What are the Boundary Conditions for Dielectric Interfaces?

[plmokn2]

Homework Statement

Say I have a boundary between two dielectrics then it's easy to show using a gaussian pillarbox that:
D(1)-D(2)=free surface charge density=s
where D(1) is the component of the first medium normal to the surface.
But suppose that there's nothing else apart from two infinite dielectrics with a constant free charge density between then, how would I work out what the actual values of D(1) and D(2) are rather than just the difference?

The Attempt at a Solution

It seems reasonable that the D field should be the same on both sides so that D=s/2 but I'm not sure how I'd prove this?

Any help appreciated.

 

[Mindscrape]

I'm having a hard time interpreting what you are asking. Could you draw it?

 

[plmokn2]

I think my question was badly worded/ I've probably misused some terminology.

I suppose my actual qustion is:
Its obvious from the symmetry that for an uncharged nonconducting non-dielectric that if you put a surface charge density s on it the field above is the same magnitude as the field below: D=s/2.

But suppose that the material is a dielectric then I'm not sure how you prove what the D field is above the surface (in air) and below the surface (in the dielectric) (under the same condition of the dielectric being infinite with surface charge s and no other free charge anywhere)

Hope this is a bit clearer, please say if it isn't.

 

[Mindscrape]

Oh, well D is going to be the same for both air and the dielectric, but E will be different because E is D/(epsilon) which will change between the two (epsilon_0 for air and dielectric epsilon in the dielectric). Is this what you are asking about?

 

[plmokn2]

That's it.

Its probably really obvious but how do you know that D has to be the same on both sides of the boundary?
Thanks

 

[Mindscrape]

It comes from the way D is defined. After its whole redefinition then you just get a Gauss's Law for materials which goes as

$$\iint \mathbf{D} \cdot d\mathbf{a} = \sigma_f$$

and that means that the only thing that you care about is the free charge, which will be the same on both sides.

 

[plmokn2]

Thanks.