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2.10: Conservation of Electroquasistatic Energy

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    30740
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    This and the next section develop a field picture of electromagnetic energy storage from fundamental definitions and principles. Results are a first step in the derivation of macroscopic force densities in Chapter 3. Energy storage in a conservative EQS system is considered first, followed by a statement of power flow. In this and the next section the macroscopic medium is at rest.

    Thermodynamics: Whether in electric or magnetic form, energy storage follows from the definition of the electric field as a force per unit charge. The work required to transport an element of charge 6q, from a reference position to a position p in the presence of the electric field intensity is

    \[ \delta_w = - \int_{ref}^P \delta q \overrightarrow{E} \cdot \overrightarrow{d}l \label{1} \]

    The integral is the work done by the external force on the electric subsystem in placing the charge at \(P\). If this process can be reversed, it can be said that the work done results in a stored energy equal to Equation \ref{1}. In an electroquasistatic system, the electric'field is irrotational. Hence, \(\overrightarrow{E} = -\nabla\phi\). Then, if \(\phi_{ref}\) is defined as zero, it follows that Equation \ref{1} becomes

    \[ \delta_w = - \int_{ref}^P \delta q \nabla \phi \cdot \overrightarrow{d}l = \delta q \phi \label{2} \]

    where use has been made of the gradient integral theorem, Equation 2.6.1. Consider now energy storage in the system abstractly represented by Figure 2.13.1. The system is perfectly insulating, except for th perfectly conducting electrodes introduced into the volume of interest, as in Sec. 2.11. It will be termed an "electroquasistatic thermodynamic subsystem."

    The electrodes have terminal variables as defined in Sec. 2.11; voltages \(v_i\) and total charges \(q_i\) But, in addition, the volume between the electrodes supports a free charge density \(\rho_f\). By definition the energy stored in assembling these charges is equal to the work required to carry the charges from a reference position to the positions of interest. Thus, the incremental energy storage associated with incremental changes in the electrode charges, \(\delta q_i\), or in the charge density, \(\delta \rho_f\), in a given neighborhood on the insulator, is

    \[ \delta_w = \sum_{i=1}^n v_i \delta q_i + \int_{V^{'}} \phi \delta \rho_f dV \label{3} \]

    The volume \(V^{'}\) is the volume excluded by the electrodes. Note that the reference electrode is not included in the summation, because the electric potential on that electrode is, by definition, zero. The work required to place a free charge at its final position correctly accounts for the polarization, because the polarization charges induced in carrying the free charges to their final position are reflected in the potential.

    clipboard_ee0cfcf9fdc8adb2bc300323ba9562eb7.png
    Figure 2.13.1 Schematic representation of electroquasistatic system composed of perfectly conducting electrodes imbedded in a perfectly insulating dielectric medium.

    Consider now the field representation of the electroquasistatic stored energy. From Gauss' law (Equation 2.3.23a), the contribution of the summation in Equation \ref{3} can be represented in terms of an integral over the surfaces \(S_i\) of the electrodes:

    \[ \delta_w = \sum_{i=1}^n \oint_{S_i} \phi_i \delta \overrightarrow{D} \cdot \overrightarrow{n} da + \int_{V^{'}} \phi \delta \rho_f dV \label{4} \]

    Here, \(\phi_i\) is the potential on the surface \(S_i\). The surfaces enclosing the electrodes can be joined together at infinity, as shown in Figure 2.13.1. The resulting simply connected surface encloses all of the electrodes, the wires as they extend to infinity, with the surface completed by a closure at infinity. Thus, the surface integration called for with the first term on the right in Equation \ref{4} can be represented by an integration over a closed surface. Gauss' theorem is then used to convert this surface integral to a volume integration. However, note that the normal vector used in Equation \ref{4} points into the volume \(V^{'}\) excluded by the electrodes and included by the surface at infinity. Thus, in using Gauss' theorem, a minus sign is introduced and Equation \ref{4} becomes

    \[ \delta_w = - \int_{V^{'}} \nabla \cdot (\phi \delta \overrightarrow{D}) dV + \int_{V^{'}} \phi \delta \rho_f dV = \int_{V^{'}} [-\phi \nabla \cdot \delta \overrightarrow{D} - \delta \overrightarrow{D} \cdot \nabla \phi + \phi \delta \rho_f] dV \label{5} \]

    In rewriting the integral, the identity \(\nabla \cdot \psi \overrightarrow{C} = \overrightarrow{C} \cdot \nabla \psi + \psi \nabla \cdot \overrightarrow{C} \) has been used.

    From Gauss' law, \(\delta \rho_f = \delta \nabla \cdot \overrightarrow{D} = \nabla \cdot \delta \overrightarrow{D} \). It follows that the first and last terms in Equation \ref{5} cancel. Also, the electric field is irrotational \((\overrightarrow{E} = - \nabla \phi)\). So Equation \ref{5} becomes

    \[ \delta_w = \int_{V} \overrightarrow{E} \cdot \delta \overrightarrow{D} dV \label{6} \]

    There is no \(\overrightarrow{E}\) inside the electrodes, so the integration is now over all of the volume \(V\).

    The integrand in Equation \ref{6} is an energy density, and it is therefore appropriate to define the incremental change in electric energy density as

    \[ \delta W = \overrightarrow{E} \cdot \delta \overrightarrow{D} \label{7} \]

    The field representation of the energy, as given by Eqs. \ref{6} and \ref{7}, should be compared to that for lumped parameters. Suppose all of the charge resided on electrodes. Then, the second term in Equation would be zero, and the incremental change in energy would be given by the first term:

    \[ \delta_w = \sum_{i=1}^n v_i \delta q_i \label{8} \]

    Comparison of Eqs. \ref{6} and \ref{8} suggests that the electric field plays a role analogous to the terminal voltage while the displacement vector is the analog of the charge on the electrodes. If the relationship between the variables \(\overrightarrow{E}\) and \(\overrightarrow{D}\), or \(v\) and \(q\), is single-valued, then the energy density and the total energy in the continuum and lumped parameter systems can be viewed, respectively, as integrals or areas under curves as sketched in Figure 2.13.2.

    If it is more convenient to have all of the voltages, rather than the charges, as independent variables, then Legendre's dual transformation can be used. That is, with the observation that

    \[ v_i \delta q_i = \delta v_i q_i - q_i \delta v_i \label{9} \]

    Equation \ref{8} becomes

    \[ \delta w^{'} = \sum_{i = 1}^{n} q_i \delta v_i; \quad w^{'} \equiv \sum_{i = 1}^{n} (v_i q_i - w) \label{10} \]

    with \(w^{'}\) defined as the coenergy function.

    In an analogous manner, a coenergy density, \(W^{'}\), is defined by writing \(\overrightarrow{E} \cdot \delta \overrightarrow{D} = \delta (\overrightarrow{E} \cdot \overrightarrow{D} - \overrightarrow{D} \cdot \delta \overrightarrow{E}\) and thus defining

    \[ \delta W^{'} = \overrightarrow{D} \cdot \delta \overrightarrow{E}; \quad W^{'} \equiv \overrightarrow{E} \cdot \overrightarrow{D} - W \label{11} \]

    The coenergy and coenergy density functions have the geometric relationship to the energy and energy density functions, respectively, sketched in Figure 2.13.2. In those systems in which there is no distribution of charge other than on perfectly conducting electrodes, Eqs. \ref{6} and \ref{8} can be regarded as equivalent ways of computing the same incremental change in electroquasistatic energy. If the charge is distributed throughout the volume, Equation \ref{6} remains valid.

    clipboard_ea907001d4aba9d6ee4f37e6aeeed69ac.png
    Figure 2.13.2 Geometric representation of energy \(w\), coenergy \(w^{'}\), energy density \(W\), and coenergy density \(W^{'}\) for electric field systems.

    With the notion of electrical energy storage goes the concept of a conservative subsystem. In the process of building up free charges on perfectly conducting electrodes or slowly conducting charge to the bulk positions (one mechanism for carrying out the process pictured abstractly by Equation \ref{3}), the work is stored much as it would be in cocking a spring. The electrical energy, like that of the spring can later be released (discharged). Included in the subsystem is storage in the polarization. For work done on polarizable entities to be stored, this polarization process must also be reversible. Here, it is profitable to think of the dipoles as internally constrained by spring-like nondissipative elements, capable of releasing energy when the polarizing field is turned off. Mathematically, this restriction on the nature of the polarization is brought in by requiring that \(\overrightarrow{P}\) and hence \(\overrightarrow{D}\) be a single valued function of the instantaneous \(\overrightarrow{E}\), or that \(\overrightarrow{E} = \overrightarrow{E} (\overrightarrow{D}) \). In lumped parameter systems, this is tantamount to \(q = q(v)\) or \(v = v(q)\).

    Power Flow: The electric and polarization energy storage subsystem is the field theory generalization of a capacitor. Just as practical circuits involve a capacitor interconnected with resistors and other types of elements, in any actual physical system the ideal energy storage subsystem is imbedded with and coupled to other subsystems. The field equations, like Kirchhoff's laws in circuit theory, encompass all of these subsystems. The following discussion is based on forming quadratic expressions from the field laws, and hence relate to the energy balance between subsystems.

    For a geometrical part of the ith subsystem, having the volume \(V\) enclosed by the surface \(S\), statement of power flow takes the integral form

    \[ \oint_{S} \overrightarrow{S_i} \cdot \overrightarrow{n} da + \int_{V} \frac{\partial{W_i}}{\partial{t}} dV = \int_{V} \phi_i dV \label{12} \]

    Here, \(S_i\) is the power flux density, \(W_i\) is the energy density, and \(\phi_i\) is the dissipation density.

    Different subsystems can occupy the same volume \(V\). In Equation \ref{12}, \(V\) is arbitrary, while i distinguishes the particular physical processes considered. The differential form of Equation \ref{12} follows by applying Gauss' theorem to the first term and (because \(V\) is arbitrary) setting the integrand to zero:

    \[ \nabla \cdot \overrightarrow{S_i} + \frac{\partial{W_i}}{\partial{t}} = \phi_i \label{13} \]

    This is a canonical form which will be used to describe various subsystems. In a given region, \(W_i\) can increase with time either because of the volumetric source \(\phi_i\) or because of a power flux \(-\overrightarrow{n} \cdot \overrightarrow{S_i}\) into the region across its bordering surfaces.

    For an electrical lumped parameter terminal pair, power is the product of voltage and current. This serves as a clue for finding a statement of power flow from the basic laws. The generalization of the voltage is the potential, while conservation of charge as expressed by Equation 2.3.25a brings in the free current density. So, the sum of Eqs. 2.3.25a and the conservation of polarization charge equation, Equation 2.8. 5, is multiplied by 0 to obtain

    \[\phi \Big[ \nabla \cdot (\overrightarrow{J_f} + \overrightarrow{J_p}) + \frac{\partial{}}{\partial{t}} (\rho_f + \rho_p) \Big] = 0 \label{14} \]

    With the objective an expression having the form of Equation \ref{13}, a vector identity (Equation \ref{15}, Appendix B) and Gauss' law, Equation 2.3.23a, convert Equation \ref{14} to

    \[ \nabla \cdot \Big[ \phi (\overrightarrow{J_f} + \overrightarrow{J_p}) \Big] + \overrightarrow{E} \cdot (\overrightarrow{J_f} + \overrightarrow{J_p}) + \phi \frac{\partial{}}{\partial{t}} \nabla \cdot \varepsilon_o \overrightarrow{E} = 0 \label{15} \]

    In the last term the time derivative and divergence are interchanged and the vector identity used again to obtain the expression

    \[ \nabla \cdot \overrightarrow{S_e} + \frac{\partial{W_e}}{\partial{t}} = \phi_e \label{16} \]

    where, with Equation 2.8.10 used for \(\overrightarrow{J_p}\),

    \[ \overrightarrow{S_e} \equiv \phi \Big( \overrightarrow{J_f} + \overrightarrow{J_p} + \frac{\partial{\varepsilon_o} \overrightarrow{E}}{\partial{t}} \Big) = \phi \Big[ \overrightarrow{J_f} + \frac{\partial{\overrightarrow{D}}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) \Big] \nonumber \]

    \[ W_e \equiv \frac{1}{2} \varepsilon_o \overrightarrow{E} \cdot \overrightarrow{E} \nonumber \]

    \[ \phi_e \equiv - \overrightarrow{E} \cdot (\overrightarrow{J_f} + \overrightarrow{J_p}) = - \overrightarrow{E} \cdot \Big[ \overrightarrow{J_f} + \frac{\partial{\overrightarrow{P}}}{\partial{t}} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) \Big] \nonumber \]

    Which terms appear where in this expression is a matter of what part of a physical system (which subsystem) is being described. Note that We does not include energy stored by polarizing the medium. Also, it can be shown that \(\nabla \cdot \overrightarrow{S_e} = \nabla \cdot (\overrightarrow{E} \times \overrightarrow{H}\), so that \(overrightarrow{S_e}\) is the poynting vector familiar from conventional classical electrodynamics. In the dissipation density, \(\overrightarrow{E} \cdot \overrightarrow{J_f}\) can represent work done on an external mechanical system due to polarization forces or, if the polarization process involves dissipation, heat energy given up to a thermal subsystem.

    The polarization terms in \(\phi_e\) can also represent energy storage in the polarization. This is illustrated by specializing Equation \ref{16} to describe a subsystem in which \(\overrightarrow{P}\) is a single-valued function of the instantaneous (\overrightarrow{E}\), the free current density is purely ohmic, \(\overrightarrow{J_f} = \sigma \overrightarrow{E}\), and the medium is at rest. Then, the polarization term from \(\phi_e\) can be lumped with the energy density term to describe power flow in a subsystem that includes energy storage in the polarization:

    \[ \nabla \cdot \overrightarrow{S_E} + \frac{\partial{W_E}}{\partial{t}} = \phi_E \label{17} \]

    where

    \[ \overrightarrow{S_E} \equiv \phi \Big[ \sigma \overrightarrow{E} + \frac{\partial{D}}{\partial{t}} \Big]; \quad W_E \equiv \int_{o}^{\overrightarrow{D}}\overrightarrow{E} \cdot \delta \overrightarrow{D}; \quad \phi_E \equiv -\sigma \overrightarrow{E} \cdot \overrightarrow{E} \nonumber \]

    Note that the integral defining the energy density \(W_E\), which is consistent with Equation \ref{7}, involves an integrand \(\overrightarrow{E}\) which is time dependent only through the time dependence of \(\overrightarrow{D}: \overrightarrow{E} = \overrightarrow{E} \Big[\overrightarrow{D}(t) \Big]\). Thus, \(\frac{\partial{W_E}}{\partial{t}} = \overrightarrow{E} \cdot (\partial{D}/\partial{t})\).

    With the power flux density placed on the right, Equation \ref{17} states that the energy density decreases because of electrical losses (note that \(\phi_E < 0\)) and because of the divergence of the power density.


    This page titled 2.10: Conservation of Electroquasistatic Energy 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.