Consider the following simple Maxwell's equations:
$$
\nabla\cdot\mathrm{D}=\rho
$$
$$
\nabla\times\mathrm{E}+i\omega\mathrm{B}=0
$$
$$
\nabla\cdot\mathrm{B}=0
$$
$$
\nabla\times\mathrm{H}=\mathrm{J}+i\omega\mathrm{D}
$$
It is well known that at the interface between two media, the boundary condition gives
$$
n\times(\mathrm{E}_1-\mathrm{E}_2)=0
$$
$$
n\cdot(\mathrm{D}_1-\mathrm{D}_2)=\sigma
$$
$$
n\times(\mathrm{H}_1-\mathrm{H}_2)=\mathrm{J}
$$
$$
n\cdot(\mathrm{B}_1-\mathrm{B}_2)=0
$$
Now my question is: are those boundary conditions independent or redundant? Could someone explain them to me? Thank you!
Best Answer
Each boundary condition comes from an independent Maxwell equation, so the four boundary conditions are independent. The right-hand side of the third boundary condition should K, the surface current. The first boundary condition could be replaced by $\phi_1=\phi_2$, which is easier to implement