I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a Dielectric Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says the following:
At an interface between two lossless dielectric materials, no charge or surface current densities will ordinarily exist. Equations (1.31), (1.32), (1.36), and (1.37) then reduce to
$$\hat{n} \cdot \bar{D}_1 = \hat{n} \cdot \bar{D}_2, \tag{1.38a}$$
$$\hat{n} \cdot \bar{B}_1 = \hat{n} \cdot \bar{B}_2, \tag{1.38b}$$
$$\hat{n} \times \bar{E}_1 = \hat{n} \times \bar{E}_2, \tag{1.38c}$$
$$\hat{n} \times \bar{H}_1 = \hat{n} \times \bar{H}_2. \tag{1.38d}$$In words, these equations state that the normal components of $\bar{D}$ and $\bar{B}$ are continuous across the interface, and the tangential components of $\bar{E}$ and $\bar{H}$ are continuous across the interface. Because Maxwell's equations are not all linearly independent, the six boundary conditions in the above equations are not all linearly independent. Thus, the enforcement of (1.38c) and (1.38d) for the four tangential field components, for example, will automatically force the satisfaction of the equations for the continuity of the normal components.
Section Fields at a General Material Interface says the following:
Fields at a General Material Interface
Consider a plane interface between two media, as shown in Figure 1.5. Maxwell's equations in integral form can be used to deduce conditions involving the normal and tangential fields at this interface.
The time-harmonic version of (1.4), where $S$ is the closed "pillbox"-shaped surface shown in Figure 1.6, can be written as
$$\oint_S \bar{D} \cdot d\bar{s} = \int_V \rho \ dv. \tag{1.29}$$
In the limit as $h \to 0$, the contribution of $D_\text{tan}$ through the sidewalls goes to zero, so (1.29) reduces to
$$\Delta SD_{2n} – \Delta SD_{1n} = \Delta S \rho_s,$$
or
$$D_{2n} – D_{1n} = \rho_s, \tag{1.30}$$
where $\rho_s$ is the surface charge density on the interface. In vector form, we can write
$$\hat{n} \cdot (\bar{D}_2 – \bar{D}_1) = \rho_s. \tag{1.31}$$
A similar argument for $\hat{B}$ leads to the result that
$$\hat{n} \cdot \bar{B}_2 = \hat{n} \cdot \bar{B}_1, \tag{1.32}$$
because there is no free magnetic charge.For the tangential components of the electric field we use the phasor form of (1.6),
$$\oint_C \bar{E} \cdot d\bar{l} = -j\omega \int_S \bar{B} \cdot d\bar{s} – \int_S \bar{M} \cdot d \bar{s}, \tag{1.33}$$
in connection with the closed contour $C$ shown in Figure 1.7. In the limit as $h \to 0$, the surface integral of $\bar{B}$ vanishes (because $S = h \Delta \mathscr{l}$ vanishes). The contribution from the surface integral of $\bar{M}$, however, may be nonzero if a magnetic surface current density $\bar{M}_s$ exists on the surface. The Dirac delta function can then be used to write
$$\bar{M} = \bar{M}_s \delta(h), \tag{1.34}$$
where $h$ is a coordinate measured normal from the surface. Equation (1.33) then gives
$$\Delta \mathscr{l} E_{t1} – \Delta \mathscr{l} E_{t2} = -\Delta \mathscr{l} M_s,$$
or
$$E_{t1} – E_{t2} = -M_s, \tag{1.35}$$
which can be generalised in vector form as
$$(\bar{E}_2 – \bar{E}_1) \times \hat{n} = \bar{M}_s. \tag{1.36}$$
A similar argument for the magnetic field leads to
$$\hat{n} \times (\bar{H}_2 – \bar{H}_1) = \bar{J}_s, \tag{1.37}$$
where $\bar{J}_s$ is an electric surface density that may exist at the interface. Equations (1.31), (1.32), (1.36), and (1.37) are the most general expressions for the boundary conditions at an arbitrary interface of materials and/or surface currents.
I'm confused by this part:
Because Maxwell's equations are not all linearly independent, the six boundary conditions in the above equations are not all linearly independent.
What are the six boundary conditions that this is referring to?
Best Answer
The six boundary equations being: two for the tangential components of each of $\mathbf{E}$ and $\mathbf{H}$, and one for the normal components of each of $\mathbf{E}$ and $\mathbf{H}$. Or $\mathbf{D}$ and $\mathbf{B}$ if you like, they are of course related by the constitutive relations $\mathbf{D} = \varepsilon \mathbf{E}$ and $\mathbf{B} = \mu \mathbf{H}$.
Maxwell's equations are not independent, as you might remember from my my answer to a previous question of yours. This means that if a result satisfies boundary condition 1.38c, it must automatically satisfy 1.38b. The same goes for 1.38d and 1.38a respectively.