Maxwell's Equations

We begin with Maxwell's Equations (ME):

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$D$} = \rho$$ (9.1)

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} - \frac{\partial \mbox{\boldmath$D$}}{\partial t} = \mbox{\boldmath$J$}$$ (9.2)

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$} = 0$$ (9.3)

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t} = 0$$ (9.4)

in SI units, where $\mbox{\boldmath$D$} = \epsilon\mbox{\boldmath$E$}$ and $\mbox{\boldmath$H$} = \mbox{\boldmath$B$}/\mu$. By this point, remembering these should be second nature, and you should really be able to freely go back and forth between these and their integral formulation, and derive/justify the Maxwell Displacement current in terms of charge conservation, etc. Note that there are two inhomogeneous (source-connected) equations and two homogeneous equations, and that the inhomogeneous forms are the ones that are medium-dependent. This is significant for later, remember it.

For the moment, let us express the inhomogeneous MEs in terms of just $\mbox{\boldmath$E$} = \epsilon \mbox{\boldmath$D$}$ and $\mbox{\boldmath$B$} = \mbox{\boldmath$H$}/\mu$, explicitly showing the permittivity $\epsilon$ and the permeability $\mu$^9.1:

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$} = \frac{\rho}{\epsilon}$$ (9.5)

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$} - \mu \epsilon \frac{\partial \mbox{\boldmath$E$}}{\partial t} = \mu \mbox{\boldmath$J$}$$ (9.6)

It is difficult to convey to you how important these four equations are going to be to us over the course of the semester. Over the next few months, then, we will make Maxwell's Equations dance, we will make them sing, we will ``mutilate'' them (turn them into distinct coupled equations for transverse and longitudinal field components, for example) we will couple them, we will transform them into a manifestly covariant form, we will solve them microscopically for a point-like charge in general motion. We will (hopefully) learn them.

For the next two chapters we will primarily be interested in the properties of the field in regions of space without charge (sources). Initially, we'll focus on a vacuum, where there is no dispersion at all; later we'll look a bit at dielectric media and dispersion. In a source-free region, $\rho = 0$ and $\mbox{\boldmath$J$} = 0$ and we obtain Maxwell's Equations in a Source Free Region of Space:

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$} = 0$$ (9.7)

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$} = 0$$ (9.8)

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t} = 0$$ (9.9)

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$} - \epsilon\mu \frac{\partial \mbox{\boldmath$E$}}{\partial t} = 0$$ (9.10)

where for the moment we ignore any possibility of dispersion (frequency dependence in $\epsilon$ or $\mu$).