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. 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 $$(\bar{E}_2 - \bar{E}_1) \times \hat{n} = \bar{M}_s. \tag{1.36}$$
(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:
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 $$(\bar{E}_2 - \bar{E}_1) \times \hat{n} = \bar{M}_s. \tag{1.36}$$
Why do we have the negative on the RHS of $\Delta \mathscr{l} E_{t1} - \Delta \mathscr{l} E_{t2} = -\Delta \mathscr{l} M_s$? It seems to me that it has something to do with having $E_{t1} - E_{t2}$, and that $E_{t2} - E_{t1}$ instead would have been positive ($\Delta \mathscr{l} E_{t2} - \Delta \mathscr{l} E_{t1} = \Delta \mathscr{l} M_s$). But I don't understand why the order of subtraction for the electric fields matters here. What does it matter if we subtract the electric field of medium 1 from the electric field of medium 2 or vice-versa? What leads to the sign change on the RHS of the equation?
