Standing wave solutions for the Free Maxwell Equations in Vacuum with azimuthal symmetry in cylindrical coordinates

Abstract

The aim of this paper is to get analytical solutions for the Free Maxwell Equations in Vacuum, using a cylindrical coordinate system, considering axial symmetry. The solutions represent steady-state electromagnetic field configurations in the form of closed magnetic surfaces where the magnetic field is time-dependent and tangential on these surfaces, and there is no electric field on them, and each of these closed surfaces includes a ring-like configuration of time-dependent electric field and tangential to the ring, without magnetic field on them. We analyse the temporal variation of the energy density together with the Poynting vector field, which describes the electromagnetic energy flow. We conclude that these configurations behave as standing wave solutions.

Appendix 1: Treatment of other solutions for the electromagnetic field in vacuum

Consider case 1 of solutions (37) and (38) of section 2.1, where $\xi =0$, $AB\ne 0$. In this case, the vector $\textbf{a}(\textbf{r})$ is

$$\begin{aligned} \textbf{a}(\textbf{r})&=\nu (c_{1}\rho +c_{2}\rho ^{-1})(-A\cos {\nu z}+B\sin {\nu z})\hspace{0.2em}\hat{\textbf{e}}_{\rho } \nonumber \\&\quad +k(c_{1}\rho +c_{2}\rho ^{-1})(A\sin {\nu z}+B\cos {\nu z})\hspace{0.2em}\hat{\textbf{e}}_{\phi } \nonumber \\&\quad +2c_{1}\rho (A\sin {\nu z}+B\cos {\nu z})\hspace{0.2em}\hat{\textbf{e}}_{z}. \end{aligned}$$

(68)

We use Eqs. (16) and (17) to separate spatial parts of the electric and magnetic fields. We obtain

$$\begin{aligned} \textbf{e}(\textbf{r})&=-\nu A(c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho } \nonumber \\&\quad +kB(c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\phi } \nonumber \\&\quad +2c_{1}A\rho \sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z}, \end{aligned}$$

(69)

and

$$\begin{aligned} \textbf{b}(\textbf{r})&=\nu B(c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho } \nonumber \\&\quad +kA(c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\phi } \nonumber \\&\quad +2c_{1}B\rho \cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z}. \end{aligned}$$

(70)

Because we want the solutions to correspond again to standing waves, we need to apply the condition $\textbf{e}(\textbf{r})\cdot \textbf{b}(\textbf{r})=0$ to these solutions. We have the two cases $A=0$ and $B=1$ or $A=1$ and $B=0$. For the former conditions

$$\begin{aligned} \textbf{e}(\textbf{r})=k(c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\phi }, \end{aligned}$$

(71)

and

$$\begin{aligned} \textbf{b}(\textbf{r})= \nu (c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }+2c_{1}\rho \cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z}. \end{aligned}$$

(72)

Therefore, the full solution for this case is

$$\begin{aligned} \textbf{E}(\textbf{r},t)=\left[ k(c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\phi } \right] \sin {\omega t}, \end{aligned}$$

(73)

and

$$\begin{aligned} \textbf{B}(\textbf{r},t)= \left[ \nu (c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }+2c_{1}\rho \cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z} \right] \cos {\omega t}. \end{aligned}$$

(74)

For the last conditions $A=1$ and $B=0$ we get

$$\begin{aligned} \textbf{e}(\textbf{r})=-\nu (c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }+2c_{1}\rho \sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z}, \end{aligned}$$

(75)

and

$$\begin{aligned} \textbf{b}(\textbf{r})=k(c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\phi }. \end{aligned}$$

(76)

These results allow us to write the full solution for the second case as

$$\begin{aligned} \textbf{E}(\textbf{r},t)= \left[ -\nu (c_{1}\rho +c_{2}\rho ^{-1})\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }+2c_{1}\rho \sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z} \right] \sin {\omega t}, \end{aligned}$$

(77)

and

$$\begin{aligned} \textbf{B}(\textbf{r},t)=k(c_{1}\rho +c_{2}\rho ^{-1})\sin {\nu z}\cos {\omega t}\hspace{0.2em}\hat{\textbf{e}}_{\phi }. \end{aligned}$$

(78)

These sets of solutions (73)–(74) and (77)–(78) diverge at the origin of coordinates as well as at infinity.

For the case 3, $A=1$, $B=0$, and $\xi \ne 0$, we have

$$\begin{aligned} \textbf{E}(\textbf{r},t) &= {} \left[ -\nu J_{1}(\xi \rho )\cos {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }+\xi J_{0}(\xi \rho )\sin {\nu z}\hspace{0.2em}\hat{\textbf{e}}_{z}\right] \sin {\omega t}, \end{aligned}$$

(79)

$$\begin{aligned} \textbf{B}(\textbf{r},t) &= {} k J_{1}(\xi \rho )\sin {\nu z}\cos {\omega t}\hspace{0.2em}\hat{\textbf{e}}_{\phi }. \end{aligned}$$

(80)

Analogous to the analysis in Sects. 3 and 4, we conclude that in this case, we obtain closed surfaces of the electric field in a hollow flat washer form, where the electric field is tangential over all the surfaces, and inside of each electric closed surface, there is only one magnetic field in the form of a ring.

The energy density for the solutions (79) and (80) is

$$\begin{aligned} W=W_{0}-\left\{ W_{0}-\frac{1}{8\pi } \nu ^{2}J_{1}^{2}(\xi \rho )\sin ^{2}{\nu z} \right\} \cos {2\omega t}, \end{aligned}$$

(81)

where

$$\begin{aligned} W_{0}=\frac{1}{16\pi } \left[ (\nu J_{1}(\xi \rho )\cos {\nu z})^{2} +(k J_{1}(\xi \rho )\sin {\nu z})^{2} +(\xi J_{0}(\xi \rho )\sin {\nu z})^{2} \right] . \end{aligned}$$

(82)

Bot expressions (81) and (82) are different from the energy density given by Eqs. (64) and (65).

The Poyntig vector field is

$$\begin{aligned} \textbf{S}=\frac{\omega }{8\pi }[\nu J_{1}^{2}(\xi \rho )\sin {\nu z}\cos {\nu z} \hspace{0.2em}\hat{\textbf{e}}_{z}-\xi J_{0}(\xi \rho )J_{1}(\xi \rho )\sin ^{2}{\nu z}\hspace{0.2em}\hat{\textbf{e}}_{\rho }]\sin {2\omega t}. \end{aligned}$$

(83)

This expression differs from Eq. (67). Expression (83) implies that the Poynting vector is zero on the electric surfaces and on the magnetic rings.

Both sets of solutions (46)–(47) and (79)–(80), which are convergent, can be taken, separately or combined, as a basis for constructing particular solutions of Maxwell’s equations with axial symmetry and convergent at the origin of coordinates and at infinity. However, the properties of these new solutions depend on the particular problem to be solved.