A2 - energy in fields and waves

energy is conserved
a system with energy density $U [J m^{- 3}]$ can do work and radiate heat, and its environment can also do work or apply heat
the change in internal energy is:

Δ U = Q - W

where, $Q$ is the heat added to the system, and $W$ is the work done by the system

force on a charged particle:

\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})

work done per unit time is:

\vec{F} \cdot \vec{v} = q (\vec{E} \cdot \vec{v} + \underset{0}{\underset{⏟}{(\vec{v} \times \vec{B}) \cdot \vec{v}}})

hence, magnetic fields do no work\
force per unit volume, $\vec{f} :$

\begin{matrix} \vec{f} \cdot \vec{v} = ρ \vec{v} \cdot \vec{E} = \vec{j} \cdot \vec{E} \\ \frac{\partial U}{\partial t} = - \vec{j} \cdot \vec{E} - \vec{\nabla} \cdot \vec{S} \end{matrix}

where, $\vec{S}$ is the Poynting vector

the Poynting vector describes the rate of flow of energy per unit area

\begin{aligned} \frac{\partial U}{\partial t} + \vec{\nabla} \cdot \vec{S} & = (\vec{\nabla} \times \vec{H} - \frac{\partial \vec{B}}{\partial t}) \cdot \vec{E} \\ = \vec{E} \cdot \frac{\partial \vec{B}}{\partial t} - \vec{H} \cdot (\vec{\nabla} \times \vec{E}) + \vec{\nabla} \cdot (\vec{E} \times \vec{H}) \\ = \vec{E} \cdot \frac{\partial \vec{B}}{\partial t} - \vec{H} \cdot \frac{\partial \vec{B}}{\partial t} + \vec{\nabla} \cdot (\vec{E} \times \vec{H}) \end{aligned}

equating div and time components on each side:

\begin{matrix} \vec{S} = \vec{E} \times \vec{H} = \frac{1}{μ_{0}} \vec{E} \times \vec{B} \\ U = \frac{1}{2} (ε_{0} E^{2} + \frac{1}{μ_{0}} B^{2}) \end{matrix}