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Next: Wave Equation in Cylindrical Up: Maxwell's Equations and a Previous: Introduction

Wave Equation from Maxwell's Equations

Maxwell's equations for a region with no charge or current are, in differential form:


$\displaystyle \nabla\cdot{{\bf B}}$ $\textstyle =$ $\displaystyle 0$ (1)
$\displaystyle \nabla\times{{\bf B}}$ $\textstyle =$ $\displaystyle -{1\over c^2}{\partial{\bf E}\over\partial t}$ (2)
$\displaystyle \nabla\cdot{{\bf E}}$ $\textstyle =$ $\displaystyle 0$ (3)
$\displaystyle \nabla\times{{\bf E}}$ $\textstyle =$ $\displaystyle {\partial{\bf B}\over\partial t}$ (4)

Here I have assumed that the the charge density $\rho$ and current density ${\bf J}$ are zero, and that the electric displacement vector can be expressed as ${\bf D} = \epsilon{\bf E}$ and the magnetic flux can be expressed as ${\bf H} = \mu{\bf B}$, which are common assumptions. Thus, the speed $c$ in the equations above refer to the speed of light in the particular medium $c = 1/\sqrt{\epsilon\mu}$, and in the case of a vacuum, $c$ is the standard speed of light in vacuum.

In this problem I will focus exclusively on the magnetic field components. Since the above equations are coupled, it suffices to solve either for the electric field ${\bf E}$ or magnetic field ${\bf B}$, and then the other vector can be inferred. This leads to a set of vector wave equation:

\begin{displaymath}
\nabla^2{\bf B}= {1\over c^2}{\partial^2{\bf B}\over\partial t^2}
\end{displaymath} (5)

which holds for each component of the vector ${\bf B}$. The above equation can be verified by using the vector differential identity,
\begin{displaymath}
\nabla^2{\bf F} = \nabla(\nabla\cdot{\bf F}) - \nabla\times{(\nabla\times{\bf F})}
\end{displaymath} (6)

When taking $\nabla^2{\bf B}$, one can use $\nabla\cdot{{\bf B}} = 0$ to eliminate the first term on the right hand side, leaving only the second curl-of-curl term. The interior curl can be found by again using Maxwell's equations, leaving
\begin{displaymath}
\nabla^2{{\bf B}} = - \nabla\times{\left(-{1\over c^2}{\partial{\bf E}\over\partial t}\right)}.
\end{displaymath} (7)

Upon exchanging the order of differentiation on the right hand side, and using Maxwell's equations one last time for $\nabla\times{{\bf E}}$, I verify the the wave equation (5) is correct.


next up previous
Next: Wave Equation in Cylindrical Up: Maxwell's Equations and a Previous: Introduction
Craig Markwardt 2001-07-30