PX284 - P3 - flux in EM waves

- W = \frac{\partial u}{\partial t} + \vec{\nabla} \cdot \vec{P}

where, $u = 1 / 2 (ε_{0} E^{2} + B^{2} / μ_{0})$ , is the energy density, and $\vec{P} = (\vec{E} \times \vec{B}) / μ_{0}$ is the poynting vector/energy flux
since $\vec{B} = \frac{1}{ω} \vec{k} \times \vec{E} :$

⟹ \vec{P} = \frac{1}{μ_{0} ω} \vec{E} \times (\vec{k} \times \vec{E}) = \frac{1}{μ_{0} ω} E^{2} \vec{k} = ε_{0} c E^{2} \hat{k}

\vec{P} = \frac{c}{μ_{0}} B^{2} \hat{k} = \frac{1}{2} c (ε_{0} E^{2} + \frac{1}{μ_{0}} B^{2}) \hat{k} = c u \hat{k}