Electric displacement field

[sindhu1610]

Hello,
I'm going through electrodynamics by griffiths..
I'm unable to understand the case
1) for no free charge and
2)when curl of P(polarisation ) is zero at the boundary
Then what can we comment about D the displacement field having both it's divergence and curl zero at the
boundary.

Please help

 

[Aaron Barnard]

When there is no free charge and the curl of P is zero at the boundary, then the divergence of the displacement field D must also be zero at the boundary. This implies that the electric field E is continuous across the boundary, since E = - ∇D. This means that the electric flux through the boundary must be zero, which is also a consequence of Gauss's Law.