The density of flux momentum of a spherical wave

The previous calculation of the flux momentum can be extended to spherical waves. The electric and magnetic field of the most simple spherical wave that verifies Maxwell equations is given by

$\displaystyle \vec{E}(r)$ $\displaystyle =$ $\displaystyle -\frac{1}{r}E_0e^{i(kr-\omega t)}\hat{\theta}=E_{\theta}\hat{\theta}$  
$\displaystyle \vec{B}(r)$ $\displaystyle =$ $\displaystyle -\frac{1}{r}B_0e^{i(kr-\omega t)}\hat{\varphi}=E_{\varphi}\hat{\varphi}$ (B.9)

where $ r$ is the distance to the origin of coordinates and $ \hat{\theta}$ and $ \hat{\varphi}$ are the unitary vectors in spherical coordinates. The amplitudes of the electric and magnetic field are related according to $ E_0=cB_0$. Eqs. B.9 represent a diverging spherical wave that emanates from the origin of coordinates. At every time, the amplitude of the wave decreases with the inverse of the distance to the origin ( $ \sim r^{-1}$), but the energy of the wavefront is spread over a spherical surface of area $ 4\pi r^2$. This is consistent with the principle of conservation of energy, since the energy per unit of area (i.e., the intensity) of the wave is proportional to the square of the amplitude. Indeed, the calculation of the Poynting vector can be done in spherical coordinates
$\displaystyle \vec{S}$ $\displaystyle =$ $\displaystyle \frac{1}{\mu_0}\vec{E}\times\vec{B}$  
  $\displaystyle =$ $\displaystyle \frac{1}{\mu_0}E_\theta B_\varphi\cdot\hat{\theta}\times\hat{\varphi}$  
  $\displaystyle =$ $\displaystyle c\epsilon_0 E_\theta^2 \hat{r}$  
  $\displaystyle =$ $\displaystyle \frac{c\epsilon_0 E_0^2}{r^2}\hat{r}$ (B.10)

where the explicit temporal dependence has been omitted ( $ e^{2i(kr-\omega t)}$). Note that the intensity is radial and follows an inversely square law. The total power of the wave ($ L$) is given by the sum of the intensity over an arbitrary surface, and $ L$ is independent of such surface. In the case of a spherical surface $ A$, the total power is independent of the radius:
$\displaystyle L$ $\displaystyle =$ $\displaystyle \int_A \vec{S}\cdot d\vec{a}$  
  $\displaystyle =$ $\displaystyle c\epsilon_0 E_0^2 \int_A \frac{1}{r^2}\hat{r}d\vec{a}$  
  $\displaystyle =$ $\displaystyle c\epsilon_0 E_0^2 \int_A \frac{1}{r^2} \hat{r} \cdot r^2 \hat{r}d\Omega$  
  $\displaystyle =$ $\displaystyle c\epsilon_0 E_0^2 \int_A d\Omega$  
  $\displaystyle =$ $\displaystyle 4\pi c\epsilon_0 E_0^2$ (B.11)

where $ d\vec{a}=r^2 \hat{r} d\Omega$ is the surface element of a spherical shell.

The calculation of the Maxwell Stress tensor can also be done in spherical coordinates. Here we use a more convenient notation to write the components of a tensor. The component $ \widehat{e_i e_j}$ represents the tensor product between the two unitary vectors $ e_i\otimes e_j$, with $ e_i,e_j=\{\hat{r},\hat{\theta},\hat{\varphi}\}$.


$\displaystyle \overleftrightarrow{T}$ $\displaystyle =$ $\displaystyle \epsilon_0\left(E_\theta\hat{\theta}\otimes E_\theta\hat{\theta}-...
...{rr}+\widehat{\theta\theta}+\widehat{\varphi\varphi}\right)E_{\theta}^2\right)+$  
    $\displaystyle +\frac{1}{\mu_0}\left(B_{\varphi}\hat{\varphi}\otimes B_{\varphi}...
...{rr}+\widehat{\theta\theta}+\widehat{\varphi\varphi}\right)B_{\varphi}^2\right)$  
  $\displaystyle =$ $\displaystyle \epsilon_0\left(E_\theta^2\widehat{\theta\theta}-\frac{1}{2}E_\th...
...eft(\widehat{rr}+\widehat{\theta\theta}+\widehat{\varphi\varphi}\right)\right)+$  
    $\displaystyle +\frac{1}{\mu_0}\left(B_{\varphi}^2\widehat{\varphi\varphi}-\frac...
...left(\widehat{rr}+\widehat{\theta\theta}+\widehat{\varphi\varphi}\right)\right)$  
  $\displaystyle =$ $\displaystyle \left(-\frac{\epsilon_0}{2}E_\theta^2-\frac{1}{2\mu_0}B_\varphi^2...
...on_0}{2}E_\theta^2+\frac{1}{2\mu_0}B_\varphi^2\right)\widehat{\varphi\varphi}~.$  

The components $ \widehat{\theta\theta}$ and $ \widehat{\varphi\varphi}$ vanish, because $ B_\varphi=E_\theta/c$ and $ \mu_0^{-1}=\epsilon_0c^2$. The only surviving term is the pressure along the radial direction, which coincides with the direction of propagation of the radiation.

$\displaystyle \overleftrightarrow{T}=-\epsilon_0 E_\theta^2 \widehat{rr}~.$ (B.12)

JM Huguet 2014-02-12