Eigenvector Fields

According to the method of eigenvalues and eigenvectors we wish to arrive at a solution to

$${\mathcal{A}}=\left[ \begin{array}{c} \phi\\ \hat A_r\\ \hat A_\t... ...n{array}{c} \rho\\ \hat J_r\\ \hat J_\theta\\ \hat J_\varphi \end{array}\right]$$ (694)

in terms of those eigenvectors of Maxwell's wave operator $\mathcal A$ which are proportional to the divergenceless sources in Eq.(6.93). We achieve this goal by first validating that the vector fields

$$\left[ \begin{array}{c} \phi\\ \hat A_r\\ \hat A_\theta\\ \hat A_... ...ay} \right]}_{\equiv\vec{\mathcal{W}}^{(3)}\Psi +\vec{\mathcal{W}}^{(4)}\Phi}~.$$ (695)

are invariant under $\mathcal A$ , i.e. there exist functions $F^{TE},F^{TM},G$ and $H$ such that

$$\mathcal{A} \vec{\mathcal{V}}^{(1)}\Phi^{TE} =\alpha \vec{\mathcal{V}}^{(1)}F^{TE}$$ (696)

$$\mathcal{A} \vec{\mathcal{V}}^{(2)}\Phi^{TM} =\beta \vec{\mathcal{V}}^{(2)}F^{TM}$$ (697)

$$\mathcal{A} [\vec{\mathcal{W}}^{(3)}\Psi+\vec{\mathcal{W}}^{(4)}\Phi] =\gamma \vec{\mathcal{W}}^{(3)}G+\delta~\vec{\mathcal{W}}^{(4)}H~,$$ (698)

where $\alpha,\beta,\gamma$ , and $\delta$ are unique coefficients which are determined by $\mathcal{A}$ . The verification of the form of these three equations and the determination of their scalars $F^{TE},F^{TM},G$ and $H$ in terms of $\Phi^{TE},\Phi^{TM},\Phi$ and $\Psi$ is not unfamiliar. It constitutes relative to spherical coordinates what was done in Section 6.2.3 on page  relative to cartesian coordinates.

There, we recall, we diagonalized the operator $\mathcal A$ by exhibiting the TE, TM, and the TEM eigenvector fields and their respective eigenvalues, all relative to cartesian coordinates. Here we shall do the same relative to spherical coordinates.

At first sight this seems like a computationally intense task, especially because one has to calculate the curl of a curl, $\nabla\times\nabla\times\vec A$ , in Eq.() relative to these coordinates. However, the task becomes managable, in fact, downright pleasant, if one extends to curvilinear coordinates the familiar determinantal expression for the curl,

$$\nabla\times\vec A=\left\vert \begin{array}{ccc} \vec i&\vec j&\vec k\\ \partial_x&\partial_y&\partial_z\\ A_x&A_y&A_z \end{array}\right\vert ~.$$

Suppose one has orthogonal curvilinear coordinates $(x^1,x^2,x^3)$ whose scale factors are $h_i(x^1,x^2,x^3),~i=1,2,3$ . Then one has

$$dx^2+dy^2+dz^2=h_1^2 (dx^1)^2+h_2^2 (dx^2)^2+h_3^2 (dx^3)^2~,$$

and the determinantal expression for the curl is⁶¹⁰

$$\nabla\times\vec A=\left\vert \begin{array}{ccc} h_1\vec e_1&h_2\... ...1\hat A_1&h_2\hat A_2&h_3\hat A_3 \end{array}\right\vert\frac{1}{h_1h_2h_3} ~.$$

Here $\vec e_1,\vec e_2,\vec e_3$ are the o.n. basis vectors tangent to the coordinate lines, and $\hat A_1,\hat A_2,\hat A_3$ are the components of $\vec A$ relative to this o.n. basis. Relative to spherical coordinates one has therefore

$$dx^2+dy^2+dz^2=dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\varphi^2$$

and

$$\nabla\times\vec A=\left\vert \begin{array}{ccc} \vec e_r&r\vec e... ...a&r\sin\theta\,\hat A_\varphi \end{array} \right\vert\frac{1}{r^2\sin\theta} ~.$$ (699)

To exhibit the TE, TM, and TEM eigenvector fields, one inserts each of the vector potential four-vector fields, Eq.() into Eqs.(6.46) on page  and uses Eq.(6.99). One also uses the corresponding sources, Eq.(6.93). The result is as follows:

TE:

$$\left[ \begin{array}{c} 0\\ 0\\ \frac{1}{r\sin\theta}~\partial_\v... ...a \end{array} \right] \left\{ -\partial_t^2 +\partial_r^2 +\frac{1}{r^2}\right. \left.\left( \frac{1}{\sin\theta} \partial_\theta \sin\theta~ \partial_\theta+ \frac{1}{\sin^2\theta}\right) \right\}\Phi^{TE}~~~~~~~~~~~~~$$

$$=4\pi \left[ \begin{array}{c} 0\\ 0\\ \frac{1}{r\sin\theta}~\partial_\varphi \\ -\frac{1}{r}~\partial_\theta \end{array} \right]S^{TE}~,$$ (6100)

TM:

$$\frac{1}{r^2} \left[ \begin{array}{c} \partial_r\\ -\partial_t\\ ... ...end{array} \right]r^2 \left\{ -\partial_t^2 +\partial_r^2 +\frac{1}{r^2}\right. \left.\left( \frac{1}{\sin\theta} \partial_\theta \sin\theta~ \partial_\theta+ \frac{1}{\sin^2\theta}\right) \right\}\Phi^{TM}~~~~~~~~$$

$$=4\pi\frac{1}{r^2} \left[ \begin{array}{c} -\partial_r\\ \partial_t\\ 0\\ 0 \end{array} \right]\left(r^2S^{TM}\right)~,$$ (6101)

TEM:

$$\left[ \begin{array}{c} 0\\ 0\\ \frac{1}{r}~\partial_\theta~ \\ \... ...i \end{array} \right] \left\{ -\partial_t^2 +\partial_r^2 \right\} (\Phi -\Psi) =4\pi \left[ \begin{array}{c} 0\\ 0\\ \frac{1}{r}~\partial_\theta~ \\ \frac{1}{r\sin\theta}~\partial_\varphi \end{array} \right] I$$ (6102)

$$\frac{-1}{r^2}\left[ \begin{array}{c} -\partial_t \\ \partial_r\\... ..._\theta \sin\theta~ \partial_\theta+ \frac{1}{\sin^2\theta}\right) (\Phi -\Psi) =4\pi \frac{1}{r^2} \left[ \begin{array}{c} -\partial_t \\ \partial_r \\ 0\\ 0 \end{array} \right] J~.$$ (6103)

These three systems of vector field equations constitute a step forward in one's understanding of Maxwell's equations. This is because each system, which can be integrated by inspection, yields three master wave equations which

1. are decoupled and hence independent,
2. are inhomogeneous scalar wave equations, each one with its own scalar source,
3. can be solved with the methods developed in Chpter 5 and 6,
4. have solutions from which one derives the three (TE, TM, and TEM) types e.m. fields corresponding to the three types of concomitant e.m. sources.

These three master scalar equations are

$$\textrm{\bf TE:} \left\{ -\partial_t^2 +\partial_r^2 +\frac{1}{r^2}\left( \frac{1}... ...rtial_\theta+ \frac{1}{\sin^2\theta}\partial_\varphi^2\right) \right\}\Phi^{TE} =-4\pi S^{TE}$$ (6104)

$$\textrm{\bf TM:} \left\{ -\partial_t^2 +\partial_r^2 +\frac{1}{r^2}\left( \frac{1}... ...tial_\theta+ \frac{1}{\sin^2\theta} \partial_\varphi^2\right) \right\}\Phi^{TM} =-4\pi S^{TM}$$ (6105)

$$\textrm{\bf TEM:} \left( -\partial_t^2 +\partial_r^2 \right)(\Phi -\Psi) =+4\pi I$$ (6106)

$$\left( \frac{1}{\sin\theta} \partial_\theta \sin\theta~ \partial_\theta+ \frac{1}{\sin^2\theta}\partial_\varphi^2\right) (\Phi -\Psi) =-4\pi J$$ (6107)

The TEM system results in a pair coupled differential equations. However, they are integrable. Their source functions satisfy

$$\left(-\partial_t^2 +\partial_r^2\right)(r^2J) = -\left( \frac{1}{\sin\theta} \partial_\theta \sin\theta~ \partial_\theta+ \frac{1}{\sin^2\theta} \right)I~.$$ (6108)

This is because the left nullspace element $\vec {\mathcal U}_\ell^T$ , Eq.(6.92), annihilates the sum of the two TEM vectors, Eqs.(6.102) and (6.103). This guarantees that one can find a function $\Phi -\Psi$ which satisfies Eqs.(6.106) and (6.107). The validation of this claim is consigned to Exercise 6.2.7 on page .

Whereas the $TE$ and the $TM$ systems are characterized by a single wave equation in 3+1 dimensions, the $TEM$ system is characterized by two different problems in the form of two independent equations:

• A potential problem expressed by Poisson's equation on the transverse ( $\theta,\phi$ )-surface, and
• a wave propagation problem expressed by the wave equation on the longitudinal ($r,t$ )-plane.

The domain of these two problems are orthogonal and independent, but their solutions are not. In fact, they are one and the same. This means that the existence of a solution $\Phi -\Psi$ implies that the source scalars $J^{~}$ and $I^{~}$ are not independent either. Instead, they are related so as to guarantee that the law of charge conservation $\vec \nabla\cdot\vec J+\frac{\partial\rho}{\partial t}=0$ , i.e. Eq.(6.108), is satisfied.

Conversely, as shown in Exercise 6.2.7, this conservation law implies the existence of a solution, $\Phi -\Psi$ , to the two differential equations.

Footnotes

... is⁶¹⁰

This result is a consequence of Stoke's theorem applied to infinitesimal elements of area expressed in terms of these curvilinear coordinates.