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Quickie Review of Chapter 6

Recall the following morphs of Maxwell's equations, this time with the sources and expressed in terms of potentials by means of the homogeneous equations. Gauss's Law for magnetism is:

\begin{displaymath}
\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}= 0
\end{displaymath} (11.3)

This is an identity if we define $\mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}$:
\begin{displaymath}
\mbox{\boldmath$\nabla$}\cdot (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}) = 0
\end{displaymath} (11.4)

Similarly, Faraday's Law is

$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$} + \frac{\partial \mbox{\boldmath$B$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (11.5)
$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$ (11.6)
$\displaystyle \mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t})$ $\textstyle =$ $\displaystyle 0$ (11.7)

and is satisfied as an identity by a scalar potential such that:
$\displaystyle \mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t}$ $\textstyle =$ $\displaystyle -\mbox{\boldmath$\nabla$}\phi$ (11.8)
$\displaystyle \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t}$ (11.9)

Now we look at the inhomogeneous equations in terms of the potentials. Ampere's Law:

$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle \mu (\mbox{\boldmath$J$}+
\epsilon\frac{\partial \mbox{\boldmath$E$}}{\partial t})$ (11.10)
$\displaystyle \mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$})$ $\textstyle =$ $\displaystyle \mu (\mbox{\boldmath$J$}+
\epsilon\frac{\partial \mbox{\boldmath$E$}}{\partial t})$ (11.11)
$\displaystyle \mbox{\boldmath$\nabla$}(\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}) - \nabla^2\mbox{\boldmath$A$}$ $\textstyle =$ $\displaystyle \mu\mbox{\boldmath$J$}+
\mu\epsilon\frac{\partial \mbox{\boldmath$E$}}{\partial t}$ (11.12)
$\displaystyle \mbox{\boldmath$\nabla$}(\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}) - \nabla^2\mbox{\boldmath$A$}$ $\textstyle =$ $\displaystyle \mu\mbox{\boldmath$J$}-
\mu\epsilon\mbox{\boldmath$\nabla$}\frac{...
...}{\partial t} - \mu\epsilon
\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$ (11.13)
$\displaystyle \nabla^2\mbox{\boldmath$A$}- \mu\epsilon \frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$ $\textstyle =$ $\displaystyle -\mu\mbox{\boldmath$J$}
+ \mbox{\boldmath$\nabla$}(\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}+ \mu\epsilon\frac{\partial \phi}{\partial t})$ (11.14)

Similarly Gauss's Law for the electric field becomes:

$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$E$}$ $\textstyle =$ $\displaystyle \frac{\rho}{\epsilon}$ (11.15)
$\displaystyle \mbox{\boldmath$\nabla$}\cdot \left(-\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t}\right)$ $\textstyle =$ $\displaystyle \frac{\rho}{\epsilon}$ (11.16)
$\displaystyle \nabla^2 \phi + \frac{\partial \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}}{\partial t}$ $\textstyle =$ $\displaystyle -\frac{\rho}{\epsilon}$ (11.17)

In the the Lorentz gauge,

\begin{displaymath}
\mbox{\boldmath$\nabla$}\cdot {\bf A} + \mu\epsilon \frac{\partial \Phi}{\partial t} = 0
\end{displaymath} (11.18)

the potentials satisfy the following inhomogeneous wave equations:
$\displaystyle \nabla^2 \Phi - \mu\epsilon \frac{\partial ^2\Phi}{\partial t^2}$ $\textstyle =$ $\displaystyle -\frac{\rho}{\epsilon}$ (11.19)
$\displaystyle \nabla^2 {\bf A} - \mu\epsilon \frac{\partial ^2{\bf A}}{\partial t^2}$ $\textstyle =$ $\displaystyle -\mu\mbox{\boldmath$J$}$ (11.20)

where $\rho$ and $\mbox{\boldmath$J$}$ are the charge density and current density distributions, respectively. For the time being we will stick with the Lorentz gauge, although the Coulomb gauge:
\begin{displaymath}
\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}= 0
\end{displaymath} (11.21)

is more convenient for certain problems. It is probably worth reminding y'all that the Lorentz gauge condition itself is really just one out of a whole family of choices.

Recall that (or more properly, observe that in its role in these wave equations)

\begin{displaymath}
\mu\epsilon = \frac{1}{v^2}
\end{displaymath} (11.22)

where $v$ is the speed of light in the medium. For the time being, let's just simplify life a bit and agree to work in a vacuum:
\begin{displaymath}
\mu_0\epsilon_0 = \frac{1}{c^2}
\end{displaymath} (11.23)

so that:
$\displaystyle \nabla^2 \Phi - \frac{1}{c^2}\frac{\partial ^2\Phi}{\partial t^2}$ $\textstyle =$ $\displaystyle -
\frac{\rho}{\epsilon_0 }$ (11.24)
$\displaystyle \nabla^2 {\bf A} - \frac{1}{c^2} \frac{\partial ^2{\bf A}}{\partial t^2}$ $\textstyle =$ $\displaystyle -\mu_0\mbox{\boldmath$J$}$ (11.25)

If/when we look at wave sources embedded in a dielectric medium, we can always change back as the general formalism will not be any different.


next up previous contents
Next: Green's Functions for the Up: Maxwell's Equations, Yet Again Previous: Maxwell's Equations, Yet Again   Contents
Robert G. Brown 2013-01-04