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Maxwell's Equations

Electrodynamics is the study of the entire electromagnetic field. We have learned four distinct differential (or integral) equations for the electric and magnetic fields: Gauss's Laws for Electricity and for Magnetism, Ampere's Law (with the Maxwell Displacement Current) and Faraday's Law. Collectively, these are known as:

Maxwell's Equations (ME)

$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$D$}$ $\textstyle =$ $\displaystyle \rho$ (9.1)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} - \frac{\partial \mbox{\boldmath$D$}}{\partial t}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$J$}$ (9.2)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle 0$ (9.3)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (9.4)

These equations are formulated above in in SI units, where $\mbox{\boldmath$D$} = \epsilon\mbox{\boldmath$E$}$ and $\mbox{\boldmath$H$} =
\mbox{\boldmath$B$}/\mu$. $\epsilon $, recall, is the permittivity of the medium, where $\mu$ is called the permeability of the medium. Either of them can in general vary with e.g. position or with frequency, although we will initially consider them to be constants. Indeed, we will often work with them in a vacuum, where $\epsilon_0 = 8.854\times 10^{-12} \frac{\rm C^2}{\rm
N-m^2}$ and $\mu_0 = 4\pi \times 10^{-7} \frac{\rm N}{\rm A^2}$ are the permittivity and permeability of free space, respectfully.

They are related to the (considerably easier to remember) electric and magnetic constants by:

$\displaystyle k_e$ $\textstyle =$ $\displaystyle \frac{1}{4\pi \epsilon_0} = 9\times 10^9 \ \frac{\rm
N-m^2}{\rm C^2}$ (9.5)
$\displaystyle k_m$ $\textstyle =$ $\displaystyle \frac{\mu_0}{4\pi} = 10^{-7} \ \frac{\rm N}{\rm A^2}$ (9.6)

so that
\begin{displaymath}
c = \frac{1}{\sqrt{\epsilon_0\mu_0}} = \sqrt{\frac{k_e}{k_m}} = 3
\times 10^8 \ \frac{\rm m}{\rm sec^2}
\end{displaymath} (9.7)

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 (source-free) equations, and that it is the inhomogeneous forms that are medium-dependent. This is significant for later, remember it. Note also that if magnetic monopoles were discovered tomorrow, we would have to make all four equations inhomogeneous, and incidentally completely symmetric.

For the moment, let us express the inhomogeneous MEs in terms of the electric field $\mbox{\boldmath$E$} = \epsilon \mbox{\boldmath$D$} $ and the magnetic induction $\mbox{\boldmath$B$} = \mbox{\boldmath$H$}/\mu$ directly:


$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle \frac{\rho}{\epsilon}$ (9.8)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$} - \mu \epsilon \frac{\partial \mbox{\boldmath$E$}}{\partial t}$ $\textstyle =$ $\displaystyle \mu \mbox{\boldmath$J$}$ (9.9)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle 0$ (9.10)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (9.11)

It is difficult to convey to you how important these four equations9.1 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 try very hard to 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:

$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle 0$ (9.12)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle 0$ (9.13)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (9.14)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$} - \epsilon\mu \frac{\partial \mbox{\boldmath$E$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (9.15)


next up previous contents
Next: The Wave Equation Up: The Free Space Wave Previous: The Free Space Wave   Contents
Robert G. Brown 2014-08-19