5.12: Transfer Relations and Boundary Conditions for Uniform Ohmic Layers

Transport Relations

In a region having uniform conductivity and permittivity, the free charge density is zero unless the material occupying the region can be traced back along a particle line to a source of charge. With the understanding that charge-free bulk regions are being described, it follows from either Gauss' Law or conservation of charge (Eqs. 5.10.4 or 5.10.5) that \(\overrightarrow{E}\) is solenoidal in the bulk of such regions. Because \(\overrightarrow{E}\) is also irrotational (Eq. 5.10.3), it follows that the distribution of potential \(\phi\) is governed by Laplace's equation. To describe the volume field distributions, the same relations are applicable as used to derive the flux-displacement relations of Sec. 2.16. The transfer relations for planar layers, cylindrical annuli and spherical shells summarized in Sec. 2.16 are also applicable to regions having uniform conductivity. Because the effect of material motion on the fields comes from the convection of the free charge density, and \(\rho_f\) is zero in the material, these relations hold even if the material is moving. For example, the planar layer of Table 2.16.1 could be moving in the \(z\) direction with an arbitrary velocity profile.

In conjunction with the transfer relations, the conduction currents normal to the bounding surfaces \((\alpha,\beta)\) are of interest, and these are simply

\[\begin{bmatrix} \tilde{J}_n^{\alpha} \\ \tilde{J}_n^{\beta} \end{bmatrix} = \begin{bmatrix} \tilde{E}_n^{\alpha} \\ \tilde{E}_n^{\beta} \end{bmatrix} \label{1}  \]

where \(n\) signifies a coordinate normal to the \((\alpha,\beta)\) surfaces and \(\sigma\) has the value appropriate to the region between.

 

Conservation of Charge Boundary Condition

A typical model involves two or more materials having uniform properties and separated by interfaces. The boundary condition implied by the requirement that charge be conserved is given with some generality by Eq. 2.10.16. With the proviso that the regions neighboring the interface have the nature described in the previous paragraph, the volume current densities are simply \(\overrightarrow{J}_f = \sigma \overrightarrow{E}\). In certain situations, the interface is itself comprised of a thin region over which the conductivity is appreciably greater than in the bulk. Then, a surface conductivity as is used to model a surface conduction and the surface current density is

\[ \overrightarrow{K}_f^{'} = \sigma_s \overrightarrow{E}_t \rightarrow \overrightarrow{K}_f \overrightarrow{=} \sigma_s \overrightarrow{E}_t + \overrightarrow{v}_t \sigma_f \label{2} \]

where the subscript \(t\) means that only components of the vector tangential to the interlace contribute and \(\sigma_f\) is the surface charge density. Incorporating the appropriate values of \(\overrightarrow{J}_f\) and \(\overrightarrow{K}_f\), the required boundary condition, Eq. 2.10.16, becomes

\[ \frac{\partial{\sigma_f}}{\partial{t}} + \nabla_{\Sigma} \cdot ( \sigma_s \overrightarrow{E}_t + \overrightarrow{v} \sigma_f) + \overrightarrow{n} \cdot [] \sigma \overrightarrow{E} [] = 0 \label{3} \]

The tangential component of \(\overrightarrow{E}\) is continuous at the interface, and so \(\overrightarrow{E}_t\) or the potential can be evaluated on either side of the interface.

As an example used in subsequent sections, suppose that the interface is planar (in the \(y-z\) plane) and moves with the uniform velocity \(U\) in the \(z\) direction. Then, for \(\overrightarrow{n} = \overrightarrow{i}_x\), Eq. \ref{3} becomes

\[ -\Big ( \frac{\partial{\sigma_f}}{\partial{t}} + U \frac{\partial{\sigma_f}}{\partial{z}} \Big ) = \sigma_s \Big ( \frac{\partial{E_y}}{\partial{y}} + U \frac{\partial{E_z}}{\partial{z}} + [] \sigma E_x [] \Big ) \label{4} \]

Physically, this expression states that, for an observer moving with the material, the rate of decrease of af with respect to time is proportional to the conduction current flowing out of the interfacial region in the plane of the interface and to the disparity between volume conduction currents leaving and entering from the bulk regions to either side of the interface.