# Electromagnetic Waves in Annular Regions
## 3 The Case of the Ring  
The classical equations of electromagnetism in vacuum are well-suited for solutions confined in steady ring-shaped domains. This is made possible by the orthogonality of \(\mathbf{E}\) and \(\mathbf{B}\), as well as by the enforcement of the two divergence-free conditions (\(\mathrm{div}\mathbf{E}=0\) and \(\mathrm{div}\mathbf{B}=0\)). Indeed, when the the two sets of lines of force remain closed and orthogonal, the toroid topology comes very natural. These are typical circumstances where all the eight Equations (1)-(4) (two of scalar type and two of vector type) in the six unknowns \(E_{1},E_{2},E_{3},B_{1},B_{2},B_{3}\) can be totally fulfilled. Other (classical) instances only lead to a partial resolution of the set of [PERSON]'s equations in vacuum. As pointed out in [27; 28; 29], there is plenty of examples where at least one of the equations may remain unsatisfied. This is mainly due to the impossibility to enforce at the same time the wave equation, the divergence-free relation and the boundary conditions. As in the previous section, the recipe is the study of the lower spectrum of a suitable elliptic operator, depending on the shape and magnitude of the sections. In cylindrical coordinates \((r,z,\phi)\), we introduce the potentials:\[\mathbf{A}\ =\ \frac{1}{c}\Big{(}a(t,r,z),\ b(t,r,z),\ 0\Big{)}\qquad\quad\Phi\ =\ -\ \left(\frac{\partial A}{\partial r}+\frac{A}{r}+\frac{\partial B}{\partial z}\right) \tag{18}\]  
where \(a\) and \(b\) are the unknowns to be computed, whereas \(A\) and \(B\) are their primitives with respect to the time variable. The Lorenz condition (6) turns out to be automatically satisfied. The functions \(a\) and \(b\) are defined in a region \(\Omega\) of the plane \((r,z)\). Differently from the previous case, \(\Omega\) is vertically oriented. Since there is no dependence on \(\phi\), the solutions can be naturally extended to a toroid \(\Sigma\) with section \(\Omega\). The axis of \(\Sigma\) will be parallel to the \(z\)-axis (see Figure 3). From (18) we deduce the electromagnetic fields:  
\[\mathbf{E}\ =\ \Big{(}-\frac{\partial\Phi}{\partial r}-\frac{1}{c^{2}}\frac{ \partial a}{\partial t},\ -\ \frac{\partial\Phi}{\partial z}-\frac{1}{c^{2}}\frac{ \partial b}{\partial t},\ 0\Big{)}\qquad\mathbf{B}\ =\ \frac{1}{c^{2}}\Big{(}0,\ 0,\ \frac{ \partial b}{\partial r}-\frac{\partial a}{\partial z}\Big{)} \tag{19}\]  
By requiring that \(\mathbf{E}\) and \(\mathbf{B}\) satisfy (1), we arrive at the following system:  
\[\frac{1}{c^{2}}\frac{\partial^{2}a}{\partial t^{2}}\ =\ \frac{\partial^{2}a}{ \partial z^{2}}\ +\ \frac{\partial}{\partial r}\left(\frac{\partial a}{\partial r}\ +\ \frac{a}{r}\right) \tag{20}\]  
\[\frac{1}{c^{2}}\frac{\partial^{2}b}{\partial t^{2}}\ =\ \frac{\partial^{2}b}{ \partial z^{2}}\ +\ \frac{\partial^{2}b}{\partial r^{2}}\ +\ \frac{1}{r}\frac{ \partial b}{\partial r} \tag{21}\]  
A possible way to couple \(a\) and \(b\) is through the boundary condition on \(\partial\Omega\):  
\[\frac{\partial a}{\partial z}\ =\ \frac{\partial b}{\partial r} \tag{22}\]  
though other options can be examined. Relation (22) amounts to require that \(\ \mathbf{B}=c^{-1}\mathrm{curl}\mathbf{A}=0\) at the contour of \(\Omega\). In this way the field \(\mathbf{E}\) is tangential to the boundary. Indeed, thanks to (18), (20) and (21), one can write:  
\[\mathbf{E}\ =\ \left(-\frac{\partial}{\partial z}\Big{(}\frac{\partial A}{ \partial z}-\frac{\partial B}{\partial r}\Big{)},\ \frac{1}{r}\frac{\partial}{\partial r}\Big{(}r\frac{ \partial A}{\partial z}-r\frac{\partial B}{\partial r}\Big{)},\ 0\right) \tag{23}\]  
which implies that \(\mathbf{E}\) is orthogonal to the gradient of the potential \(\ r\left(\partial A/\partial z-\partial B/\partial r\right)\). Since (22) is valid for all \(t\), such a potential vanishes at the boundary of \(\Omega\), so that its gradient is orthogonal to \(\partial\Omega\). As a consequence \(\mathbf{E}\) is tangential.