## Maxwell equation recap ### Maxwell equation in vacuum Maxwell equation is a set of equations describing electric field $E$ and magnetic field $B$. The general form (in vacuum) is given as $\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_{0}}$ $\mathbf{\nabla} \cdot \mathbf{B} = 0$ $\mathbf{\nabla} \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$ $\mathbf{\nabla} \times \mathbf{B} = \mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}$ The parameters presented here are: - $\rho$, charge density - $\epsilon_{0}$, permittivity of vacuum, $8.854\times 10^{-6} F / m$ - $\mu_{0}$, permeability of vacuum, $1.257\times 10^{-6} Vs / Am$ - $\mathbf{J}$, current density The first equation shows that the divergence of $\mathbf{E}$ is $\frac{\rho}{\epsilon_{0}}$, gives the "source" of an electric field, or say electric field diverges from a point source (Gauss). The second eq means that no magnetic "charge" exist (monopole). The third equation means the changes in magnetic field induce circular current, which is given as the curl of $\mathbf{E}$ (Faraday). And the last equation means currents and changing electric fields cause circular magnetic fields. The term $\mu_{0}\varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}$ is an addition by Maxwell, added to Ampère Law to properly describe the behavior of $\mathbf{B}$ (Ampère-Maxwell). ### Maxwell equation in matter In matter, the case is slightly different because we have to consider the behavior of materials. Therefore, we introduce electric and magnetic susceptibility, which go into $\mathbf{E}$ and $\mathbf{B}$ terms as **polarization** and **magnetization**, and eventually turn them into $\mathbf{D}$ and $\mathbf{H}$, **electric displacement** and **magnetic field intensity**. The equations are as below $\mathbf{\nabla} \cdot \mathbf{D} = \rho_{f}$ $\mathbf{\nabla} \cdot \mathbf{B} = 0$ $\mathbf{\nabla} \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$ $\mathbf{\nabla} \times \mathbf{H} = \mathbf{J_{f}}+ \frac{\partial \mathbf{D}}{\partial t}$ Here$\rho_{f}$ is free charge density, $\mathbf{J_{f}}$ is free current density. $\mathbf{J_{f}}$ may also be written as $\mathbf{J_{ext}}$. $\mathbf{D}$ and $\mathbf{H}$ are considered as auxiliary field for they provide a simplification on original field and material behavior. They are connected by $\mathbf{D} = \varepsilon_{0}\mathbf{E} + \mathbf{P}$ $\mathbf{H} = \frac{1}{\mu_{0}} \mathbf{B} - \mathbf{M}$ $\mathbf{P}$ is polarization and $\mathbf{M}$ is magnetization. These two equations are also termed as *the first and second material equation*. >[!Question]- >When we use the term *electric displacement*, one may have a question that, what this quantity really describe? And one may feel very strange about the direction of $\mathbf{P}$ and $\mathbf{E}$, that for polarization, it has the same direction of [electrical dipole moment](https://en.wikipedia.org/wiki/Electric_dipole_moment), whose direction (with respect to charges) is ***negative to positive***, while for electric field the direction is ***positive to negative***. How can they be added directly with the same + sign? > >The answer is that, $\mathbf{D}$ is just a quantity cannot be measured but show how electric field would lead to **inside** a material. It shows how material affect the electric field, and the result is a combination of polarization and field in vacuum. So the effect is added to combine both parts. >As for the second question, the plus sign, is caused by the field caused by the electrical dipole moment is **not the same as individual charges**. It has the field direction from negative to positive charger, if view it as a point. See the gif below (from https://commons.wikimedia.org/wiki/File:VFPt_dipole_animation_electric.gif). ><img src="https://upload.wikimedia.org/wikipedia/commons/a/aa/VFPt_dipole_animation_electric.gif"> >So the combination is the field in vacuum, added by the polarization of a material. > >The potential of an ideal point dipole is >$V_{\text{ideal dipole}} (r, \theta) = \frac{\mathbf{p_d} \cdot \mathbf{r}}{4 \pi \varepsilon_0 \varepsilon_{\text{d}} r^3} = \frac{p_d \cos \theta}{4 \pi \varepsilon_0 \varepsilon_{\text{d}} r^2}$ Their value is determined by field $\mathbf{E}$ and $\mathbf{H}$ also electric and magnetic susceptibility $\chi_{e}$ and $\chi_{m}$. If we consider only the **linear dielectrics**, we have $\mathbf{P} = \varepsilon_{0} \chi_{e} \mathbf{E}$ $\epsilon = \varepsilon_{0} (1+\chi_{e})$ $\mathbf{D} = \epsilon \mathbf{E}$ $\epsilon_{r} = \frac{\varepsilon}{\varepsilon_{0}}$ $\chi = \varepsilon_{r} -1$ Here we call $\epsilon$ and $\epsilon_{0}$ permittivity of material and relative permittivity of material. Similarly, for linear magnetics, the relations are $\mathbf{M} = \chi_{m} \mathbf{H}$ $\mu = \mu_{0} (1+\chi_{m})$ $\mathbf{H} = \frac{1}{\mu} \mathbf{B}$ $\mu_{r} = \frac{\mu}{\mu_{0}} \approx 1$ $\mu$ and $\mu_{r}$ are permeability of material and relative permeability. Since $\mu_{r}$ has a typical value, we do not often use it as a variable. >[!Notice] >In most case, we will just ignore $\mu_{r}$ not only numerically, but also in equations. >[!Note] >Even here we do not have the linear material assumption, we may also write $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{H} = \frac{1}{\mu} \mathbf{B}$. But here the material constant $\epsilon$ and $\mu$ (or say the relative form $\epsilon_{r}$ and $\mu_{r}$ ) would be field dependent. If write in this way, the polarization term $\mathbf{P}$ would go into this $\epsilon_{r}$ and $\mu_{r}$, and $\epsilon_{r} = \epsilon_{r}(\mathbf{E})$. >The $\epsilon_{r}(\mathbf{E})$ include the nonlinearity properties inside the material property. Therefore, if we wanna study the nonlinear behavior, we should not use $\epsilon_{r}(\mathbf{E})$ but write the polarization term **explicitly**. That is what we do when we discuss nonlinear optics in later chapters. If the nonlinearity is not the primary target, writing the permittivity in this way could simplify the expressions. And this is only the linear case, for nonlinear case, the problem is more complicated and will be discussed extensively in later chapters, see [[Introduction to polarization]]. (The nonlinearity means $\mathbf{P} = \varepsilon_{0} \chi_{e} \mathbf{E}+\varepsilon_{0} \chi_{e}^{(2)} \mathbf{E}^{2} +\varepsilon_{0} \chi_{e}^{(3)} \mathbf{E}^{3}+\cdots$) ### Wave equations and continuity of charges One may know that from Maxwell equations, wave equations of magnetic and electric field could be deduced. Take electric field in vacuum as an example, calculate curl for the third equation $\mathbf{\nabla} \times (\mathbf{\nabla} \times \mathbf{E}) = - \frac{\partial }{\partial t} (\mathbf{\nabla} \times \mathbf{B})$ use $\nabla \times (\nabla \times A) = \nabla(\nabla \cdot A) - \nabla^{2} A$ and the forth equation, we may get $\begin{align} \nabla^2 \mathbf{E} - \nabla(\nabla \cdot \mathbf{E}) &= \mu_0\varepsilon_0\frac{\partial^2 \mathbf{E}}{\partial t^2} \\ \nabla^2 \mathbf{E} &= \mu_0\varepsilon_0\frac{\partial^2 \mathbf{E}}{\partial t^2} \end{align} $ ^79091d We have the wave equation for electric field. Magnetic field would be similar. The detail will be discussed in the next chapter [[Nonlinear wave equation]]. Also, current density $J$ and charge density $\rho$ follows the continuity equation, which means we have $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$This could also be the boundary condition when doing device simulation. ### Boundary conditions for waves across an interface To match fields across a material interface, we should follow the relations below. ![[Drawing 2024-03-30 17.31.28.excalidraw.svg]]$\begin{aligned} D_{\perp,1} &= D_{\perp,2}\\ (\varepsilon_{r,1} E_{\perp,1} &= \varepsilon_{r,2} E_{\perp,2}) \\ E_{\parallel,1} &= E_{\parallel,2} \end{aligned}$$\begin{aligned}H_{\parallel,1} &= H_{\parallel,2}\\ \Bigg(\frac{1}{\mu_{\parallel,1}} B_{\parallel,1} &= \frac{1}{\mu_{\parallel,2}} B_{\parallel,2}\Bigg) \\ B_{\perp,1} &= B_{\perp,2} \end{aligned}$ >[!info] >Read more at https://en.wikipedia.org/wiki/Maxwell%27s_equations >See notes on tensor operation in [[Basic operations]] >In the future, an updated version describing the Maxwell relations under [Lorenz gauge condition](https://en.wikipedia.org/wiki/Lorenz_gauge_condition) are likely to be given.