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,

$$\nabla \times {{\bf B}\over \mu} = {\bf J} + {\partial \over \partial t}\left (\epsilon {\bf E}\right ),$$ (3.1)

$$\nabla \times {\bf E} = -{\partial {\bf B}\over \partial t},$$ (3.2)

$$\nabla \cdot {\bf B} = 0,$$ (3.3)

$$\nabla \cdot {\bf E} = {\rho_{c}\over \epsilon},$$ (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

$${\bf J} = \sigma {\bf E},$$

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.