[NOTES/EM-11002]-Energy Density of Free Electromagnetic Waves

Free electromagnetic wave

1.Energy density

$$ U = \frac{1}{2}\left(\epsilon_0\vec{E}^2+\frac{1}{\mu_0}~\vec{B}^2\right) $$ Poynting vector $$ \vec{N} = \frac{1}{\mu_0}~(\vec{E}\times\vec{B}) $$ Momentum density $$ \vec{p} = \mu_0\epsilon_0\vec{N} = \epsilon_0(\vec{E}\times\vec{B}) $$ Angular momentum density \begin{align*} \vec{\mathcal{L}} = &\vec{r}\times\vec{p}\\ =& \epsilon_0\vec{r}\times(\vec{E}\times\vec{B}) \end{align*} Though it is easy to work with complex form, $\vec{E},\vec{B}$ must be real \begin{align*} \vec{E}=&\vec{E}_0\cos(\vec{k}\cdot\vec{r}-\omega t+\delta)\\ \vec{B} = & \vec{B}_0 \cos(\vec{k}\cdot\vec{r}-\omega t+\delta)\\ |\vec{B}_0| = & |\vec{E}_0|/C\\ \vec{B} = & \frac{\vec{k}\times\vec{E}}{\omega} \end{align*} For plane waves it is simpler to work with complex exponentials.\\ But $\vec{E},\vec{B}$ must be real \begin{align*} \vec{E} = & \vec{E}_0\cos(\vec{k}\cdot\vec{r}-\omega t+\delta)\\ \vec{B} = & \vec{B}_0\cos(\vec{k}\cdot\vec{r}-\omega t+\delta)\\ |\vec{B}_0| = & |\vec{E}_0|/c\qquad\qquad \text{use}~~\bar{\nabla}\times\bar{E}= -~\frac{\partial\vec{B}}{\partial t}\\ \vec{B} = & (\vec{k}\times\vec{E})\\ \end{align*}

2. Energy density of electromagnetic waves

\begin{align*} U =& \frac{1}{2}\left(\epsilon_0\vec{E}^2+\frac{1}{\mu_0}~\vec{B}^2\right)\\ =& \frac{1}{2}\left(\epsilon_0\vec{E}^2+\frac{1}{\mu_0}~c^2\vec{E}^2\right)\\ =&\epsilon_0\vec{E}^2\\ =&\epsilon_0|\vec{E}_0|^2\cos^2(\vec{k}\cdot\vec{r}-\omega t+\delta)\\ =&\frac{1}{2}~\epsilon_0|\vec{E}_0|^2\qquad \text{average over one time period} \end{align*}

3. Flow of energy --- Poynting vector

\begin{align*} \vec{N} = & \frac{1}{\mu_0}(\vec{E}\times\vec{E}) =\frac{1}{\mu_0}|\vec{E}\times\vec{B}|\hat{n}\qquad\quad\hat{n}=\text{unit vector along}~\vec{k}\\ =& \frac{1}{\mu_0}|\vec{E}|~|\vec{B}|~\hat{n} =\frac{1}{\mu_0}|\vec{E}|^2~\frac{1}{c}~\hat{n}\qquad\qquad <\vec{N}>=\frac{1}{2}(\mu_0c)^{-1}|E|^2\\ =&\frac{1}{\mu_0\epsilon_0c }~U\, \hat{n} =(c\,U)\hat{n}\\ =&\text{energy density $\times$ velocity $\times \hat{n}$}\\ &<\vec{N}> = ~c\hat{n} \end{align*}

4. Other physical quantities

Momentum density \begin{align*} \vec{p} = & \epsilon(\vec{E}\times\vec{B}) =\mu_0\epsilon_0\times\vec{N}\\ =&~\frac{1}{c}~\hat{n} \end{align*} Angular momentum density $$ \vec{\mathcal{L}} = \vec{r}\times\vec{p} $$

References: 

1. Sec 27-2  Energy conservation in electromagnetism Sec 27-3 Energy density and energy flow in the electromagnetic fieldR. P. Feynman, Robert B. Leighton and Mathew Sands Lectures on Physics, vol-II, B.I. Publications (1964)

2. Sec 9.2.3 Energy and momentum in electromagnetic waves David Griffiths, Introduction to Electrodynamics , 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).