PX284 - P2 - energy in EM fields

for a conservation law, there must be a continuity equation:

\frac{\partial u}{\partial t} + \underset{energy flux}{\underset{⏟}{\vec{\nabla} \cdot \vec{P}}} = - W

where, $u$ is the energy density, $W$ is the work done per unit volume per unit time

the lorentz force:

\vec{F} = Q \vec{E} = ρ Δ V \vec{E}

where, $ρ$ is the charge density, and $Δ V$ is the volume element

the rate of doing work on $Δ V :$

\begin{matrix} W Δ V = ρ Δ V \vec{E} \cdot \vec{v} = Δ V \vec{E} \cdot (ρ \vec{v}) \\ \vec{J} = ρ \vec{v} \\ ⟹ - W = - \vec{E} \cdot \vec{J} = - \vec{E} \cdot (\frac{\vec{\nabla} \times \vec{B}}{μ_{0}} - ε_{0} \frac{\partial \vec{E}}{\partial t}) \end{matrix}

using the vector identity:

\vec{\nabla} \cdot (\vec{E} \times \vec{B}) = \vec{B} \cdot (\vec{\nabla} \times \vec{E}) - \vec{E} \cdot (\vec{\nabla} \times \vec{B})

\begin{aligned} ⟹ - W & = \frac{\vec{\nabla} \cdot (\vec{E} \times \vec{B}) - \vec{B} \cdot (\vec{\nabla} \times \vec{E})}{μ_{0}} + ε_{0} \vec{E} \cdot \frac{\partial \vec{E}}{\partial t} \\ = \vec{\nabla} \cdot \frac{\vec{E} \times \vec{B}}{μ_{0}} + \frac{1}{μ_{0}} \vec{B} \cdot \frac{\partial \vec{B}}{\partial t} + ε_{0} \vec{E} \frac{\partial \vec{E}}{\partial t} \\ = \vec{\nabla} \cdot \underset{P}{\underset{⏟}{\frac{\vec{E} \times \vec{B}}{μ_{0}}}} + \frac{\partial}{\partial t} \underset{u}{\underset{⏟}{(\frac{B^{2}}{2 μ_{0}} + ε_{0} \frac{E^{2}}{2})}} \\ ∴ - W & = \frac{\partial u}{\partial t} + \vec{\nabla} \cdot \vec{P} \end{aligned}