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Maxwell's Equations

We may describe the behaviour of electric and magnetic fields, ${\bf E}$ and ${\bf B}$ through the following differential equations, where $\mu$ and $\epsilon$ are constants,
$\displaystyle \nabla \times {{\bf B}\over \mu}$ $\textstyle =$ $\displaystyle {\bf J} + {\partial \over
\partial t}\left (\epsilon {\bf E}\right ),$ (3.1)
$\displaystyle \nabla \times {\bf E}$ $\textstyle =$ $\displaystyle -{\partial {\bf B}\over \partial t},$ (3.2)
$\displaystyle \nabla \cdot {\bf B}$ $\textstyle =$ $\displaystyle 0,$ (3.3)
$\displaystyle \nabla \cdot {\bf E}$ $\textstyle =$ $\displaystyle {\rho_{c}\over \epsilon},$ (3.4)

where the electric displacement is given in terms of the electric field as ${\bf D} = \epsilon {\bf E}$, $\epsilon$ is the dielectric constant, with a vacuum value of $\epsilon_{0}=8.84\times 10^{-12}farad/m$ and $\rho _{c}$ is the charge density.

The magnetic induction ${\bf B}$ (often referred to as the magnetic field) is related to the magnetic field ${\bf H}$ through ${\bf B} =
\mu {\bf H}$. $\mu$ is the magnetic permeability with a vacuum value of $\mu_{0} = 4\pi \times 10^{-7}henry/m$.

Finally, the electric current ${\bf J}$ is related to the electric field ${\bf E}$ through Ohm's law

\begin{displaymath}
{\bf J} = \sigma {\bf E},
\end{displaymath}

where $\sigma$ is the electrical conductivity.

We have already seen that (3.3) implies that there are no sources of magnetic field (no magnetic monopoles) and flux tubes have a constant strength.


next up previous
Next: Electrostatics Up: Electromagnetism Previous: Introduction
Prof. Alan Hood
2000-11-06