It is very important to specify boundary conditions at material interfaces and physical boundaries in order to get a full description of an electromagnetic problem. However the boundary conditions between two media is difficult to memorize. Here we present a way to memorize them.
The maxwell’s equations are
\(\nabla \times H=J+\partial_t D\)
\(\nabla \times E=-\partial_t H\)
\(\nabla \cdot D=\rho\)
\(\nabla \cdot B=0\)
and equation of continuity is
\(\nabla \cdot J+\partial_t \rho=0\)
Now we will get the boundary according to the maxwell’s equations with the following rule instead of thinking the integration near the boundary.
Step 1: replace \(\nabla\) with \( n_2\)
Step 2: replace the electromagnetic field F on the left side with \(F_1-F_2\), while let the electromagnetic field on the ride be zero.
Step 3: add a subscript s to all the source term.
Then the boundary condition would be expressed mathematically as
\(n_2 \times (H_1-H_2)=J_s\)
\(n_2 \times (E_1-E_2)=0\)
\(n_2 \cdot (D_1-D_2)=\rho_s\)
\(n_2 \cdot (B_1-B_2)=0\)
where \(\rho_s\) and \(J_s\) denote surface charge density and surface current density respectively and \(n_2\) is the outward normal from medium 2 to medium 1.
A perfect conductor has infinite electrical conductivity and thus there is no internal electric field. Otherwise, it would produce an infinite energy according to \(J=\sigma E\). As a result, the boundary condition is simplied as the following when media 1 a perfect conductor.
\(-n_2 \times H_2=J_s\)
\(-n_2 \times E_2=0\)
\(-n_2 \cdot D_2=\rho_s\)
\(-n_2 \cdot B_2=0\)
That is all. Hope you enjoy this method.