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Potentials

We begin our discussion of potentials by considering the two homogeneous equations. For example, if we wish to associate a potential with $\mbox{\boldmath$B$}$ such that $\mbox{\boldmath$B$}$ is the result of differentiating the potential, we observe that we can satisfy GLM by construction if we suppose a vector potential $\mbox{\boldmath$A$}$ such that:

\begin{displaymath}
\mbox{\boldmath$B$}= \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}
\end{displaymath} (8.28)

In that case:
\begin{displaymath}
\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}= \mbox{\bo...
...cdot (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}) = 0
\end{displaymath} (8.29)

as an identity.

Now consider FL. If we substitute in our expression for $\mbox{\boldmath$B$}$:

$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$}}{\partial t}$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle \mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t})$ $\textstyle =$ $\displaystyle 0$ (8.30)

We can see that if we define:
\begin{displaymath}
\mbox{\boldmath$E$}+ \frac{\partial \mbox{\boldmath$A$}}{\partial t} = -\mbox{\boldmath$\nabla$}\phi
\end{displaymath} (8.31)

then
\begin{displaymath}
\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$E$}+ \frac{...
...box{\boldmath$\nabla$}\times \mbox{\boldmath$\nabla$}\phi = 0
\end{displaymath} (8.32)

is also an identity. This leads to:
\begin{displaymath}
\mbox{\boldmath$E$}= -\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t}
\end{displaymath} (8.33)

Our next chore is to transform the inhomogeneous MEs into equations of motion for these potentials - motion because MEs (and indeed the potentials themselves) are now potentially dynamical equations and not just static. We do this by substituting in the equation for $\mbox{\boldmath$E$}$ into GLE, and the equation for $\mbox{\boldmath$B$}$ into AL. We will work (for the moment) in free space and hence will use the vacuum values for the permittivity and permeability.

The first (GLE) yields:

$\displaystyle \mbox{\boldmath$\nabla$}\cdot (-\mbox{\boldmath$\nabla$}\phi - \frac{\partial \mbox{\boldmath$A$}}{\partial t})$ $\textstyle =$ $\displaystyle \frac{\rho}{\epsilon_0 }$  
$\displaystyle \nabla^2\phi + \frac{\partial (\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$})}{\partial t}$ $\textstyle =$ $\displaystyle -
\frac{\rho}{\epsilon_0 }$ (8.34)

The second (AL) is a bit more work. We start by writing it in terms of $\mbox{\boldmath$B$}$ instead of $\mbox{\boldmath$H$}$ by multiplying out the $\mu_0 $:

$\displaystyle \mbox{\boldmath$\nabla$}\times \mbox{\boldmath$B$}$ $\textstyle =$ $\displaystyle \mu_0 \mbox{\boldmath$J$}+ \mu_0 \epsilon_0 \frac{\partial \mbox{\boldmath$E$}}{\partial t}$  
$\displaystyle \mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$A$})$ $\textstyle =$ $\displaystyle \mu_0 \mbox{\boldmath$J$}+ \mu_0 \epsilon_0 \frac{\partial }{\par...
...\mbox{\boldmath$\nabla$}\phi -
\frac{\partial \mbox{\boldmath$A$}}{\partial t})$  
$\displaystyle -\nabla^2\mbox{\boldmath$A$}+ \mbox{\boldmath$\nabla$}(\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$})$ $\textstyle =$ $\displaystyle \mu_0 \mbox{\boldmath$J$}- \frac{1}{c^2}\mbox{\boldmath$\nabla$}
...
...{\partial t} - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$  
$\displaystyle \nabla^2\mbox{\boldmath$A$}+ - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$ $\textstyle =$ $\displaystyle - \mu_0 \mbox{\boldmath$J$}
+ \mbox{\boldmath$\nabla$}(\mbox{\bo...
...ath$A$})+ \mbox{\boldmath$\nabla$}\frac{1}{c^2}\frac{\partial \phi}{\partial t}$  
$\displaystyle \nabla^2\mbox{\boldmath$A$}+ - \frac{1}{c^2}\frac{\partial^2 \mbox{\boldmath$A$}}{\partial t^2}$ $\textstyle =$ $\displaystyle - \mu_0 \mbox{\boldmath$J$}
+ \mbox{\boldmath$\nabla$}\left (\mb...
...\cdot \mbox{\boldmath$A$}+ \frac{1}{c^2}\frac{\partial \phi}{\partial t}\right)$ (8.35)



Subsections
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Next: Gauge Transformations Up: Maxwell's Equations Previous: The Maxwell Displacement Current   Contents
Robert G. Brown 2013-01-04