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2.3: Quasistatic Laws and the Time-Rate Expansion

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    With a quasistatic model, it is recognized that relevant time rates of change are sufficiently low that contributions due to a particular dynamical process are ignorable. The objective in this section is to give some formal structure to the reasoning used to deduce the quasistatic field equations from the more general Maxwell's equations. Here, quasistatics specifically means that times of interest are long compared to the time, \(\tau_{em}\), for an electromagnetic wave to propagate through the system.

    Generally, given a dynamical process characterized by some time determined by the parameters of the system, a quasistatic model can be used to exploit the comparatively long time scale for processes of interest. In this broad sense, quasistatic models abound and will be encountered in many other contexts in the chapters that follow. Specific examples are:

    1. processes slow compared to wave transit times in general; acoustic waves and the model is one of incompressible flow, Alfven and other electromechanical waves and the model is less standard;
    2. processes slow compared to diffusion (instantaneous diffusion models). What diffuses can be magnetic field, viscous stresses, heat, molecules or hybrid electromechanical effects;
    3. processes slow compared to relaxation of continua (instantaneous relaxation or constant potential models). Charge relaxation is an important example.

    The point of making a quasistatic approximation is often to focus attention on significant dynamical processes. A quasistatic model is by no means static. Because more than one rate process is often imbedded in a given physical system, it is important to agree upon the one with respect to which the dynamics are quasistatic.

    Rate processes other than those due to the transit time of electromagnetic waves enter through the dependence of the field sources on the fields and material motion. To have in view the additional characteristic times typically brought in by the field sources, in this section the free current density is postulated to have the dependence

    \[\overrightarrow{J_f} = \sigma (\overrightarrow{r}) \overrightarrow{E} + \overrightarrow{J_v} (\overrightarrow{v},\rho_f,\overrightarrow{H}) \label{1} \]

    In the absence of motion, \(\overrightarrow{J_v}\) is zero. Thus, for media at rest the conduction model is ohmic, with the electrical conductivity a in general a function of position. Examples of \(\overrightarrow{J_v}\) are a convection current \(\rho_f v\), or an ohmic motion-induced current \(\sigma(\overrightarrow{v} \times \mu_o\overrightarrow{H})\). With an underbar used to denote a normalized quantity, the conductivity is normalized to a typical (constant) conductivity \(\sigma_o\) :

    \[ \sigma = \sigma_o \underline{\sigma}(\overrightarrow{r},t) \label{2} \]

    To identify the hierarchy of critical time-rate parameters, the general laws are normalized. Coordinates are normalized to one typical length \(l\), while \(\tau\) represents a characteristic dynamical time:

    \[ (x,y,z) = (l\underline{x},l\underline{y},l\underline{z}); \quad t = \tau \underline{t} \label{3} \]

    In a system sinusoidally excited at the angular frequency \(\omega\), \(\tau = \omega^{-1}\).

    The most convenient normalization of the fields depends on the specific system. Where electromechanical coupling is significant, these can usually be categorized as "electric-field dominated" and "magnetic-field dominated." Anticipating this fact, two normalizations are now developed "in parallel," the first taking \(\mathscr{E}\) as a characteristic electric field and the second taking \(\mathscr{H}\) as a characteristic magnetic field:

    \[\begin{align}\overrightarrow{E} &= \mathscr{E}\underline{\overrightarrow{E}}, \quad \overrightarrow{P} = \varepsilon_v\mathscr{E}\underline{\overrightarrow{P}}, \quad \overrightarrow{v} = (l/\tau) \underline{\overrightarrow{v}}, \quad \overrightarrow{J_v} = \frac{\varepsilon_o \mathscr{E}}{\tau} \underline{\overrightarrow{J}}_v \nonumber \quad \quad & \overrightarrow{H} &= \mathscr{H}\underline{\overrightarrow{H}}, \quad \overrightarrow{M} = \varepsilon_v\mathscr{H}\underline{\overrightarrow{M}}, \quad \overrightarrow{v} = (l/\tau) \underline{\overrightarrow{v}}, \quad \overrightarrow{J_v} = \frac{\mathscr{H}}{l} \underline{\overrightarrow{J}}_v \nonumber \\ \rho_f &= \frac{\mathscr{E}\varepsilon_o}{l} \underline{\rho}_f, \quad \overrightarrow{H} = \frac{\varepsilon_o l \mathscr{E}}{\tau} \underline{\overrightarrow{H}}, \quad\overrightarrow{M} = \frac{\varepsilon_o l \mathscr{E}}{\tau} \underline{\overrightarrow{M}} \nonumber \quad \quad & \overrightarrow{E} &= \frac{\mu_o l \mathscr{H}}{\tau} \underline{\overrightarrow{E}}, \quad \rho_f = \frac{\varepsilon_o\mu_o\mathscr{H}}{\tau}\underline{\rho}_f , \quad\overrightarrow{P} = \frac{\varepsilon_o \mu_o l \mathscr{E}}{\tau} \underline{\overrightarrow{P}} \end{align} \label{4} \]

    It might be appropriate with this step to recognize that the material motion introduces a characteristic (transport) time other than \(\tau\). For simplicity, Equation \ref{4} takes the material velocity as being of the order of \(l/\tau\).

    The normalization used is arbitrary. The same quasistatic laws will be deduced regardless of the starting point, but the normalization will determine whether these laws are "zero-order" or higher order in a sense to now be defined.

    The normalizations of Equation \ref{4} introduced into Eqs. 2.2.1-5 result in

    \[ \begin{align} &\nabla \cdot \overrightarrow{E} = -\nabla \cdot \overrightarrow{P} + \rho_f \quad \quad & &\nabla \cdot \overrightarrow{E} = -\nabla \cdot \overrightarrow{P} + \rho_f \quad \quad \label{5} \\ &\nabla \cdot \overrightarrow{H} = -\nabla \cdot \overrightarrow{M} \quad \quad & &\nabla \cdot \overrightarrow{H} = -\nabla \cdot \overrightarrow{M}\label{6} \\ &\nabla \times \overrightarrow{H} = \frac{\tau}{\tau_e} \sigma \overrightarrow{E} + \overrightarrow{J_v} + \frac{\partial\overrightarrow{E}}{\partial{t}} + \frac{\partial\overrightarrow{P}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) \quad \quad & &\nabla \times \overrightarrow{H} = \frac{\tau_m}{\tau} \sigma \overrightarrow{E} + \overrightarrow{J_v} + \beta\Big[\frac{\partial\overrightarrow{E}}{\partial{t}} + \frac{\partial\overrightarrow{P}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v})\Big] \label{7} \\ &\nabla \times \overrightarrow{E} = -\beta \Big[\frac{\partial\overrightarrow{H}}{\partial{t}} + \frac{\partial\overrightarrow{M}}{\partial{t}} + \nabla \times (\overrightarrow{M} \times \overrightarrow{v})\Big] \quad \quad & &\nabla \times \overrightarrow{E} = -\frac{\partial\overrightarrow{H}}{\partial{t}} - \frac{\partial\overrightarrow{M}}{\partial{t}} - \nabla \times (\overrightarrow{M} \times \overrightarrow{v}) \label{8} \\ &\nabla \cdot \sigma \overrightarrow{E} +\frac{\tau_e}{\tau} \Big[\nabla \cdot \overrightarrow{J_v} + \frac{\partial{\rho_f}}{\partial{t}} \Big] = 0 \quad \quad & &\nabla \cdot \sigma \overrightarrow{E} +\frac{\tau}{\tau_m} \nabla \cdot \overrightarrow{J_v} + \beta \frac{\tau}{\tau_m} \frac{\partial{\rho_f}}{\partial{t}} = 0 \label{9}\end{align} \]

    where underbars on equation numbers are used to indicate that the equations are normalized and

    \[ \tau_m \equiv \mu_o \sigma_o l^2, \quad \tau_e \equiv \varepsilon_o/\sigma_o \nonumber \]

    and

    \[ \beta = \Big(\frac{\tau_{em}}{\tau}\Big)^2 ; \quad \tau_{em} \equiv \sqrt{\mu_o \varepsilon_o} l = l/c \label{10} \]

    In Chapter 6, \(\tau_m\) will be identified as the magnetic diffusion time, while in Chapter 5 the role of the charge-relaxation time \(\tau_e\) is developed. The time required for an electromagnetic plane wave to propagate the distance \(l\) at the velocity \(c\) is \(tau_{em}\). Given that there is just one characteristic length, there are actually only two characteristic times, because as can be seen from Equation \ref{10}

    \[ \sqrt{\tau_m \tau_e} = \tau_{em} \label{11} \]

    Unless \(\tau_e\) and \(\tau_m\), and hence \(\tau_{em}\), are all of the same order, there are only two possibilities for the relative magnitudes of these times, as summarized in Figure 2.3.1.

    clipboard_e35cff5320754af5a17604ecef6359681.png
    Figure 2.3.1. Possible relations between physical time constants on a time scale T which typifies the dynamics of interest.

    By electroquasistatic (EQS) approximation it is meant that the ordering of times is as to the left and that the parameter \(\beta = (\tau_{em}/\tau)^2 \) is much less than unity. Note that \(\tau\) is still arbitrary relative to \(\tau_e\) In the magnetoquasistatic (MQS) approximation, \(\beta\) is still small, but the ordering of characteristic ti is as to the right. In this case, \(\tau\) is arbitrary relative to \(\tau_m\).

    To make a formal statement of the procedure used to find the quasistatic approximation, the normalized fields and charge density are expanded in powers of the time-rate parameter \(\beta\).

    \[ \begin{align} &\overrightarrow{E} = \overrightarrow{E_o} + \beta \overrightarrow{E_1} + \beta^2 \overrightarrow{E_2} + ... \nonumber \\ &\overrightarrow{H} = \overrightarrow{E_o} + \beta \overrightarrow{H_1} + \beta^2 \overrightarrow{H_2} + ... \nonumber \\ &\overrightarrow{J_v} = (\overrightarrow{J_v})_o + \beta (\overrightarrow{J_v})_1 + \beta^2 (\overrightarrow{J_v})_2 + ...\nonumber \\ &\overrightarrow{\rho_f} = (\overrightarrow{\rho_f})_o + \beta (\overrightarrow{\rho_f})_1 + \beta^2 (\overrightarrow{\rho_f})_2 + ... \nonumber \end{align} \label{12} \]

    In the following, it is assumed that constitutive laws relate \(\overrightarrow{P}\) and \(\overrightarrow{M}\) to \(\overrightarrow{E}\) and \(\overrightarrow{H}\), so that these densities are similarly expanded. The velocity \(\overrightarrow{v}\) is taken as given. Then, the series are substituted into Eqs. \ref{5} - \ref{9} and the resulting expressions arranged by factors multiplying ascending powers of \(\beta\). The "zero order" equations are obtained by requiring that the coefficients of \(\beta^0\) vanish. These are simply Eqs. \ref{5} - \ref{9} with \(\beta = 0\):

    \[ \begin{align} &\nabla \cdot \overrightarrow{E_o} = -\nabla \cdot \overrightarrow{P_o} + (\rho_f)_o \quad \quad & &\nabla \cdot \overrightarrow{E} = -\nabla \cdot \overrightarrow{P_o} + (\rho_f)_o \quad \quad \label{13} \\ &\nabla \cdot \overrightarrow{H_o} = -\nabla \cdot \overrightarrow{M_o} \quad \quad & &\nabla \cdot \overrightarrow{H_o} = -\nabla \cdot \overrightarrow{M_o}\label{14} \\ &\nabla \times \overrightarrow{H_o} = \frac{\tau}{\tau_e} \sigma \overrightarrow{E_o} + (\overrightarrow{J_v})_o + \frac{\partial\overrightarrow{E_o}}{\partial{t}} + \frac{\partial\overrightarrow{P_o}}{\partial{t}} + \nabla \times (\overrightarrow{P_o} \times \overrightarrow{v}) \quad \quad & &\nabla \times \overrightarrow{H_o} = \frac{\tau_m}{\tau} \sigma \overrightarrow{E_o} + (\overrightarrow{J_v})_o \label{15} \\ &\nabla \times \overrightarrow{E_o} = 0 \quad \quad & &\nabla \times \overrightarrow{E_o} = -\frac{\partial\overrightarrow{H_o}}{\partial{t}} - \frac{\partial\overrightarrow{M_o}}{\partial{t}} - \nabla \times (\overrightarrow{M_o} \times \overrightarrow{v}) \label{16} \\ &\nabla \cdot \sigma \overrightarrow{E_o} +\frac{\tau_e}{\tau} \Big[\nabla \cdot (\overrightarrow{J_v})_o + \frac{\partial{(\rho_f})_o}{\partial{t}} \Big] = 0 \quad \quad & &\nabla \cdot \sigma \overrightarrow{E_o} +\frac{\tau}{\tau_m} \nabla \cdot (\overrightarrow{J_v})_o = 0 \label{17}\end{align} \]

    The zero-order solutions are found by solving these equations, augmented by appropriate boundary conditions. If the boundary conditions are themselves time dependent, normalization will turn up additional characteristic times that must be fitted into the hierarchy of Figure 2.3.1.

    Higher order contributions to the series of Equation \ref{12} follow from a sequential solution of the equations found by making coefficients of like powers of \(\beta\) vanish. The expressions resulting from setting the coefficients of \(\beta^n\) to zero are:

    \[ \begin{align} &\nabla \cdot \overrightarrow{E_n} +\nabla \cdot \overrightarrow{P_n} - (\rho_f)_n = 0 \quad \quad & &\nabla \cdot \overrightarrow{E_n} + \nabla \cdot \overrightarrow{P_n} - (\rho_f)_n = 0 \label{18} \\ &\nabla \cdot \overrightarrow{H_n} + \nabla \cdot \overrightarrow{M_n} = 0 \quad \quad & &\nabla \cdot \overrightarrow{H_n} + \nabla \cdot \overrightarrow{M_n} = 0 \label{19} \\ &\nabla \times \overrightarrow{H_n} - \frac{\tau}{\tau_e} \sigma \overrightarrow{E_n} - (\overrightarrow{J_v})_n - \frac{\partial{\overrightarrow{E_n}}}{\partial{t}} - \frac{\partial{\overrightarrow{P_n}}}{\partial{t}} - \nabla \times (\overrightarrow{P_n} \times \overrightarrow{v}) = 0 \quad \quad & &\nabla \times \overrightarrow{H_n} - \frac{\tau_m}{\tau} \sigma \overrightarrow{E_n} - (\overrightarrow{J_v})_n = \Big[ \frac{\partial{\overrightarrow{E}_{n-1}}}{\partial{t}} + \frac{\partial{\overrightarrow{P}_{n-1}}}{\partial{t}} + \nabla \times (\overrightarrow{P}_{n-1} \times \overrightarrow{v})\Big] \label{20} \\ &\nabla \times \overrightarrow{E_n} = -\Big[ \frac{\partial{\overrightarrow{H}_{n-1}}}{\partial{t}} + \frac{\partial{\overrightarrow{M}_{n-1}}}{\partial{t}} + \nabla \times (\overrightarrow{M}_{n-1} \times \overrightarrow{v})\Big] \quad \quad & &\nabla \times \overrightarrow{E_n} + \frac{\partial{\overrightarrow{H_n}}}{\partial{t}} + \frac{\partial{\overrightarrow{M_n}}}{\partial{t}} + \nabla \times (\overrightarrow{M_n} \times \overrightarrow{v}) \label{21} \\ &\nabla \cdot \sigma \overrightarrow{E_n} +\frac{\tau_e}{\tau} \Big[\nabla \cdot (\overrightarrow{J_v})_n + \frac{\partial{(\rho_f)}_n}{\partial{t}} \Big] = 0 \quad \quad & &\nabla \cdot \sigma \overrightarrow{E_n} +\frac{\tau}{\tau_m} \nabla \cdot (\overrightarrow{J_v})_n = -\frac{\tau}{\tau_m} \frac{\partial{(\rho_f)}_{n-1}}{\partial{t}} \label{22} \end{align} \]

    To find the first order contributions, these equations with \(n=1\) are solved with the zero order solutions making up the right-hand sides of the equations playing the role of known driving functions. Boundary conditions are satisfied by the lowest order fields. Thus higher order fields satisfy homogeneous boundary conditions.

    Once the first order solutions are known, the process can be repeated with these forming the "drives" for the \(n=2\) equations.

    In the absence of loss effects, there are no characteristic times to distinguish MQS and EQS systems. In that limit, which set of normalizations is used is a matter of convenience. If a situation represented by the left-hand set actually has an EQS limit, the zero order laws become the quasistatic laws. But, if these expressions are applied to a situation that is actually MQS, then firstorder terms must be calculated to find the quasistatic fields. If more than the one characteristic time \(\tau_{em}\) is involved, as is the case with finite \(\tau_{em}\) and \(\tau_{m}\), then the ordering of rate parameters can contribute to the convergence of the expansion.

    In practice, a formal derivation of the quasistatic laws is seldom used. Rather, intuition and experience along with comparison of critical time constants to relevant dynamical times is used to identify one of the two sets of zero order expressions as appropriate. But, the use of normalizations to identify critical parameters, and the notion that characteristic times can be used to unscramble dynamical processes, will be used extensively in the chapters to follow.

    Within the framework of quasistatic electrodynamics, the unnormalized forms of Eqs. \ref{13}-\ref{17} comprise the "exact" field laws These equations are reordered to reflect their relative importance:

    \[ \begin{align} &Electroquasistatic \quad (EQS) \quad \quad \quad & &Magnetoquasistatic \quad (MQS) \nonumber \\ \nonumber \\ &\nabla \cdot \varepsilon_o \overrightarrow{E} = -\nabla \cdot \overrightarrow{P} + \rho_f \quad \quad & &\nabla \times \overrightarrow{H} = \overrightarrow{J_f} \label{23} \\ &\nabla \times \overrightarrow{E} = 0 \quad \quad & &\nabla \cdot \mu_o \overrightarrow{H} = -\nabla \cdot \mu_o \overrightarrow{M} \label{24} \\ &\nabla \cdot \overrightarrow{J_f} + \frac{\partial{\rho_f}}{\partial{t}} = 0 \quad \quad & &\nabla \times \overrightarrow{E} = -\frac{\partial{\mu_o \overrightarrow{H}}}{\partial{t}} -\frac{\partial{\mu_o \overrightarrow{M}}}{\partial{t}} -\mu_o \nabla \times (\overrightarrow{M} \times \overrightarrow{v}) \label{25} \\ &\nabla \times \overrightarrow{H} = \overrightarrow{J_f} + \frac{\partial{\varepsilon_o \overrightarrow{E}}}{\partial{t}} + \frac{\partial{\overrightarrow{P}}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) \quad \quad & &\nabla \cdot \overrightarrow{J_f} = 0 \label{26} \\ &\nabla \cdot \mu_o \overrightarrow{H} = -\nabla \mu_o \overrightarrow{M} \quad \quad & &\nabla \cdot \varepsilon_o \overrightarrow{E} = -\nabla \cdot \overrightarrow{P} + \rho_f \label{27}\end{align} \]

    The conduction current \(\overrightarrow{J_f}\) has been reintroduced to reflect the wider range of validity of these equations than might be inferred from Equation \ref{1}. With different conduction models will come different characteristic times,exemplified in the discussions of this section by \(\tau_e\) and \(\tau_m\). Matters are more complicated if fields and media interact electromechanically. Then, \(\overrightarrow{v}\) is determined to some extent at least by the fields themselves and must be treated on a par with the field variables. The result can be still more characteristic times.

    The ordering of the quasistatic equations emphasizes the instantaneous relation between the respective dominant sources and fields. Given the charge and polarization densities in the EQS system, or given the current and magnetization densities in the MQS system, the dominant fields are known and are functions only of the sources at the given instant in time.

    The dynamics enter in the EQS system with conservation of charge, and in the MQS system with Faraday's law of induction. Equations \ref{26}a and \ref{27}a are only needed if an after-the-fact determination of \(\overrightarrow{H}\) is to be made. An example where such a rare interest in \(\overrightarrow{H}\) exists is in the small magnetic field induced by electric fields and currents within the human body. The distribution of internal fields and hence currents is determined by the first three EQS equations. Given \(\overrightarrow{E}\), \(\overrightarrow{P}\), and \(\overrightarrow{J_f}\), the remaining two expressions determine \(\overrightarrow{H}\). In the MQS system, Equation \ref{27}b can be regarded as an expression for the after-the-fact evaluation of \(\rho_f\) which is not usually of interest in such systems.

    What makes the subject of quasistatics difficult to treat in a general way,even for a system of fixed ohmic conductivity, is the dependence of the appropriate model on considerations not conveniently represented in the differential laws. For example, a pair of perfectly conducting plates, shorted on one pair of edges and driven by a sinusoidal source at the opposite pair, will be MQS at low frequencies. The same pair of plates, open-circuited rather than shorted, will be electroquasistatic at low frequencies. The difference is in the boundary conditions.

    Geometry and the inhomogeneity of the medium (insulators, perfect conductors and semiconductors) one are also essential to determining the appropriate approximation. Most systems require more than one characteristic dimension and perhaps conductivity for their description, with the result that more than two time constants are often involved. Thus, the two possibilities identified in Figure 2.3.1 can in principle become many possibilities. Even so, for a wide range of practical problems, the appropriate field laws are either clearly electroquasistatic or magnetoquasistatic.

    Problems accompanying this section help to make the significance of the quasistatic limits more substantive by considering cases that can also be solved exactly.


    This page titled 2.3: Quasistatic Laws and the Time-Rate Expansion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.