I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says the following:
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.
(1.6) is given as follows:
Applying Stokes' theorem (B.16) to (1.1a) gives
$$\oint_C \bar{\mathcal{E}} \cdot d\bar{l} = – \dfrac{\partial}{\partial{t}} \int_S \bar{\mathcal{B}} \cdot d \bar{s} – \int_S \bar{\mathcal{M}} \cdot d \bar{s}, \tag{1.6}$$
which, without the $\bar{\mathcal{M}}$ term, is the usual form of Faraday's law and forms the basis for Kirchhoff's voltage law.
Chapter 1.2 Maxwell's Equations introduces Maxwell's equations as follows:
The general form of time-varying Maxwell equations, then, can be written in "point," or differential, form as
$$\nabla \times \overline{\mathcal{E}} = \dfrac{-\partial{\overline{\mathcal{B}}}}{\partial{t}} – \overline{\mathcal{M}}, \tag{1.1a}$$
$$\nabla \times \overline{\mathcal{H}} = \dfrac{\partial{\overline{\mathcal{D}}}}{\partial{t}} + \overline{\mathcal{J}}, \tag{1.1b}$$
$$\nabla \cdot \overline{\mathcal{D}} = \rho, \tag{1.1c}$$
$$\nabla \cdot \overline{\mathcal{B}} = 0 \tag{1.1d}$$
The MKS system of units is used throughout this book. The script quantities represent time-varying vector fields and are real functions of spatial coordinates $x$, $y$, $z$, and the time variable $t$. These quantities are defined as follows:$\overline{\mathcal{E}}$ is the electric field, in volts per meter $(\text{V}/\text{m})$.
$\overline{\mathcal{H}}$ is the magnetic field, in empires per meter $(\text{A}/\text{m})$.
$\overline{\mathcal{D}}$ is the electric flux density, in coulombs per meter squared ($\text{Coul}/\text{m}^2$).
$\overline{\mathcal{B}}$ is the magnetic flux density, in webers per meter squared ($\text{Wb}/\text{m}^2$).
$\overline{\mathcal{M}}$ is the (fictitious) magnetic current density, in volts per meter $(\text{V}/\text{m}^2)$.
$\overline{\mathcal{J}}$ is the electric current density, in amperes per meter squared ($\text{A}/\text{m}^2$).
$\rho$ is the electric charge density, in coulombs per meter cubed ($\text{Coul}/\text{m}^3$).
I'm curious about this part:
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.
What does it mean to say that "$h$ is a coordinate measured normal from the surface"? How does this work in practice?
Best Answer
Let's try to make a parallel with a more common concept. $\overline{\mathcal{M}}$ is just the magnetic current equivalent of charge density $\rho$. One writes $\rho$ as charge divided by volume. So how would you write this if the charge is in a very thin layer on top on some flat surface, say the $x-y$ plane? $\rho=\sigma \delta(z)$, where $\sigma$ is the surface charge density. To see that this makes sense we use $$\int_{-\infty}^{\infty}\delta(z)dz=1$$ and the fact that integrating the charge density over the entire space gives the total charge: $$\rho(x,y,z)=\frac{dQ}{dxdydz}\\\sigma(x,y)=\frac{dQ}{dxdy}\\Q=\int_{\mathbb R^3}\rho(x,y,z) dxdydz=\int_{\mathbb R^3}\sigma(x,y)\delta(z) dxdydz=\int_{\mathbb R^2}\sigma(x,y) dxdy$$ Note that the units for $\sigma$ and $\rho$ differ by a unit of length.
Note: in my formulas above I've used a flat surface. How would these be different if the surface is curved? If you just use a coordinate system that changes from point to point, such as that two axes are in the plane tangent to the surface at the given point and $\hat h$ is the direction along the normal to this plane, then the formulas are exactly the same.
Extending the same concept to magnetism, you just need to change some letters. Just note that this assumes that there are magnetic monopoles. The correct way would be to write equation (1.1d) as $$\nabla \overline{\mathcal{B}}=\rho_m$$ where $\rho_m$ is the magnetic monopole density.