Quiz 1.6

$\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ are given in spherical coordinates by Eq. (1.24). Let us calculate the different terms of Maxwell's equations.

$$\divg\vec{E} =\frac{c}{r^2}\frac{1}{\tan\theta}\cos\phi \divg\vec{B} =0$$

$$\curl\vec{E} =-\uvec{\varphi}\frac{k}{r}\sin\phi \curl\vec{B} =\frac{\uvec{r}}{r^2\tan\theta}\cos\phi +\uvec{\theta}\frac{k}{r}\sin\phi$$

$$\pdiff{t}\vec{E} =\frac{\omega c}{r}\uvec{\theta}\sin\phi \pdiff{t}\vec{B} =\frac{\omega}{r}\uvec{\varphi}\sin\phi$$

For large $r$, we identify

$$\quad:\quad\rho=0$$

$$\quad\uvec{\varphi}:\quad kc=\omega$$

$$\quad\uvec{r}:\quad j_r=0,\quad \uvec{\theta}:\quad j_\theta=0,\quad \uvec{\varphi}:\quad j_\varphi=0$$

In addition

$$\vec{k}\cross\vec{E}_0=k\uvec{r}\cross\frac{c}{r}\uvec{\theta}= \... ...vec{B}_0,\qquad \vec{k}\cdot\vec{E}_0=k\uvec{r}\cdot\frac{c}{r}\uvec{\theta}=0$$

In conclusion, for large $r$ and if the dispersion relation Eq. (1.17) holds, Eq. (1.24) is a solution of Maxwell's equations in a source free region of space.

This is a spherical wave since the phase at any given time is constant when $kr=$const which is the equation of a sphere in spherical coordinates.