Solve Gauss's Law for D: Electric Displacement of Sphere with Polarization kr

[fayled]

Homework Statement

We have a sphere with a polarization kr. I need to show that the electric displacement D=0 everywhere.

Homework Equations

∫_{closed surface}D.dS=q_free

The Attempt at a Solution

q_free=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.

 

[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 $\int D \hat{r}\cdot dS\hat r=0 \Rightarrow D\int dS=0 \Rightarrow D 4 \pi R^2=0 \Rightarrow D=0$.

 

[fayled]

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 $\int D \hat{r}\cdot dS\hat r=0 \Rightarrow D\int dS=0 \Rightarrow D 4 \pi R^2=0 \Rightarrow D=0$.

Ah that was a bit silly of me, thanks.

Another question regarding the D field. My book says that in a homogenous linear dielectric, ∇.D=ρ_f (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 D=ε₀E_vac (where E_vac 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.