# 5.10: Charge Relaxation in Deforming Ohmic Conductors

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If it is taken as an empirically substantiated fact that a material at rest is an ohmic conductor,then, moving in an inertial (primed) frame of reference, it is described by the constitutive law

\[ \overrightarrow{J}_f^{'} = \sigma \overrightarrow{E}^{'} \label{1} \]

The conductivity, \(\sigma(\overrightarrow{r},t)\), is a parameter characterizing (and hence tied to) the material. The electroquasistatic transformation laws require that \(\overrightarrow{E}^{'} = \overrightarrow{E}\) but that \( \overrightarrow{J}_f^{'} = \overrightarrow{J}_f - \rho_f \overrightarrow{v} \) (Eqs. 2.5.9a and 2.5.12a) and show that in terms of laboratory-frame variables, the constitutive law implied by Eq. \ref{1} is

\[ \overrightarrow{J}_f = \sigma \overrightarrow{E} + \rho_f \overrightarrow{v} \label{2} \]

With the use of Eq. \ref{2} to describe an accelerating material goes the postulate that the conduction process is not altered by material accelerations. Because of the high collision frequency between charge carriers and the molecules comprising the material, this is usually an excellent assumption.

In this section, it is further assumed that polarization can be modeled in terms of a permittivity \(\varepsilon(\overrightarrow{r},t)\), in general a function of space and time. Like the conductivity \(\varepsilon\) is a property tied to the material. Also, the given material deformations are incompressible: \(\nabla \cdot \overrightarrow{v} = 0\).

The fundamental laws required to define the relaxation process picture \(\overrightarrow{E}\) as irrotational, relate \(\rho_f\) to \(\overrightarrow{E}\) through Gauss' law \(\nabla \cdot \varepsilon \overrightarrow{E} = \varepsilon \nabla \cdot \overrightarrow{E} + \overrightarrow{E} \cdot \nabla \varepsilon\) and envoke conservation of charge:

\[ \overrightarrow{E} = - \nabla \phi \label{3} \]

\[ \nabla \cdot \overrightarrow{E} = \frac{\rho_F}{\varepsilon} - \frac{\overrightarrow{E} \cdot \nabla \varepsilon}{\varepsilon} \label{4} \]

\[ \nabla \cdot \overrightarrow{J}_f + \frac{\partial{\rho_f}}{\partial{t}} = 0 \label{5} \]

The charge relaxation equation is obtained by entering \(\overrightarrow{J}_f\) from Eq. \ref{2} into Eq. \ref{5}, using Eq. \ref{4} to replace the divergence of \(\overrightarrow{E}\) and remembering that \(\overrightarrow{v}\) is solenoidal,

\[ \frac{\partial{\rho_f}}{\partial{t}} + \overrightarrow{v} \cdot \nabla \rho_f = - \frac{\sigma}{\varepsilon} \rho_f - \overrightarrow{E} \cdot \nabla \sigma + \frac{\sigma}{\varepsilon} \overrightarrow{E} \cdot \nabla \varepsilon \label{6} \]

For a material of uniform permittivity, this is the same expression as Eq. 5.9.19, a fact that emphasizes the multispecies contribution to the conduction process necessary to justify the use of the ohmic model.

If characteristic lines are defined as the trajectories of fluid elements, then

\[ \frac{d \overrightarrow{r}}{dt} = \overrightarrow{v} \label{7} \]

and time is measured for an observer moving along a line satisfying Eq. \ref{7}, the charge relaxation equation, Eq. \ref{6}, becomes

\[ \frac{d \rho_f}{dt} = - \frac{\sigma}{\varepsilon} \rho_f - \overrightarrow{E} \cdot \nabla \sigma + \frac{\sigma}{\varepsilon} \overrightarrow{E} \cdot \nabla \varepsilon \label{8} \]

For an observer moving with the material, the three terms on the right are the possible contributors to a time rate of change of the charge density. Respectively, they represent the relaxation of the charge due to its self-field, the possible accumulation of charge where the electrical conductivity varies, and where the permittivity is inhomogeneous. Typically, these latter two terms are at interfaces, and hence are singular.

## Region of Uniform Properties

In this case, the last two terms in Eq. \ref{8} are zero, and the equation can be integrated without regard for details of geometry and boundary conditions:

\[ \rho_f = \rho_o (\overrightarrow{r}) e^{-t/ \tau_e}; \, \tau_e \equiv \varepsilon/ \sigma \label{9} \]

For the neighborhood of a given material particle, \(\rho_o\) is the charge density when \(t=0\). With Eq. \ref{9}, it has been deduced that at a given location within a deforming material having uniform conductivity and permittivity, the free charge density is zero unless that point can be traced backward in time along a particle line to a source of free charge density.

The general solution summarized by Eq. \ref{9} has a physical significance which is best emphasized by considering two typical situations, one where the initial charge distribution is known, and the other involving a condition on the charge density where characteristic lines enter the volume of interest.

Suppose that the charge distribution is to be determined in an ohmic fluid as it passes between plane-parallel walls in the planes \(x = 0\) and \(x = d\). The flow is in the steady state with a velocity profile that is consistent with fully developed laminar flow:

\[ \overrightarrow{v} = \frac{4x}{d} (1- \frac{x}{d}) U \overrightarrow{i}_z \label{10} \]

## Initial Value Problem

When \(t = 0\), the charge distribution throughout the flow is known to be \(\rho_f (x,0) = \rho_t \, sin(kz) \label{11} \]

This distribution is sketched in Fig. 5.10.1a. For the given steady velocity distribution, it is simple to integrate Eq. \ref{7} to find the characteristic lines \(x = x_o\), \(y = y_o\) and

\[ z = \frac{4x}{d} (1- \frac{x}{d}) Ut + z_o \label{12} \]

The integration constant, \(z_o\), is the \(z\) intercept of the characteristic line with the \(t = 0\) plane.Figure 5.10.1b represents these characteristic lines in the \(x-z-t\) space. In the channel center, the characteristic line has its greatest slope \((U)\) in the \(z-t\) plane, while at the channel edges the slope is zero. The lines take the same geometric shape regardless of \(z_o\), and therefore other families of lines are generated by simply translating the picture shown along the \(z\) axis.

Now according to Eq. \ref{9}, the charge density at any time \(t > 0\) is found by evaluating the initial charge density at the root of a characteristic line, when \(t = 0\), and following that line to the point in question. The charge decays along this line by an amount predicted by the exponential equation using the elapsed time. If \((x,z,t)\) represent the coordinates where the solution is required at some later time, then these coordinates are related to \(z_o\) through Eq. \ref{12}, and the initial charge density appropriate to the point in question is given by Eq. \(11\) with \(z \rightarrow z_o\). Thus, the required solution is

\[ \rho_f (x,z,t) = \rho_t \, sin \, k [ z - \frac{4x}{d} ( 1- \frac{x}{d}) Ut] e^{-t/ \tau_e} \label{13} \]

This distribution is the one sketched in Fig. 5.10.1c.

The consequences of a boundary-value transient serve to provide further background for establishing the point of this section.

## Injection from a Boundary

It is possible to inject charge into the bulk of an ohmic fluid so that a steady-state condition can be established with a space charge in the material volume. However, the position of interest in the material bulk must then be joined by a characteristic line to a source of charge. As an illustration, consider the case where, initially, there is no charge in the material. Again, the fluid flow of Eq. \ref{10} is considered. However, now charge is introduced by a source in the plane \(z = 0\). When \(t = 0\), this source is turned on and provides a volume charge density \(\rho_s\) henceforth at \(z = 0\). The problem is then one of finding the resulting downstream charge distribution. The boundary condition is shown graphically in Fig. 5.10.2a.

For this type of problem, the characteristic lines of Eq. \ref{12} are more conveniently used if written in terms of the time \(t = t_a\) when a given characteristic intercepts the \(z = 0\) plane, where the source of charge is located, and it is known that for \(t > 0\), the charge density is \(\rho_s\). Then

\[ z = \frac{4x}{d} ( 1 - \frac{x}{d}) U (t - t_a) \label{14} \]

The family of characteristics having roots in the \(z = 0\) plane when \(t = t_a\) is sketched in Fig. 5.10.2b.

From the characteristic lines of the sketch, Fig. 5.10.2b, it follows that the distribution of charge can be divided into two regions, the surface of demarcation between the two being the surface formed by the characteristic lines with \(t_a = 0\). For \(z\) greater than the envelope of these characteristic lines there is no response, because the characteristic lines originate from the \(z = 0\) plane at a time when the charge density is constrained to be zero. For \(z\) less than the envelope, the initial charge distribution at \(z = 0\) is the constant \(\rho_s\). Thus, there is a wavefront between the two regions, as sketched in Fig. 5.10.2c. The charge density at any point behind the wavefront is determined by multiplying \(exp[(t-t_a)/ \tau_e]\) times the charge density at \(z=0\). That is, the appropriate evaluation of Eq. \ref{9} is

\[ \rho_f = \rho_s e^{-(t-t_a)/\tau_e} \label{15} \]

and in view of the relation between a point in question \((x,z,t)\) and the time of origination from the \(z = 0\) plane, \(t_a\) (given by Eq. \ref{14}), the charge distribution of Eq. \ref{15} can be written in terms of \((x,z,t)\) as

\[ \rho_f = \rho_s e^{-z/[v(x) \tau_e]|; \, v(x) = \frac{4x}{d} ( 1- \frac{x}{d}) U \label{16} \]

This stationary distribution of charge is shown in Fig. 5.10.2c.

Because of the dependence of the velocity on \(x\), the spatial rate of decay behind the front depends on the transverse position \(x\). At the center of the channel, where the velocity is \(U\), the spatial rate of decay is determined by the ratio of the relaxation time to the time required for the material to transport the charge to the given \(z\) position in question. This ratio is a measure of the influence of the material motion on the charge distribution: for a characteristic length \(l\) in the \(z\) direction, it is convenient to define the electric Reynolds number of an ohmic conductor as

\[ R_e \equiv (\varepsilon/\sigma)/ (l/U) = \frac{\varepsilon U}{\sigma l} \label{17} \]

and Eq. \ref{16}, written for the channel center where \(x = d/2\), becomes

\[ \rho_f (\frac{d}{2},z,t) = \rho_s e^{-\frac{z}{l} (1/R_e)} \label{18} \]

At a given location \(z\), once the wavefront has passed, the response represented in general by Eq. \ref{9} is independent of time.