4.5: Constrained-Charge Transfer Relations

For field sources constrained in their relative distribution, the transfer relation approach cannot only be used for sources confined to boundaries, but can also be used to describe interactions with sources distributed through the bulk of a subregion. The objective in this section is to develop the principles underlying this generalization of the transfer relations for electro quasistatic fields and to summarize useful relations. The method is extended to certain magnetoquasistatic systems in Sec. 4.7.

In a region having a given net charge density \(\rho\) and uniform permittivity \(\varepsilon\), Gauss' law.and the requirement of irrotationality for \(\overrightarrow{E}\) (Eqs. 2.3.23a and 2.3.23b) show that the electric potential \(\phi\) must satisfy Poisson's equation:

\[ \Delta^2 \phi = - \frac{\rho}{\varepsilon} \label{1} \]

In solving this linear equation, consider the solution to be a superposition of a homogeneous part \(\phi_H\) satisfying Laplace's equation and a particular solution \(\phi_P\) which, at each point in the volume of interest, has a Laplacian \(-\rho/ \varepsilon\):

\[ \phi = \phi_H + \phi_P \label{2} \]

It is this latter component that balances the "drive" provided by the charge density when the total solution \(\phi\) is inserted into Equation \ref{1}. By definition

\[ \Delta^2 \phi_P = - \frac{\rho}{\varepsilon} \label{3} \]

\[ \Delta^2 \phi_H = 0 \label{4} \]

In the three standard coordinate systems, the particular solution can be written as a superposition of the same variable-separable solutions used in Sec. 2.16 for the homogeneous solution. Thus,

\[ \phi_P = \begin{cases} Re \, \tilde{\phi}_P (x,t) \, exp[-j(k_y y + k_z z)] & \text{(Cartesian)} \\\\ Re \, \tilde{\phi}_P (r,t) \, exp[-j(m \theta + kz)] & \text{(Cylindrical)} \\\\ Re \, \tilde{\phi}_P (r,t) P_n^m \, (cos \, \theta) exp[-jm \phi] & \text{(Spherical)} \label{5} \end{cases} \]

With \(n\) used to denote the normal component at the respective bounding surfaces of the region described by the transfer relations, the homogeneous transfer relations of Tables 2.16.1, 2.16.2 and 2.16.3relate the components of the homogeneous part of the solutions evaluated at the respective surfaces.Thus, in these relations, the substitution is made

\[ \begin{align} &\tilde{\phi}^{\alpha} \rightarrow \tilde{\phi}^{\alpha}_H = \tilde{\phi}^{\alpha} - \tilde{\phi}^{\alpha}_P; \quad \tilde{\phi}^{\beta} \rightarrow \tilde{\phi}^{\beta}_H = \tilde{\phi}^{\beta} - \tilde{\phi}^{\beta}_P \nonumber \\ &\tilde{D}^{\alpha}_n \rightarrow \tilde{D}^{\alpha}_{nH} = \tilde{D}^{\alpha}_n - \tilde{D}^{\alpha}_{nP}; \quad \tilde{D}^{\beta}_n \rightarrow \tilde{D}^{\beta}_{nH} = \tilde{D}^{\beta}_n - \tilde{D}^{\beta}_{nP} \nonumber \end{align} \label{6} \]

The transfer relations, which take the general form of Eq. 2.17.6, therefore relate the new surface variables and the particular solution evaluated at the surfaces:

\[\begin{bmatrix} \tilde{\phi}^{\alpha} - \tilde{\phi}^{\alpha}_P \\ \tilde{\phi}^{\beta} - \tilde{\phi}^{\beta}_P \end{bmatrix} = \begin{bmatrix} -A_{11} & A_{12} \\ -A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} \tilde{D}_n^{\alpha} - \tilde{D}_{nP}^{\alpha} \\ \tilde{D}_n^{\beta} - \tilde{D}_{nP}^{\beta} \end{bmatrix} \label{7} \]

Multiplied out, the transfer relations for regions with a bulk distribution of charge are

\[\begin{bmatrix} \tilde{\phi}^{\alpha} \\ \tilde{\phi}^{\beta} \end{bmatrix} = \begin{bmatrix} -A_{11} & A_{12} \\ -A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} \tilde{D}_n^{\alpha} \\ \tilde{D}_n^{\beta} \end{bmatrix} + \begin{bmatrix} \tilde{h}^{\alpha} \\ \tilde{h}^{\beta} \end{bmatrix} \label{8} \]

where

\[\begin{bmatrix} \tilde{h}^{\alpha} \\ \tilde{h}^{\beta} \end{bmatrix} = \begin{bmatrix} \tilde{\phi}^{\alpha}_P \\ \tilde{\phi}^{\beta}_P \end{bmatrix} + \begin{bmatrix} A_{11} \tilde{D}_{nP}^{\alpha} & -A_{12} \tilde{D}_{nP}^{\beta} \\ A_{21} \tilde{D}_{nP}^{\alpha} & -A_{22} \tilde{D}_{nP}^{\beta} \end{bmatrix} \label{9} \]

Associated with the surface variables related by these transfer relations are the bulk distributions of potential. These are obtained from the distributions of potential for no charge density by again using the substitutions summarized by Equation \ref{6}. For example, in Cartesian coordinates, the potential distribution is the sum of Eq. 2.16.15 with \(( \tilde{\phi}^{\alpha} , \tilde{\phi}^{\beta})\) replaced by \(( \tilde{\phi}^{\alpha} - \tilde{\phi}^{\alpha}_P , \tilde{\phi}^{\beta} - \tilde{\phi}^{\beta}_p )\) and the particular solution.

\[ \tilde{\phi} = ( \tilde{\phi}^{\alpha} - \tilde{\phi}_P^{\alpha}) \, \frac{sinh \, \gamma x}{ sinh \, \gamma \Delta} - ( \tilde{\phi}^{\beta} - \tilde{\phi}_P^{\beta}) \, \frac{ sinh \, \gamma(x - \Delta)}{ sinh \, \gamma \Delta} + \tilde{\phi}_p(x) \label{10} \]

The same substitution generalizes the cylindrical coordinate potentials, Eqs. 2.16.20, 2.16.21 and2.16.25 as well as those in spherical coordinates, Eq. 2.16.36.

Particular Solutions (Cartesian Coordinates): Any \(\phi_p\) having the form of Equation \ref{5} can be used in Eqs. \ref{8} and \ref{9}. "Inspection" yields solutions in many cases. However, it is often true that the most useful solutions belong to a class that can be generated by the procedure now illustrated in Cartesian coordinates.

Within the planar region (shown in Table 2.16.1) there is a charge distribution that has an arbitrary dependence on the transverse coordinate x but the y-z dependence of Equation \ref{5} a for complex amplitude,Fourier series or Fourier transform representations:

\[ \rho = Re \, \sum_{i = 0}^{\infty} \, \tilde{\rho}_i (t) \Pi_i(x) e^{-j(k_y y + k_z z)} \label{11} \]

Here, the distribution has been represented as a superposition of modes \(\Pi_i (x)\) having individual complex amplitudes \(\tilde{\rho}_i (t)\). These as yet to be determined modes are defined such that the particular solution can be written as a superposition of the same modes:

\[ \phi_P = Re \, \sum_{i = 0}^{\infty} \, \tilde{\phi}_i (t) \Pi_i (x) e^{-j(k_y y + k_z z)} \label{12} \]

The same functions are used for both \(\rho\) and \(\phi_p\) because then substitution into Poisson's equation, Equation \ref{3}, shows that a particular solution has been found, provided that the modes satisfy the Helmholtz equation:

\[ \frac{d^2 \Pi_i}{dx^2} + \nu_i^2 \Pi_i = 0; \quad \nu_i^2 = \frac{\tilde{\rho}_i}{\varepsilon \tilde{\phi}_i} - k_y^2 - k_z^2 \label{13} \]

It follows from Equation \ref{13} that \(\Pi_i\) is a linear combination of \(sin(\nu_i x)\) and \(cos(\nu_i x)\). Boundary conditions, selected as a matter of convenience and to give orthogonal modes that can be used to expand an arbitrary charge distribution in a quickly convergent series, complete the specification of the modes. For example, the transfer relations, Eqs. \ref{8} and \ref{9}, are simplified if

\[ \tilde{D}_{nP}^{\alpha} \equiv - \varepsilon \, \frac{d \tilde{\phi}_P}{dx} \Big|_{\alpha} = 0; \quad \tilde{D}_{nP}^{\beta} \equiv - \varepsilon \, \frac{d \tilde{\phi}_P}{dx} \Big|_{\beta} = 0 \label{14} \]

so these will be used as boundary conditions in solving Equation \ref{13}. It follows that for a layer with \(\alpha\) and \(\beta\) surfaces at \(x = \Delta\) and \(x = 0\), respectively,

\[ \Pi_i = cos \, \nu_i x; \quad \nu_i = \frac{i \pi}{\Delta}; \quad i = 0,1,2,... \label{15} \]

From the definition of \(\nu_i\), Equation \ref{13}, the potential and charge-density amplitudes called for in Eqs. \ref{11} and \ref{12} are related by

\[ \tilde{\phi}_i = \frac{\tilde{\rho}_i}{ \varepsilon ( \nu_i^2 + k_y^2 + k_z^2) } \label{16} \]

The charge-density amplitudes are determined from a given distribution \(Re \, \tilde{\rho}(x,t) \, exp[-j(k_y y + k_z z)]\) by a Fourier analysis. That is, Equation \ref{11} is multiplied by \(\Pi_k\), integrated from \(0 \rightarrow A\), solved for \(\tilde{\rho}_k\) and \(k \rightarrow i\):

\[ \tilde{\rho}_i = \frac{2}{\Delta} \int_o^{\Delta} \, \tilde{\rho}(x,t) \Pi_i (\nu_i x) dx; \quad i \neq 0: \quad \tilde{\rho}_o = \frac{1}{\Delta} \int_o^{\Delta} \, \tilde{\rho} (x,t) dx \label{17} \]

The associated transfer relations, Eqs. \ref{8} and \ref{9} evaluated using Eqs. \ref{12}, \ref{15} and \ref{16}, with \(A_{ij}\)'s from Table 2.16.1, become

\[\begin{bmatrix} \tilde{\phi}^{\alpha} \\ \tilde{\phi}^{\beta} \end{bmatrix} = \frac{1}{\varepsilon_{\gamma}} \begin{bmatrix} -coth \, \gamma \Delta & \frac{1}{sinh \, \gamma \Delta} \\ \frac{-1}{sinh \, \gamma \Delta} & coth \, \gamma \Delta \end{bmatrix} \begin{bmatrix} \tilde{D}^{\alpha}_x \\ \tilde{D}^{\beta}_x \end{bmatrix} + \sum_{i=0}^{+ \infty} \, \frac{ \tilde{\rho}_i}{ \varepsilon ( \nu_i^2 + \gamma^2)} \begin{bmatrix} (-1)^i \\ 1 \end{bmatrix} \label{18} \]

The potential distribution is given in terms of these amplitudes and the particular solution (Eqs. \ref{12},\ref{15} and \ref{16}) by Equation \ref{10}. Note that to make use of Equation \ref{10} the origin of the \(x\) axis need not be coincident with the \(\beta\) surface. The equation applies to a region with the \(\beta\) surface at \(x = a\) if the substitution is made \(x \rightarrow x + a\).

Cylindrical Annulus: In cylindrical coordinates, the given charge distribution and particular solution take the form

\[ \rho = Re \, \sum_{i=0}^{\infty} \, \tilde{\rho}_i (t) \Pi_i (r) e^{-j(m \theta + kz)}; \quad \phi_P = Re \, \sum_{i = 0}^{\infty} \, \tilde{\phi}_i (t) \Pi_i (r) e^{-j(m \theta + kz)} \label{19} \]

Thus, Poisson's equation, Equation \ref{1}, requires that

\[ \frac{d^2 \Pi_i}{dr^2} + \frac{1}{r} \frac{d \Pi_i}{dr} + (\nu_i^2 - \frac{m^2}{r^2}) \, \Pi_i = 0; \quad \nu_i^2 \equiv \frac{\tilde{\rho}_i}{\varepsilon \tilde{\phi}_i} - k^2 \label{20} \]

and the potential amplitudes are related to the charge density amplitudes by

\[ \tilde{\phi}_i = \frac{\tilde{\rho}_i}{\varepsilon (\nu_i^2 + k^2)} \label{21} \]

Boundary conditions used in selecting solutions to Equation \ref{20} might be selected analogous to those of Equation \ref{14}.This would simplify the transfer relations, but require solution of a relatively complicated transcendental equation for the \(nu_i\)'s. Instead, the particular solution is required to vanish on the outer surface only and solutions that are singular at the origin are excluded. In cylindrical coordinates this is sufficient to result in a complete set of orthogonal modes:

\[ \tilde{D}_{rP}^{\alpha} = - \varepsilon \frac{d \tilde{\phi}_P}{dr} \Big |_{\alpha} = 0 \label{22} \]

Comparison of Equation \ref{20} to Eq. 2.16.19 shows that the solutions that are not singular at the originare Bessel's functions of first kind and order \(m\):

\[ \Pi_i = J_m (\nu_ir) \label{23} \]

To satisfy the boundary condition, Equation \ref{22}, the \(nu_i\)'s must be roots of

\[ \nu_i J_m^{'} (\nu_i \alpha) = 0 \label{24} \]

In now evaluating the transfer relations, Eqs. \ref{8} and \ref{9}, the normal flux density is zero at the \(\alpha\) surface, but otherwise all of the particular solution entries make a contribution:

\[\begin{bmatrix} \tilde{\phi}^{\alpha} \\ \tilde{\phi}^{\beta} \end{bmatrix} = \frac{1}{\varepsilon} \begin{bmatrix} F_m ( \beta, \alpha) & G_m (\alpha, \beta) \\ G_m ( \beta, \alpha) & F_m (\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{D}^{\alpha}_r \\ \tilde{D}^{\beta}_r \end{bmatrix} + \sum_{i=0}^{+ \infty} \, \frac{ \tilde{\rho}_i}{ \varepsilon ( \nu_i^2 + \gamma^2)} \begin{bmatrix} J_m ( \nu_i \alpha) + \nu_i G_m (\alpha, \beta) J_m^{'} (\nu_i \beta) \\ J_m ( \nu_i \beta) + \nu_i F_m (\alpha, \beta) J_m^{'} (\nu_i \beta) \end{bmatrix} \label{25} \]

An important limiting case is \(\beta \rightarrow 0\) so that the region is a "solid" cylinder. This limit is most conveniently taken by first using the limiting form of the transfer relation, Eq. (b) of Table 2.16.2,which becomes

\[ \tilde{\phi}^{\alpha} - \tilde{\phi}_P^{\alpha} = \frac{1}{\varepsilon} F_m (0, \alpha) [ \tilde{D}^{\alpha}_r - \tilde{D}^{\alpha}_{rP}] \label{26} \]

Put in the form of Equation \ref{25}, the transfer relation for a solid cylinder is

\[ \tilde{\phi}^{\alpha} = \frac{1}{\varepsilon} F_m (0, \alpha) \tilde{D}_r^{\alpha} + \sum_{i = 0}^{\infty} \, \frac{\tilde{\rho}_i}{\varepsilon (\nu_i^2 + k^2)} J_m ( \nu_i \alpha) \label{27} \]

The charge-density amplitudes \(\tilde{\rho}_i\) are evaluated in terms of the given charge distribution by exploiting the orthogonality of the \(\Pi_i\)'s.

Orthogonality of \(\Pi_i\)'s and Evaluation of Source Distributions: The given transverse distribution of \(\rho\) is used to evaluate the mode amplitudes, \(\Pi_i(x)\) or \(\Pi_i(r)\) and hence \(\tilde{\rho}_i\). Because the particular solutions are in each case a superposition of solutions to the Helmholtz equation, with appropriate boundary conditions, the eigen modes Hi are orthogonal. In the Cartesian coordinate cases, this means that

\[ \int_o^{\Delta} \, \Pi_i (\nu_i x) \Pi_j (\nu_j x) dx = \frac{\Delta}{2} \delta_{ij} \label{28} \]

This relation is the basis for evaluating the Fourier coefficients, for example Equation \ref{17}. Proof of orthogonality and determination of the coefficients is possible in this case by direct integration. But, in the circular geometry, a more powerful method is needed, one based on the properties of \(\Pi_i (\nu_i r)\) that can be deduced from the differential equation and boundary conditions. The proof of orthogonality and determination of the normalizing factor is as follows.

Multiply Equation \ref{20} by \(r \Pi_j\) and integrate from the origin to the outer radius. The first term can then be integrated by parts to obtain

\[ r \Pi_j (\nu_j r) \frac{d \Pi (\nu_i r)}{dr} \Bigg |_o^{\alpha} - \int_o^{\alpha} r \, \frac{d \Pi_i (\nu_i r)}{dr} \frac{d \Pi_i (\nu_i r)}{dr} dr + \int_o^{\alpha} r (\nu_i^2 - \frac{m^2}{r^2}) \Pi_i \Pi_j dr = 0 \label{29} \]

This expression also holds with \(i\) and \(j\) reversed. The latter equation, subtracted from Equation \ref{29}, gives

\[ (\nu_i^2 - \nu_j^2) \int_o^{\alpha} r \Pi_i \Pi_j dr = r \Pi_i \frac{d \Pi_j}{dr} \Bigg |_o^{\alpha} - r \Pi_j \frac{d \Pi_i}{dr} \Bigg |_o^{\alpha} \label{30} \]

Thus, it is clear that either for \(\Pi_i = 0\) or \(d \Pi_i/dr = 0\) at \(r = \alpha\), the function \(\Pi_i\) and \(\Pi_j\) are orthogonal in the sense that the integral in Equation \ref{30} vanishes provided \(i \neq j\).

The value of the integral for \(i = j\) is required in evaluating the coefficients in the charge density expansion, and is deduced by taking the limit where \(\nu_j \rightarrow \nu_i\), or \( \Delta \nu \rightarrow 0\) in \(\nu_j = \nu_i + \Delta \nu\)

\[ \Pi_j (\nu_j r) = \Pi_j [ \nu_i r + (\Delta \nu)r] \simeq \Pi_j (\nu_ir) + [ \Pi_j^{'} (\nu_i r)] r \Delta \nu \label{31} \]

Again, the prime indicates a derivative with respect to the argument \((\nu_j r)\). Expansion of Equation \ref{31} to first order in \(\Delta \nu\) shows that in the limit \( \Delta \nu \rightarrow 0\),

\[ \int_o^{\alpha} r \Pi_i \Pi_j dr = \delta_{ij} \frac{\alpha^2}{2} \Bigg \{ [ \Pi_i^{'} (\nu_i \alpha)]^2 + [ 1 - \frac{m^2}{(\nu_i \alpha)^2}] \Pi_i^2 (\nu_i \alpha) \Bigg \} \label{32} \]

In obtaining this result, the fact that \(\Pi_i\) satisfies Bessel's equation, Equation \ref{20}, has again been used to substitute for \(\Pi_i^{''}\) in terms of \(\Pi_i\) and \(\Pi_i^{'}\).

An example exploiting the cylindrical constrained-charge transfer relations and orthogonality relations is developed in Sec. 4.6.