If we are having a dielectric medium with $\epsilon > \epsilon_0$ and vacuum $\epsilon_0$. I want to find the boundary conditions in this case. Therefore I do:
$$\nabla \cdot \vec D(\vec r)=\rho_{\text{free}}(\vec r)$$
By integrating this and also using Gauss's law you can find:
$$D_{n_{\text{outside}}} – D_{n_{\text{inside}}}= \eta_{\text{free}}(\vec r)$$
Then if we have no free charges: $$D_{n_{\text{outside}}} – D_{n_{\text{inside}}}= 0$$ and from here we find out: $$ D_{n_{\text{outside}}}=D_{n_{\text{inside}}}$$
where in all the above equations: $D_{n_{\text{outside}}}$ normal component of the D-Field outside the dielectric and $D_{n_{\text{outside}}}$ the normal component inside the dielectric.
I have 2 questions about two things:
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What is the free charge density here? The charge inside the dielectric or a charge distribution (characterized by a charge density function) in front of the dielectric?
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What do we mean when we say that we have no free charge density? That we lack charges inside the dielectric or outside of it?
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If you lack free charges, which are the source of the D-field how can you speak about the components of the D-field, when you have no D-field in the first place, since you lack the source that generates this field?
Best Answer
Not sure what you mean by the charge inside the dielectric? The dielectric will be net neutral. The free charge density here refers to the density of mobile conduction charges on the interface between the media. It excludes the (static) charges associated with the induced dipoles in the media.
That there are no mobile conduction charges at the interface and so $\eta_{\rm free}=0$.
The D-field begins and ends on mobile conduction charges that are not on the interface. They could be anywhere else. It is not necessary to have a charge present at the position where you measure the D-field. It just means that the D-field must have zero divergence at that point.