spherical symmetry. Because of that, D is radial and can only depend on r and because you're integrating on a sphere, you're not integrating w.r.t. r and so the integrand is a constant.So we have $\int D \hat{r}\cdot dS\hat r=0 \Rightarrow D\int dS=0 \Rightarrow D 4 \pi R^2=0 \Rightarrow D=0$.

3. Aug 28, 2014

### fayled

Ah that was a bit silly of me, thanks.

Another question regarding the D field. My book says that in a homogenous linear dielectric, .Df (free charge density) and xD=0 (I'm fine with that). Then it says D can be found from the free charge just as though the dielectric were not there so D0Evac (where Evac is the field the same free charge distribution would produce in the absence of any dielectric). Then it goes on from here to prove that in such a medium, the vacuum field is reduced by a factor of the relative permittivity, which I'm fine with. I really don't get the reasoning behind the jump from the divergence and curl to D being found as though no dielectric were there. It sort of comes after a discussion about the parallel between E and D.