Boundary conditions

Let us review the general boundary conditions on the field vectors at a surface between medium 1 and medium 2:

$${\bfm n}\!\cdot\! ({\bfm D}_1 - {\bfm D}_2) = \tau,$$ (1034)

$${\bfm n}\wedge ({\bfm E}_1-{\bfm E}_2) = 0,$$ (1035)

$${\bfm n}\!\cdot\!({\bfm B}_1 - {\bfm B}_2) = 0,$$ (1036)

$${\bfm n}\wedge ({\bfm H}_1-{\bfm H}_2) = {\bfm K},$$ (1037)

where $\tau$ is used for the surface change density (to avoid confusion with the conductivity), and ${\bfm K}$ is the surface current density. Here, ${\bfm n}$ is a unit vector normal to the surface, directed from medium 2 to medium 1. We have seen in Section 4.4 that for normal incidence an electromagnetic wave falls off very rapidly inside the surface of a good conductor. Equation (4.35) implies that in the limit of perfect conductivity ( $\sigma\rightarrow\infty$) the tangential component of ${\bfm E}$ vanishes, whereas that of ${\bfm H}$ may remain finite. Let us examine the behaviour of the normal components.

Let medium 1 be a good conductor for which $\sigma /\epsilon\epsilon_0\omega \gg 1$, whilst medium 2 is a perfect insulator. The surface change density is related to the currents flowing inside the conductor. In fact, the conservation of charge requires that

$${\bfm n}\!\cdot\!{\bfm j} = \frac{\partial\tau}{\partial t} = -{\rm i}\, \omega\,\tau.$$ (1038)

However, ${\bfm n}\!\cdot\!{\bfm j} = {\bfm n}\!\cdot\!\sigma {\bfm E}_1$, so it follows from Eq. (6.1)(a) that

$$\left(1+ \frac{{\rm i}\,\omega\epsilon_0\epsilon_1}{\sigma} ... ...a \epsilon_0\epsilon_2}{\sigma} \,{\bfm n}\!\cdot\!{\bfm E}_2.$$ (1039)

It is clear that the normal component of ${\bfm E}$ within the conductor also becomes vanishingly small as the conductivity approaches infinity.

If ${\bfm E}$ vanishes inside a perfect conductor then the curl of ${\bfm E}$ also vanishes, and the time rate of change of ${\bfm B}$ is correspondingly zero. This implies that there are no oscillatory fields whatever inside such a conductor, and that the boundary values of the fields outside are given by

$${\bfm n}\!\cdot\! {\bfm D} = -\tau,$$ (1040)

$${\bfm n}\wedge {\bfm E} = 0,$$ (1041)

$${\bfm n}\!\cdot\!{\bfm B} = 0,$$ (1042)

$${\bfm n}\wedge {\bfm H} = -{\bfm K}.$$ (1043)

Here, ${\bfm n}$ is a unit normal at the surface of the conductor pointing into the conductor. Thus, the electric field is normal and the magnetic field tangential at the surface of a perfect conductor. For good conductors these boundary conditions yield excellent representations of the geometrical configurations of external fields, but they lead to the neglect of some important features of real fields, such as losses in cavities and signal attenuation in wave guides.

In order to estimate such losses it is useful to see how the tangential and normal fields compare when $\sigma$ is large but finite. Equations (4.5) and (4.34) yield

$${\bfm H} = \frac{1+{\rm i}}{\sqrt{2}} \sqrt{\frac{\sigma}{\mu_0\omega}} \,{\bfm n}\wedge{\bfm E}$$ (1044)

at the surface of a conductor (provided that the wave propagates into the conductor). Let us assume, without obtaining a complete solution, that a wave with ${\bfm H}$ very nearly tangential and ${\bfm E}$ very nearly normal is propagated along the surface of the metal. According to the Faraday-Maxwell equation

$$\vert H_{\parallel}\vert \simeq \frac{k}{\mu_0\omega} \,\vert E_\perp\vert$$ (1045)

just outside the surface, where $k$ is the component of the propagation vector along the surface. However, Eq. (6.5) implies that a tangential component of ${\bfm H}$ is accompanied by a small tangential component of ${\bfm E}$. By comparing these two expressions, we obtain

$$\frac{\vert E_\parallel\vert}{\vert E_\perp\vert}\simeq k \s... ...0 \omega\sigma}} = \frac{d}{{\mathchar'26\mskip-10mu\lambda}},$$ (1046)

where $d$ is the skin depth (see Eq. (4.36)) and ${\mathchar'26\mskip-10mu\lambda}\equiv 1/k$. It is clear that the ratio of the tangential component of ${\bfm E}$ to its normal component is of order the skin depth divided by the wavelength. It is readily demonstrated that the ratio of the normal component of ${\bfm H}$ to its tangential component is of this same magnitude. Thus, we can see that in the limit of high conductivity, which means vanishing skin depth, no fields penetrate the conductor, and the boundary conditions are those given by Eqs. (6.4). Let us investigate the solution of the homogeneous wave equation subject to such boundary conditions.