Draft:Maxwell's True Current

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Maxwell discovered displacement current ${\varepsilon }_r{\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{\ }}$ when he identified light as electromagnetic radiation. $\boldsymbol{\mathrm{E}}\mathrm{\ }$is the electric field. ${\varepsilon }_0$ is the electrical constant. ${\varepsilon }_r\ $is the dielectric constant, known formally as the relative permittivity. The partial differential equations, that bear Maxwell's name, account for radiation even in a vacuum where there is no charge that might carry current because they include displacement current as a source for the curl of the magnetic field. Wikipedia articles on Maxwell equations, and their history, the displacement current, radiation, vector algebra and vector analysis discuss these issues and the mathematics used to deal with them.

Maxwell [1] included the displacement current in his definition of the true or total current that must be used to estimate the total movement of electricity

\noindent \textit{``One of the chief peculiarities of this treatise is the doctrine which it asserts, that the true electric current, that on which the electromagnetic phenomena depend, is not the same thing as the current of conduction, but that the time-variation of the electric displacement, must be considered in estimating the total movement of electricity, so that we must write, [the sum] .... as an equation of true currents.}

\noindent

\noindent The text is from [1] Vol. 2, Section 610, p. 232. The \textit{`equation of true currents'} is in, eq H, Vol.2, Sect 610, p. 233 of [1]. Maxwell uses the name \textit{`Total current'} throughout his analysis 327-341 of Vol~1. The names are interchangeable. Bromberg [2] describes the evolution of Maxwell's thinking on these subjects in some detail.

Maxwell emphasizes ``that [the true current] must be used to estimate the total movement of electricity. That statement can be written using modern notation for the Maxwell (partial differential) equations that describe the properties of electric and magnetic forces.

The fundamental equation of the true current is the Maxwell-Ampere law that describes how the magnetic field is created. One version of the Maxwell Ampere law is

\[{{\frac{\boldsymbol{\mathrm{1}}}{{\boldsymbol{\mu }}_{\boldsymbol{\mathrm{0}}}}}}~\boldsymbol{\mathrm{curl}}~\boldsymbol{\mathrm{B}}\mathrm{=}{\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}}\mathrm{=}\boldsymbol{\mathrm{J}}\mathrm{+}{\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{\ }}\]

Universal Displacement Current$\ ={\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{\ }}$

Note that the current defined by $\boldsymbol{\mathrm{J}}$ has two components. One is the conduction current which is strictly proportional to the flux of charge carriers with mass. The other is an idealized representation of the dielectric properties of materials, using the material dielectric current ${(\varepsilon }_r{-1)\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t.\mathrm{\ }}\ \ $ The idealization is a poor approximation in many cases and can be said to be over approximated in those cases. In those cases, a separate theory of the material dielectric current is needed, that is solved along with the Maxwell equations themselves as shown in [3].

Because this equation is a differential equation boundary conditions in space and time must be included to define the magnetic $\boldsymbol{\mathrm{B}}$ and electric fields $\boldsymbol{\mathrm{E}}\boldsymbol{\mathrm{\ }}$that are the solution to the equations. The boundary conditions reflect the shape of the system and the particular properties of the fields along the boundary. Those conditions are imposed by the physics of the system and typically include regions where measurements are made, or currents or electric potentials are imposed on the system. Initial conditions are boundary conditions applied at time zero. A common initial condition starts an electric field at time zero with a discontinuity. $E=0$ for $t<0\ $at some place or in some region, and $E=constant\ in\ time$ for $t>0$ at the same place or in the same region$.$

The current appearing in the Maxwell Ampere differential equation can be defined in several ways. A conduction current ${\boldsymbol{\mathrm{J}}}^{\boldsymbol{*}}$ can be used that displays the over approximated material dielectric current ${(\varepsilon }_r{-1)\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}\ $as well as the exact universal displacement current ${\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t.\ \mathrm{\ }}$

An alternative version of the Maxwell Ampere Law is then

\[{{\frac{\boldsymbol{\mathrm{1}}}{{\boldsymbol{\mu }}_{\boldsymbol{\mathrm{0}}}}}}~\boldsymbol{\mathrm{curl}}~\boldsymbol{\mathrm{B}}\mathrm{=}{\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}}\mathrm{=}{\boldsymbol{\mathrm{J}}}^{\boldsymbol{*}}\mathrm{+}\left({\varepsilon }_r-1\right){\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{\ }}\mathrm{+}{\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\ \mathrm{\ }}\]

Material Displacement Current$\ =\left({\varepsilon }_r-1\right){\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{=}}\ \ $arises in ideal dielectrics

\noindent Universal Displacement Current$\ ={\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\mathrm{\ }}$ is a property of space-time found everywhere

\noindent Composite Displacement Current$\ ={{\varepsilon }_r\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t\ \mathrm{=}}$ the sum of material and universal currents

The definition of conduction current as $\boldsymbol{\mathrm{J}}$ or ${\boldsymbol{\mathrm{J}}}^{\boldsymbol{*}}$is not important because the physical quantities that can be easily observed are defined by the true current, as the quotation from Maxwell makes clear and as is shown mathematically in the Maxwell current law shown below. The use of different symbols $\boldsymbol{\mathrm{J}}$ or ${\boldsymbol{\mathrm{J}}}^{\boldsymbol{*}}$ is important to minimize the chances of double counting of components.

\textbf{These field equations imply a current law. Take the divergence of $\boldsymbol{\mathrm{curl}}\boldsymbol{\mathrm{\ }}\boldsymbol{\mathrm{B}}$ and use the general mathematical result that divergence of the curl of a vector field is zero. }

\textbf{ Maxwell Current Law    $\boldsymbol{\mathrm{div}}\boldsymbol{\ }{\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}}\boldsymbol{\mathrm{=}}\boldsymbol{\mathrm{div}}\boldsymbol{\mathrm{\ }}\boldsymbol{\mathrm{curl}}\boldsymbol{\mathrm{\ }}\boldsymbol{\mathrm{B}}\boldsymbol{=}\boldsymbol{0}$}

\textbf{It should be clearly understood that the Maxwell Ampere law implies that derivative terms appear in the laws that govern the movement of electricity. Any law that does not include the derivative term is inconsistent with the Maxwell Ampere law. }

\textbf{Wikipedia articles explain the vector operators, divergence and curl, and discuss their properties. Simply substituting the differential expressions for divergence and curl verifies the statement that the divergence of the curl is zero. }

\noindent \textbf{$\boldsymbol{\mathrm{\ \ \ \ \ \ \ \ \ }}$The equation for the Maxwell Current Law says in mathematics what Maxwell said in words in the quotation above. True current does not accumulate. It has zero divergence. It has the special properties of solenoidal vortex fields that Maxwell mentions in his Presidential Address of 1870 [4] discussed by Dyson [5] and that receive extensive discussion in Vol.2 of his Treatise [1]. The Wikipedia articles on incompressible fluids and solenoidal vector fields describe those properties.}

\textbf{Vector fields that do not accumulate have many special properties that simplify analysis no matter what forces the field describes. The magnetic field $\mathrm{B}$ is solendoidal because magnetic charges have not been found experimentally. The true current ${\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}}$ is solenoidal as the unavoidable mathematical consequence of the Maxwell Ampere law. Many of the special properties of magnetism and true current arise from their solenoidal nature as Maxwell describes in many chapters of Volume 2 of his Treatise [1].}

\noindent \underbar{One dimensional systems}. \textbf{The most striking property of solenoidal vector fields is found in simple one dimensional systems in which elements and wires are in series. The equation shows that if current can be confined to one dimension, as in such series circuits, the true current is exactly the same everywhere along the series system at any time. In other words, the total current is the same everywhere in the series system for currents carried by any type of material charge. The conduction current $\boldsymbol{\mathrm{J}}$ is not the same everywhere in such series systems. It is the true current defined by Maxwell that is the same everywhere. The solenoidal nature of ${\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}},$ at any time, determines many of its properties, which are particularly striking in one dimensional systems and circuits (see Section 404 p. 27 of Volume 2 of [1]. }

\textbf{Circuits are one dimensional systems with branches. The behavior of current in such systems is described by a reduced form of the Maxwell current law called Kirchhoff's law. }

\begin{tabular}{|p{1.8in}|p{1.3in}|p{1.2in}|p{0.3in}|} \hline

\textbf{Classical Kirchhoff Current Lawfor ${\boldsymbol{i}}^{\boldsymbol{th}}\boldsymbol{\ }\boldsymbol{N}$ode in Circuits} & \textbf{$\sum{{\boldsymbol{\mathrm{J}}}_{\boldsymbol{i}\boldsymbol{(}\boldsymbol{k}\boldsymbol{)}}}\boldsymbol{=0}$} & \textbf{${\boldsymbol{\mathrm{J}}}_{\boldsymbol{i}}$ conduction current} & \textbf{} \\ \hline

\end{tabular}

\textbf{Wikipedia articles on circuit theory and Kirchhoff's law describe the analysis and behavior of such circuits.}

{\bf Kirchhoff's law in texts of circuit design and analysis is often said more clearly in words than in equations: `all the currents that flow into a node---positive quantities---flow out as negative quantities [6-23]. Or `the sum of all currents at a node is zero'. The meaning of `Currents' in Kirchhoff's law is usually not discussed in the texts of circuit design.}

Kirchhoff's current law can be written in vector form to emphasize its relation to the Maxwell equations.

\textbf{Kirchhoff Current Law for Fields }$\boldsymbol{\mathrm{div}}\boldsymbol{\mathrm{\ }}\boldsymbol{\mathrm{J}}\boldsymbol{\mathrm{\ =\ \ }}\boldsymbol{\mathrm{0}}\boldsymbol{\mathrm{;}}\boldsymbol{\mathrm{note}}\boldsymbol{\mathrm{\ \ }}\boldsymbol{\mathrm{J}}\boldsymbol{\mathrm{\neq }}\boldsymbol{\mathrm{\ }}{\boldsymbol{\mathrm{J}}}_{\boldsymbol{true}}$\textbf{}

\noindent The Kirchhoff current law in either form does not include a derivative term and thus is in itself inconsistent with the Maxwell Ampere law. It does not satisfy Maxwell's admonition: ``\textit{that the time-variation of the electric displacement, must be considered in estimating the total movement of electricity, so that we must write, [the sum] .... as an equation of true currents.} cited previously

The question then is how to reconcile the Kirchhoff law used throughout engineering with the Maxwell Ampere law that is one of the fundamental laws of electrodynamics.

\noindent \textbf{\underbar{Reconciliation by Approximation}.} The Kirchhoff Current Law is viewed as an approximation to the Maxwell Current Law in most of the textbooks cited. At low frequencies, when true current, and conduction current (however defined) is much larger than displacement current $\boldsymbol{\mathrm{J}}\mathrm{\gg }{\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}$, true current is mostly conduction current and the Kirchhoff law is a good approximation. Not all applications are at low frequencies, for example in the high speed circuits of our computers, so a different reconciliation may be useful.

\noindent \textbf{\underbar{Mathematical Note}} about initial conditions. Even when the low frequency approximation is adequation, difficulties can occur, because the initial condition associated with the time derivative ${\partial \boldsymbol{\mathrm{E}}}/{\partial t}\ $is lost in the approximation that ${\partial \boldsymbol{\mathrm{E}}}/{\partial t=0}$. Losing initial conditions causes difficulties in almost any system, as described in the Wikipedia articles on singular perturbation theory and matched asymptotic expansions.

Losing the initial condition in electrical systems can lead to paradoxes that make the role of charge hard to understand. All the conduction current that flows into a resistor flows out, so one must wonder how the charge that drives the conduction current accumulates in the first place. The charge arises, of course, in an initial condition that is not included in a treatment where the electric field never changes, i.e., in a treatment where ${\partial \boldsymbol{\mathrm{E}}}/{\partial t=0}\ $always. The initial charge can be included in an extra hypothesis added to Kirchhoff's law, but the initial charge is not present in Kirchhoff's Current Law as usually derived with ${\partial \boldsymbol{\mathrm{E}}}/{\partial t\ \to 0.\ }$ Adding initial charges in the complex circuits of a computer or the complex system of a mitochondria might not be easy.

\noindent

{\bf Reconciliation by Redefinition. Kirchhoff's current law and the Maxwell Ampere law can be reconciled without approximation by defining true current, as Maxwell advocates in the quotation that begins this article. Maxwell's definition of true current is easy to use. It requires little change in the treatment of existing idealized circuits or existing analyses of field problems in applied mathematics, that use the dielectric constant approximation, as they usually do. }

\textbf{Maxwell's definition of true current is appealing because of its author. The generality of the definition of true current is esthetically attractive. It applies to circuits. It also applies to systems that involve three dimensions, like stray capacitances and systems that are not circuits, like mitochondria, as well as biological systems that are more or less circuits, for example, nerve fibers. It depends on the most general properties of vortex fields highlighted by Maxwell in many sections of his Treatise [1], particularly Vol. 2, and in [4] discussed by Dyson [5]. }

\noindent

{\bf Reconciliation by Stray Capacitance. The Maxwell reconciliation is avoided without ceremony or discussion by engineers involved in practical circuit design. Practical circuit designers add stray capacitances to the idealized circuits of their initial designs [21, 22, 24] adjusting their values to get sensible results compatible with experiments. The approximation works quite well , if used with caution and care even when dealing with microwave circuits [25-28], but its mathematical and physical basis is not clear. Difficulties occur when circuits have complex layouts, as found in our computers. Difficulties occur in systems in which current flow in three dimensions and the definition of a circuit is tricky. And the values of the stray capacitances are not determined in most cases. Engineers rarely discuss the relation of the stray capacitances and the Maxwell definition of true current or the Maxwell-ampere law. }

{\bf Without these stray capacitances, the idealized circuits do not behave the way real circuits behave when they are actually wired up and measured. Stray capacitances are needed to describe the material displacement currents $\left({\varepsilon }_r-1\right){\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}$ and the vacuum displacement current ${\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}$ that flow in three-dimensional space not in the (branched one-dimensional) circuits themselves. The Maxwell definition of true current includes these currents in its definition of true current. These stray capacitances have unknown values. The displacement currents of the Maxwell definition can be computed by solving the Maxwell partial differential equations in three dimensions with suitable boundaries and boundary conditions. Engineers include these currents by modifying their idealized circuits with stray capacitances as described in Feynman, [29], Vol 2, Section 22-23. They are not included in idealized circuits because they depend on the layout of the circuit---the way it is wired up. Indeed, circuit diagrams and current laws are not enough to build circuits that work as intended. They are not sufficient for realization of circuits. Nor is the displacement current. }

{\bf Other practical issues must be dealt with to build functional circuits. Nonideal properties of the circuit components, and the location of components are important. Indeed, the location and properties of components are the subject of separate engineering disciplines [17-19, 23-25, 30, 31]. }

{\bf The layout is not shown in idealized circuit diagrams. The stray capacitances include material displacement current ${\left({\varepsilon }_r-1\right)\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t.\mathrm{\ }}$ They also include the Maxwell displacement current ${\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t~}$ that is everywhere, according to eq. \eqref{ZEqnNum204366}. The universal existence of the displacement current requires the presence of stray capacitances in any circuit if the circuit is analyzed with the Kirchhoff Current Law. If the Maxwell Current law is used, the stray capacitances are not needed to describe the vacuum displacement current ${\varepsilon }_0{\partial \boldsymbol{\mathrm{E}}}/{\partial t}$ but stray capacitances are still needed to describe the three-dimensional material displacement current$.$}

\noindent \textbf{\underbar{Current in Biological Systems}.} Electrical properties and current flow have an important role throughout biology because life occurs in ionic solutions and the movement of ions is a current [32-36] controlled by specialized proteins that form ion channels in almost all membranes of cells and organelles within cells [37-41].

The Maxwell redefinition of current is necessary when describing biological systems because they are not wired circuits. Long cells like nerve and muscle fibers have been approximated as circuits for a very long time [42-45] but that representation must be extended to apply to ordinary cells, and subcellular components like mitochondria, and is not well known perhaps for that reason.

The representation of long cells as circuits can be derived from the equations of electrodynamics [46, 47]. The derivation exploits the special biological characteristic of membranes. Membranes allow little charge to flow from inside a cell to outside: biological membranes have very high impedance. The derivations provide a textbook  example of singular perturbation theory and matched asymptotics, pp 218-238 [48] and can be checked against Taylor expansions that give the same result [49].

A circuit representation of short cells can be derived the same way [46]. Biological cells contain organelles defined and delimited by membranes. Among those organelles, mitochondria are among the most important and widespread. Mitochondria are the organelles responsible for the synthesis of ATP (in animals). ATP is the nearly universal storehouse of chemical energy in living systems as described in numerous articles in Wikipedia. Current flow in mitochondria couples a variety of protein complexes and so generates ATP. Analysis of that current flow benefits from Maxwell's current law that describes currents in general whether carried by ions, electrons, positively charged water (called `protons' in the biological literature [50-52]) or membrane polarization and capacitance. The current carriers in mitochondria switch from electrons, to ions, to protons (or hydronium ions [53]) even as the protons protonate weak acids and bases (found throughout organelles and cell interiors). The Maxwell true current is what drives the generation of ATP in Protein Complex 5 of mitochondria also called ATP Synthase. Wikipedia includes many articles describing mitochondria, proteins complexes in mitochondrial membranes and the role and mechanism of ATP synthesis.\textbf{References}

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