Electrodynamics is almost always written using a polarization vector field to describe the response of matter to an electric field, or more specifically, to describe the change in distribution of charges as an electric field is applied or changed. This approach does not allow unique specification of a polarization field from measurements of the electric and magnetic fields and electrical current.
Many polarization fields will produce the same electric and magnetic fields, and current, because only the divergence of the polarization enters Maxwellโs first equation, relating charge and electric field. The curl of any function can be added to a polarization field without changing the electric field at all. The divergence of the curl is always zero. Models of structures that produce polarization cannot be uniquely determined from electrical measurements for the same reason. Models must describe charge distribution not just distribution of polarization to be unique.
I propose a different approach, using a different paradigm to describe field dependent charge, i.e., to describe the phenomena of polarization. I propose an operational definition of polarization that has worked well in biophysics where a field dependent, time dependent polarization provides the gating current that makes neuronal sodium and potassium channels respond to voltage. The operational definition has been applied successfully to experiments for nearly fifty years. Estimates of polarization have been computed from simulations, models, and theories using this definition and they fit experimental data quite well.
I propose that the same operational definition be used to define polarization charge in experiments, models, computations, theories, and simulations of other systems. Charge movement needs to be computed from a combination of electrodynamics and mechanics because โeverything interacts with everything elseโ.
The classical polarization field need not enter into that treatment at all.
This document discusses chapter 3 of a textbook on electromagnetic field theory for second year electrical engineering students. The chapter covers electrostatic field theorems, including the electric field intensity, scalar potential, electric flux density, Poisson and Laplace equations, capacitance, point charges, dipoles, the method of images, homogeneous fields, and the field of two arbitrary point charges. It provides examples of the field and potential of point charges near a conducting sphere and the consideration of multi-dielectric mediums.
This document discusses electric currents and magnetostatic fields. It introduces key concepts like current density, Ohm's law, Kirchhoff's laws, and Joule's law. It also discusses magnetic fields produced by steady electric currents using Biot-Savart law, Gauss's law for magnetism, and Ampere's circuital law. The document covers boundary conditions for current density and magnetic fields at material interfaces.
This document describes a laboratory experiment conducted by a student to plot electric and displacement fields using the curvilinear square method for different conductor configurations. In the first part, equipotential lines and flux lines were drawn between two conductors set to 0-60V. The electric field intensity was calculated. In the second part, the capacitance of a coaxial capacitor was calculated theoretically and by plotting field lines, yielding similar results within human and method errors. The purpose of plotting fields graphically and comparing calculations to theory was achieved.
This document discusses electrostatics and related concepts. It begins by outlining what will be covered, including finding electrostatic fields for various charge distributions, the energy density of electrostatic fields, and how fields behave at media interfaces. It then defines Coulomb's law and the electric field, and discusses Gauss's law and how to use it to find electric fields from symmetric charge distributions. Finally, it covers electric potential, boundary value problems, and the electrostatic energy of charge distributions.
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# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
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Field energy correction with discrete chargesSergio Prats
ย
This document introduces a correction in the classical electromagnetic field energy density when there are discrete charges instead of a whole density of charge based in the fact that the EM field induces by a particle does not affect itself and does not contribute to the potential in the hamiltonian.
A deeper analysis is done on how to deal with the radiated field energy, the reaction force and its analogy with quan Vacuum
This document discusses chapter 3 of a textbook on electromagnetic field theory for second year electrical engineering students. The chapter covers electrostatic field theorems, including the electric field intensity, scalar potential, electric flux density, Poisson and Laplace equations, capacitance, point charges, dipoles, the method of images, homogeneous fields, and the field of two arbitrary point charges. It provides examples of the field and potential of point charges near a conducting sphere and the consideration of multi-dielectric mediums.
This document discusses electric currents and magnetostatic fields. It introduces key concepts like current density, Ohm's law, Kirchhoff's laws, and Joule's law. It also discusses magnetic fields produced by steady electric currents using Biot-Savart law, Gauss's law for magnetism, and Ampere's circuital law. The document covers boundary conditions for current density and magnetic fields at material interfaces.
This document describes a laboratory experiment conducted by a student to plot electric and displacement fields using the curvilinear square method for different conductor configurations. In the first part, equipotential lines and flux lines were drawn between two conductors set to 0-60V. The electric field intensity was calculated. In the second part, the capacitance of a coaxial capacitor was calculated theoretically and by plotting field lines, yielding similar results within human and method errors. The purpose of plotting fields graphically and comparing calculations to theory was achieved.
This document discusses electrostatics and related concepts. It begins by outlining what will be covered, including finding electrostatic fields for various charge distributions, the energy density of electrostatic fields, and how fields behave at media interfaces. It then defines Coulomb's law and the electric field, and discusses Gauss's law and how to use it to find electric fields from symmetric charge distributions. Finally, it covers electric potential, boundary value problems, and the electrostatic energy of charge distributions.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission โ Simplifying Students Life
Our Belief โ โThe great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.โ
Like Us - https://www.facebook.com/FellowBuddycom
Field energy correction with discrete chargesSergio Prats
ย
This document introduces a correction in the classical electromagnetic field energy density when there are discrete charges instead of a whole density of charge based in the fact that the EM field induces by a particle does not affect itself and does not contribute to the potential in the hamiltonian.
A deeper analysis is done on how to deal with the radiated field energy, the reaction force and its analogy with quan Vacuum
Introduction to Laplace and Poissons equationhasan ziauddin
ย
This document provides an introduction to electromagnetism and discusses several key concepts:
- Electromagnetism involves the study of electromagnetic forces between charged particles carried by electric and magnetic fields.
- Charges at rest or in uniform motion do not radiate, but accelerating charges do radiate electromagnetic waves like light.
- The divergence and curl of electric fields are examined for different charge configurations.
- Electric potential is defined for point charges and charge distributions.
- Laplace's and Poisson's equations are derived and used to solve boundary value problems for electric fields and potentials between surfaces with specified potentials.
The document discusses Z-matrices and potential energy surfaces. It defines a Z-matrix as a way to represent molecules using internal coordinates like bond lengths, angles, and dihedrals. It then gives an example Z-matrix for methane. It also defines potential energy surfaces as describing the energy of a system, like a collection of atoms, in terms of parameters like atomic positions. It discusses how potential energy surfaces are used to theoretically explore molecular properties and chemical reactions.
Hamiltonian Approach for Electromagnetic Field in One-dimensional Photonic Cr...IRJET Journal
ย
1) The document presents a novel Hamiltonian approach for determining the classical electromagnetic field distribution in one-dimensional photonic crystals.
2) The approach starts from a microscopic Hamiltonian describing the interaction between quantized electromagnetic fields and medium oscillators. Approximations are made to derive a macroscopic Hamiltonian in terms of averaged field operators and material susceptibilities.
3) Using the macroscopic Hamiltonian and coherent states of the electromagnetic field, the electric field operator for the photonic crystal is obtained. The expectation value of this operator gives the classical electric field distribution inside the photonic crystal.
4) As an example, the electric field distribution in a one-dimensional photonic crystal of alternating dielectric layers is determined and a phot
Potential Energy Surface Molecular Mechanics ForceField Jahan B Ghasemi
ย
This document provides an overview of potential energy surfaces (PES) in computational chemistry. It defines a PES as the relationship between a molecule's energy and its geometry. A PES is an n-dimensional surface that relates potential energy to n degrees of freedom in a molecule. Key points made include:
1) A PES allows visualization of how energy changes with molecular structure. Minima correspond to stable structures like reactants and products, while transition states are saddle points along the reaction coordinate.
2) Slices of multidimensional PES can be plotted against one or two geometric parameters to qualitatively represent the full hypersurface.
3) Stationary points on a PES satisfy dE/dq
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
The document summarizes a study investigating the dielectric susceptibility of magnetoelectric thin films containing magnetic vortex-antivortex pairs. The study models the magnetic subsystem using the XY model and incorporates a magnetoelectric coupling term. Magnetic vortices acquire electric charges proportional to their topological charge. Vortex-antivortex pairs form electric dipoles that contribute to the dielectric susceptibility. In the approximation of non-interacting dipole pairs, the susceptibility diverges as temperature approaches the Berezinskii-Kosterlitz-Thouless transition temperature. At low temperatures, the susceptibility takes an exponential form reflecting the thermal activation of vortex pairs.
Maxwell's equations and their derivations.Praveen Vaidya
ย
Being the partial differential equations along with the Lorentz law the Maxwell's equation laid the foundation for classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.[note 1] One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to ฮณ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
This document summarizes research on the dielectric susceptibility of magnetoelectric thin films containing magnetic vortex-antivortex dipole pairs. Key points:
1) A model is proposed where magnetic vortices in the thin film possess electric charges due to magnetoelectric coupling, forming electric dipoles with antivortices.
2) Below the Berezinskii-Kosterlitz-Thouless transition temperature, vortex-antivortex pairs dominate over individual vortices due to lower energy.
3) The contribution of these magnetic vortex-antivortex dipoles to the dielectric susceptibility of the material is calculated. It is shown that the susceptibility diverges as the temperature approaches the BKT
1. Michael Faraday discovered electromagnetic induction in 1831 through experiments showing that a changing magnetic field can induce an electric current in a nearby conductor.
2. Faraday's law of induction states that the induced electromotive force (emf) in a conductor is equal to the rate of change of magnetic flux through the conductor.
3. This discovery established the basis for technologies such as electric generators, transformers, electric motors, and inductors which are crucial components of modern electric power systems and electronics.
This document provides an overview of Maxwell's equations in free space and various coordinate systems. It discusses:
1) Maxwell's equations in differential and integral forms, including Gauss' law, Gauss' law for magnetism, Faraday's law, and Ampere's law.
2) The relationships between the differential and integral forms using theorems like the divergence theorem and Stokes' theorem.
3) How Maxwell's equations coupled the electric and magnetic fields and led to the prediction of electromagnetic waves traveling at the speed of light.
4) The equations of electrostatics, magnetostatics, electroquasistatics and magnetoquasistatics which describe situations where fields vary slowly or are time-
This document summarizes several quantum mechanics methods for calculating molecular properties, including semi-empirical, density functional theory (DFT), and correlation methods. It discusses how semi-empirical methods approximate integrals to speed up calculations compared to Hartree-Fock. DFT is described as an alternative to wavefunction methods that uses the electron density. Popular DFT functionals and how they include exchange and correlation are outlined. Geometry optimization and vibrational frequency calculations are also summarized.
This document discusses Maxwell's equations and time-harmonic electromagnetic fields. It begins by presenting Maxwell's four equations describing electric and magnetic fields. It then discusses the constitutive relations relating the fields to material properties. Maxwell's equations can describe fields in linear and nonlinear media. The equations are also presented in integral form. Examples are provided on applying Maxwell's equations to derive the diffusion equation and skin depth. Poynting's theorem is then introduced, relating power flow to energy storage and dissipation. Finally, time-harmonic fields are discussed, with Maxwell's equations expressed using phasors. Power flow is defined for time-harmonic fields using complex conjugates.
Lecture 3: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Relativistic formulation of Maxwell equations.dhrubanka
ย
This document discusses the relativistic formulation of Maxwell's equations. It begins by introducing the key concepts of special relativity that are needed, including Lorentz transformations and four-vectors. It then shows how the electric and magnetic fields transform under Lorentz transformations and how they can be combined into the electromagnetic field tensor. The document also discusses how charge and current densities transform and satisfy the continuity equation as a four-vector. Finally, it presents Maxwell's equations in their compact relativistic form in terms of the field tensor and its derivatives.
Density functional theory (DFT) provides an alternative approach to calculate properties of molecules by working with electron density rather than wave functions. DFT relies on two theorems linking the ground state energy and electron density. Approximations must be made for the exchange-correlation functional, with popular approximations including LDA, GGA, and hybrid functionals. DFT calculations can determine properties like molecular geometries, energies, vibrational frequencies, and more using software packages. While computationally efficient, DFT has limitations such as its reliance on approximate exchange-correlation functionals.
[1] Maxwell's equations describe the fundamental interactions between electric and magnetic fields and how they propagate as electromagnetic waves.
[2] In vacuum, Maxwell's equations simplify and predict that changing electric fields produce magnetic fields and vice versa, resulting in electromagnetic waves that propagate at the speed of light.
[3] The document derives the wave equations for electric and magnetic fields in vacuum, showing that disturbances in these fields oscillate and regenerate themselves as they travel at the speed of light, representing electromagnetic waves.
The document summarizes Maxwell's equations:
1) Maxwell's equations explain how electric charges and currents produce magnetic and electric fields. Maxwell also calculated the speed of electromagnetic waves to be the speed of light.
2) Maxwell's second equation states that the net magnetic flux through a closed surface is always zero.
3) Maxwell's third equation shows that a changing magnetic field produces an electric field. It is derived from Faraday's law of induction.
4) Maxwell's fourth equation relates the curl of the magnetic field to the electric current density and displacement current density. It is based on Ampere's law.
Core Maxwell Equations are Scary April 11, 2021Bob Eisenberg
ย
The document discusses Maxwell's core equations, which describe electricity and electrodynamics exactly and universally from inside atoms to between stars. The equations treat the dielectric constant as 1 and instead describe polarization and how charge redistributes under an electric field through stress-strain relations for charge. This makes the total current conserved exactly, independent of material properties. However, solving the differential equations requires complex mathematics. While the equations provide an exact theory, scientists are rightly skeptical of sweeping claims of perfection due to possible implied assumptions not being obvious. The document challenges the audience to identify such assumptions.
1) The document discusses the structure of fundamental particles and proposes that they are adjusted to a discontinuous metric in drax spaces with fixed forces that form variable structures.
2) Calculations are shown for the conformative energy of protons, electrons, and photons. If a proton's energy is divided between its constituent impulses, the values obtained align with theoretical predictions.
3) The interactions between fundamental particles like protons are described as producing numerous short-lived particles and resonances, challenging conventional descriptions of forces between particles.
The Evansโs ECE Theory and the Einsteinโs Tetrad Fieldinventionjournals
ย
In this note, it is shown that the electromagnetic field defined as the torsion form of differential geometry proposed by Myron W. Evans, in his EinsteinโCartanโEvans theory, is in contradictions with the Einstein Tetrad Field. Mathematically in the sector Gravitation-Electromagnetism, we found a severe refutation when it is compared with the Einstein tetrad Field
Updating maxwell with electrons and charge version 6 aug 28 1Bob Eisenberg
ย
Maxwellโs equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant.
Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant ฮต_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant ฮต_(0 ).
Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwellโs equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!
After updating,
1) Maxwellโs equations seem universal and exact.
2) Polarization must be described explicitly to use Maxwellโs equations in applications.
3) Conservation of total current (including ฮต_0 โEโโt) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.
4) Kirchhoffโs current law becomes as exact as Maxwellโs equations themselves.
5) Conservation of total current needs to be satisfied in a wide variety of systems where it has not traditionally received much attention.
6) Classical chemical kinetics is seen to need revision to conserve current.
This lecture provides an overview of electromagnetic fields and Maxwell's equations. It introduces key concepts including electric and magnetic fields, Maxwell's equations in integral and differential form, electromagnetic boundary conditions, and electromagnetic fields in materials. Maxwell's equations are the fundamental laws of classical electromagnetics and govern all electromagnetic phenomena. The lecture also discusses phasor representation for time-harmonic fields.
Introduction to Laplace and Poissons equationhasan ziauddin
ย
This document provides an introduction to electromagnetism and discusses several key concepts:
- Electromagnetism involves the study of electromagnetic forces between charged particles carried by electric and magnetic fields.
- Charges at rest or in uniform motion do not radiate, but accelerating charges do radiate electromagnetic waves like light.
- The divergence and curl of electric fields are examined for different charge configurations.
- Electric potential is defined for point charges and charge distributions.
- Laplace's and Poisson's equations are derived and used to solve boundary value problems for electric fields and potentials between surfaces with specified potentials.
The document discusses Z-matrices and potential energy surfaces. It defines a Z-matrix as a way to represent molecules using internal coordinates like bond lengths, angles, and dihedrals. It then gives an example Z-matrix for methane. It also defines potential energy surfaces as describing the energy of a system, like a collection of atoms, in terms of parameters like atomic positions. It discusses how potential energy surfaces are used to theoretically explore molecular properties and chemical reactions.
Hamiltonian Approach for Electromagnetic Field in One-dimensional Photonic Cr...IRJET Journal
ย
1) The document presents a novel Hamiltonian approach for determining the classical electromagnetic field distribution in one-dimensional photonic crystals.
2) The approach starts from a microscopic Hamiltonian describing the interaction between quantized electromagnetic fields and medium oscillators. Approximations are made to derive a macroscopic Hamiltonian in terms of averaged field operators and material susceptibilities.
3) Using the macroscopic Hamiltonian and coherent states of the electromagnetic field, the electric field operator for the photonic crystal is obtained. The expectation value of this operator gives the classical electric field distribution inside the photonic crystal.
4) As an example, the electric field distribution in a one-dimensional photonic crystal of alternating dielectric layers is determined and a phot
Potential Energy Surface Molecular Mechanics ForceField Jahan B Ghasemi
ย
This document provides an overview of potential energy surfaces (PES) in computational chemistry. It defines a PES as the relationship between a molecule's energy and its geometry. A PES is an n-dimensional surface that relates potential energy to n degrees of freedom in a molecule. Key points made include:
1) A PES allows visualization of how energy changes with molecular structure. Minima correspond to stable structures like reactants and products, while transition states are saddle points along the reaction coordinate.
2) Slices of multidimensional PES can be plotted against one or two geometric parameters to qualitatively represent the full hypersurface.
3) Stationary points on a PES satisfy dE/dq
This document summarizes a student project that used the finite difference method and the Laplace equation to model the electrostatic properties of a non-symmetrical surface. The student created an Excel model of an infinitely long magnetic strip surrounded by a conducting box. The model was used to calculate potential, electric field, surface charge density, and capacitance per unit length for different node amounts. The results showed higher potential and charge density near the strip, and flux lines directed towards the box edges rather than another plate. Overall, the model behaved similarly to a parallel-plate capacitor except for non-symmetrical flux lines.
The document summarizes a study investigating the dielectric susceptibility of magnetoelectric thin films containing magnetic vortex-antivortex pairs. The study models the magnetic subsystem using the XY model and incorporates a magnetoelectric coupling term. Magnetic vortices acquire electric charges proportional to their topological charge. Vortex-antivortex pairs form electric dipoles that contribute to the dielectric susceptibility. In the approximation of non-interacting dipole pairs, the susceptibility diverges as temperature approaches the Berezinskii-Kosterlitz-Thouless transition temperature. At low temperatures, the susceptibility takes an exponential form reflecting the thermal activation of vortex pairs.
Maxwell's equations and their derivations.Praveen Vaidya
ย
Being the partial differential equations along with the Lorentz law the Maxwell's equation laid the foundation for classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.[note 1] One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to ฮณ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
This document summarizes research on the dielectric susceptibility of magnetoelectric thin films containing magnetic vortex-antivortex dipole pairs. Key points:
1) A model is proposed where magnetic vortices in the thin film possess electric charges due to magnetoelectric coupling, forming electric dipoles with antivortices.
2) Below the Berezinskii-Kosterlitz-Thouless transition temperature, vortex-antivortex pairs dominate over individual vortices due to lower energy.
3) The contribution of these magnetic vortex-antivortex dipoles to the dielectric susceptibility of the material is calculated. It is shown that the susceptibility diverges as the temperature approaches the BKT
1. Michael Faraday discovered electromagnetic induction in 1831 through experiments showing that a changing magnetic field can induce an electric current in a nearby conductor.
2. Faraday's law of induction states that the induced electromotive force (emf) in a conductor is equal to the rate of change of magnetic flux through the conductor.
3. This discovery established the basis for technologies such as electric generators, transformers, electric motors, and inductors which are crucial components of modern electric power systems and electronics.
This document provides an overview of Maxwell's equations in free space and various coordinate systems. It discusses:
1) Maxwell's equations in differential and integral forms, including Gauss' law, Gauss' law for magnetism, Faraday's law, and Ampere's law.
2) The relationships between the differential and integral forms using theorems like the divergence theorem and Stokes' theorem.
3) How Maxwell's equations coupled the electric and magnetic fields and led to the prediction of electromagnetic waves traveling at the speed of light.
4) The equations of electrostatics, magnetostatics, electroquasistatics and magnetoquasistatics which describe situations where fields vary slowly or are time-
This document summarizes several quantum mechanics methods for calculating molecular properties, including semi-empirical, density functional theory (DFT), and correlation methods. It discusses how semi-empirical methods approximate integrals to speed up calculations compared to Hartree-Fock. DFT is described as an alternative to wavefunction methods that uses the electron density. Popular DFT functionals and how they include exchange and correlation are outlined. Geometry optimization and vibrational frequency calculations are also summarized.
This document discusses Maxwell's equations and time-harmonic electromagnetic fields. It begins by presenting Maxwell's four equations describing electric and magnetic fields. It then discusses the constitutive relations relating the fields to material properties. Maxwell's equations can describe fields in linear and nonlinear media. The equations are also presented in integral form. Examples are provided on applying Maxwell's equations to derive the diffusion equation and skin depth. Poynting's theorem is then introduced, relating power flow to energy storage and dissipation. Finally, time-harmonic fields are discussed, with Maxwell's equations expressed using phasors. Power flow is defined for time-harmonic fields using complex conjugates.
Lecture 3: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Relativistic formulation of Maxwell equations.dhrubanka
ย
This document discusses the relativistic formulation of Maxwell's equations. It begins by introducing the key concepts of special relativity that are needed, including Lorentz transformations and four-vectors. It then shows how the electric and magnetic fields transform under Lorentz transformations and how they can be combined into the electromagnetic field tensor. The document also discusses how charge and current densities transform and satisfy the continuity equation as a four-vector. Finally, it presents Maxwell's equations in their compact relativistic form in terms of the field tensor and its derivatives.
Density functional theory (DFT) provides an alternative approach to calculate properties of molecules by working with electron density rather than wave functions. DFT relies on two theorems linking the ground state energy and electron density. Approximations must be made for the exchange-correlation functional, with popular approximations including LDA, GGA, and hybrid functionals. DFT calculations can determine properties like molecular geometries, energies, vibrational frequencies, and more using software packages. While computationally efficient, DFT has limitations such as its reliance on approximate exchange-correlation functionals.
[1] Maxwell's equations describe the fundamental interactions between electric and magnetic fields and how they propagate as electromagnetic waves.
[2] In vacuum, Maxwell's equations simplify and predict that changing electric fields produce magnetic fields and vice versa, resulting in electromagnetic waves that propagate at the speed of light.
[3] The document derives the wave equations for electric and magnetic fields in vacuum, showing that disturbances in these fields oscillate and regenerate themselves as they travel at the speed of light, representing electromagnetic waves.
The document summarizes Maxwell's equations:
1) Maxwell's equations explain how electric charges and currents produce magnetic and electric fields. Maxwell also calculated the speed of electromagnetic waves to be the speed of light.
2) Maxwell's second equation states that the net magnetic flux through a closed surface is always zero.
3) Maxwell's third equation shows that a changing magnetic field produces an electric field. It is derived from Faraday's law of induction.
4) Maxwell's fourth equation relates the curl of the magnetic field to the electric current density and displacement current density. It is based on Ampere's law.
Core Maxwell Equations are Scary April 11, 2021Bob Eisenberg
ย
The document discusses Maxwell's core equations, which describe electricity and electrodynamics exactly and universally from inside atoms to between stars. The equations treat the dielectric constant as 1 and instead describe polarization and how charge redistributes under an electric field through stress-strain relations for charge. This makes the total current conserved exactly, independent of material properties. However, solving the differential equations requires complex mathematics. While the equations provide an exact theory, scientists are rightly skeptical of sweeping claims of perfection due to possible implied assumptions not being obvious. The document challenges the audience to identify such assumptions.
1) The document discusses the structure of fundamental particles and proposes that they are adjusted to a discontinuous metric in drax spaces with fixed forces that form variable structures.
2) Calculations are shown for the conformative energy of protons, electrons, and photons. If a proton's energy is divided between its constituent impulses, the values obtained align with theoretical predictions.
3) The interactions between fundamental particles like protons are described as producing numerous short-lived particles and resonances, challenging conventional descriptions of forces between particles.
The Evansโs ECE Theory and the Einsteinโs Tetrad Fieldinventionjournals
ย
In this note, it is shown that the electromagnetic field defined as the torsion form of differential geometry proposed by Myron W. Evans, in his EinsteinโCartanโEvans theory, is in contradictions with the Einstein Tetrad Field. Mathematically in the sector Gravitation-Electromagnetism, we found a severe refutation when it is compared with the Einstein tetrad Field
Updating maxwell with electrons and charge version 6 aug 28 1Bob Eisenberg
ย
Maxwellโs equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant.
Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant ฮต_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant ฮต_(0 ).
Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwellโs equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!
After updating,
1) Maxwellโs equations seem universal and exact.
2) Polarization must be described explicitly to use Maxwellโs equations in applications.
3) Conservation of total current (including ฮต_0 โEโโt) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.
4) Kirchhoffโs current law becomes as exact as Maxwellโs equations themselves.
5) Conservation of total current needs to be satisfied in a wide variety of systems where it has not traditionally received much attention.
6) Classical chemical kinetics is seen to need revision to conserve current.
This lecture provides an overview of electromagnetic fields and Maxwell's equations. It introduces key concepts including electric and magnetic fields, Maxwell's equations in integral and differential form, electromagnetic boundary conditions, and electromagnetic fields in materials. Maxwell's equations are the fundamental laws of classical electromagnetics and govern all electromagnetic phenomena. The lecture also discusses phasor representation for time-harmonic fields.
Updating maxwell with electrons and charge Aug 2 2019Bob Eisenberg
ย
Maxwellโs equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant.
Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant ฮต_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant ฮต_(0 ).
Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure ahead of time that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwellโs equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!
After updating,
Maxwellโs equations seem universal and exact.
Polarization must be described explicitly to use Maxwellโs equations in applications.
Conservation of total current (including ฮต_0 โEโโt) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.
Kirchhoffโs current law becomes as exact as Maxwellโs equations themselves.
Classical chemical kinetics is seen to need revision to conserve current.
Dielectric Dilemma 1901.10805 v2 feb 4 2019Bob Eisenberg
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A dielectric dilemma faces scientists because Maxwell's equations are poor approximations as usually written, with a single dielectric constant. Maxwell's equations are then not accurate enough to be useful in many applications. The dilemma can be partially resolved by a rederivation of conservation of current, where current is defined now to include the ยepolarization of the vacuumยf ..0 .......... Conserveration of current becomes Kirchoff's current law with this definition, in the one dimensional circuits of our electronic technology. With this definition, Kirchoff's laws are valid whenever Maxwell's equations are valid, explaining why those laws reliably describe circuits that switch in nanoseconds.
Electromagnetic theory deals with the study of charges at rest and in motion and is fundamental to electrical engineering and physics. It is applicable to areas like communications, electrical machines, and quantum electronics. Electromagnetic theory can be considered a generalization of circuit theory and is necessary for understanding situations that cannot be analyzed solely through circuit theory. It involves the study of electric and magnetic fields as well as electric charges, which act as sources of electromagnetic fields that can redistribute other charges. Vector analysis is a useful mathematical tool for expressing and understanding electromagnetic concepts involving three spatial dimensions and time.
This document discusses a theoretical study of the Casimir torque that can arise between plates with discontinuous dielectric properties. Specifically, it considers plates that are bisected or quadrisected into regions of different dielectric constants. It provides background on the Casimir effect and derives an expression for the Casimir torque. The rest of the document discusses how this system differs from other Casimir systems that have been studied and outlines the specific plate configurations that are analyzed.
Lovneesh Kumar completed a project on electromagnetic induction for their class 12 curriculum. The project involved using a copper wire wound around an iron rod and a magnet to demonstrate Faraday's law of electromagnetic induction. It summarizes the theory behind electromagnetic induction, including magnetic flux, Faraday's law, and the Maxwell-Faraday equation. It concludes that Faraday's law has had a profound impact on modern technology and our daily lives.
Maxwell's equations unified electricity, magnetism, and light by showing that electromagnetic waves propagate through space at a speed c. The equations predicted that changing electric and magnetic fields produce transverse waves that transport energy and momentum. Maxwell's work established that light is an electromagnetic wave oscillating perpendicular to the direction of propagation.
1) In the 19th century, James Clerk Maxwell combined Gauss's law, Ampere's law, and Faraday's law with his own modification to Ampere's law to fully describe electromagnetism.
2) Maxwell's equations relate electric and magnetic fields to electric charges and currents.
3) The document goes on to describe various electromagnetic concepts like current density, conduction and convection currents, and introduces Maxwell's equations in both differential and integral form.
This document is a project report by Samuel Kumar on electromagnetic induction. It includes an introduction discussing Faraday's discovery of electromagnetic induction and its importance. The aim is to determine Faraday's law of electromagnetic induction using a copper wire coil and magnet. The theory section discusses magnetic flux and Faraday's law, which states that the induced EMF is equal to the time rate of change of magnetic flux through the circuit. The required apparatus and Maxwell's equation are also described. The conclusion reinforces the significance of Faraday's law and its applications.
This document provides an introduction to the physics of dielectrics. It discusses electric dipoles, polarization, and the dielectric constant. It defines permanent and induced dipole moments. It introduces the electric field E and displacement field D inside dielectrics. Polarization P is introduced as the difference between D and E, and is proportional to the dielectric susceptibility ฯ and permittivity ฮต. Different types of polarization including electronic, atomic, orientation and ionic polarization are also described.
Electron's gravitational and electrostatic force test.RitikBhardwaj56
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This innovative ebook by John A. Macken delves deep into the fascinating world of electron interactions, shedding light on wave-based models and their impact on gravitational and electrostatic forces. Macken's theory 'Oscillating Spacetime: The Foundation of the Universe' revolutionizes our understanding of fundamental forces at the quantum level. From explaining key parameters without dimensions to comparing gravitational and electrostatic forces, this ebook offers a thorough exploration of wave-based models in electron interactions.
With a fresh perspective, this ebook is a key player in advancing physics education. It simplifies complex concepts through Macken's theory, providing valuable insights into fundamental forces for students and educators alike. The inclusion of natural units and dimensionless constants makes quantum mechanics more accessible, improving the learning process for students at every level. As a result, this ebook is an essential tool for classrooms, self-study, and research, contributing to the ongoing development of scientific knowledge in the field of physics.
The document discusses different types of polarization that can occur in dielectric materials when an external electric field is applied. It defines polarization as the electric dipole moment per unit volume induced by the field. There are four main types of polarization discussed: (1) electronic polarization due to displacement of electrons and nuclei, (2) atomic polarization involving displacement of whole atoms or groups, (3) orientation polarization where the field aligns permanent dipoles, and (4) ionic polarization in ionic materials caused by displacement of positive and negative ions in opposite directions. The polarization is proportional to the applied electric field strength.
The document discusses different types of polarization that can occur in dielectric materials when an external electric field is applied. It defines polarization as the electric dipole moment per unit volume induced by the field. There are four main types of polarization discussed: electron polarization due to displacement of electrons within an atom; atomic polarization involving displacement of whole atoms or groups; orientation polarization where the field aligns permanent dipoles in the material; and ionic polarization that results from displacement of ions in an ionic lattice. The different polarization mechanisms contribute to the overall polarization of dielectric materials in electric fields.
The document discusses dielectric materials and their properties. It introduces dielectric materials and defines electric dipoles as systems of charges with zero net charge. It describes permanent dipole moments that give rise to polarization in materials and induced dipole moments that arise in external electric fields. It then discusses different types of polarization including electronic, atomic, orientation and ionic polarization.
The document discusses dielectric materials and their properties. It introduces dielectric materials and defines electric dipoles as systems of charges with zero net charge. It describes permanent dipole moments that give rise to polarization in materials and induced dipole moments that arise in external electric fields. It then discusses different types of polarization including electronic, atomic, orientation and ionic polarization.
Maxwell's equations describe the fundamental interactions between electric and magnetic fields. The four equations are:
1) Gauss's law relates electric charge to the electric field it produces
2) Gauss's law for magnetism states that magnetic monopoles do not exist
3) Faraday's law describes how changing magnetic fields produce electric fields
4) Ampere's law relates electric currents and changing electric fields to magnetic fields
Together, Maxwell's equations form the foundation of classical electromagnetism and optics.
Maxwell's four equations describe the fundamental interactions between electricity, magnetism, and light. James Clerk Maxwell published these equations in 1865. The equations show that electric and magnetic fields propagate as waves travelling at the speed of light. The four equations are: Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of induction, and Ampere's law with Maxwell's correction. The document further explains these four equations in both differential and integral form, and describes the physical significance of each one.
1. The document discusses the derivation of the permittivity constant (ฮต0) and permeability constant (ฮผ0) from first principles, which has never been done before.
2. It shows that ฮต0 and ฮผ0 can be derived from dimensional analysis and have dimensions of meters per second, corresponding to the speed of light.
3. The product of ฮต0 and ฮผ0 is inversely proportional to the square of the speed of light, allowing the speed of light to be calculated from Maxwell's equations.
Similar to Maxwell equations without a polarization field august 15 1 2020 (20)
Maxwell's equations can be written without a dielectric constant. They are then universal and exact. They are also useful because they imply a universal and exact conservation of total current (including the displacement current). Kirchhoff's current law for circuits then also becomes universal and exact.
Maxwell's Equations can be written without a dielectric constant.
They then imply conservation of total current as a universal and exact physical law. In circuits they imply a generalization of Kirchhoff's current law that is universal and exact
Thermostatics vs. electrodynamics preprints 10.20944.v1Bob Eisenberg
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Thermodynamics has been the foundation of many models of biological and technological systems. But thermodynamics is static and is misnamed. A more suitable name is thermostatics.
Thermostatics does not include time as a variable and so has no velocity, flow or friction. Indeed, as usually formulated, thermostatics does not include boundary conditions. Devices require boundary conditions to define their input and output. They usually involve flow and friction. Thermostatics is an unsuitable foundation for understanding technological and biological devices. A time dependent generalization of thermostatics that might be called thermal dynamics is being
developed by Chun Liu and collaborators to avoid these limitations. Electrodynamics is not restricted like thermostatics, but in its classical formulation involves drastic assumptions about polarization and an over-approximated dielectric constant. Once the Maxwell equations are rewritten without a dielectric constant, they are universal and exact. Conservation of total current,including displacement current, is a restatement of the Maxwell equations that leads to dramatic simplifications in the understanding of one dimensional systems, particularly those without branches, like the ion channel proteins of biological membranes and the two terminal devices of electronic systems. The Brownian fluctuations of concentrations and fluxes of ions become the spatially independent total current, because the displacement current acts as an unavoidable low pass filter, a consequence of the Maxwell equations for any material polarization. Electrodynamics and thermal dynamics together form a suitable foundation for models of technological and biological systems.
Molecular Mean Field Theory of ions in Bulk and ChannelsBob Eisenberg
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Life and most of chemistry occurs in ionic solutions, but ionic solutions have only recently been recognized as the complex fluids that they are. The molecular view shows ions interacting with surrounding water and nearby ions. Everything is correlated in a complex way because ions and water have diameters comparable to their interaction length. The molecular scale shows only a small part of the correlation enforced by electrodynamics. Current defined as Maxwell did to include the ethereal current is exactly conserved, and therefore correlated, over all scales reaching to macroscopic boundary conditions some 10^9ร larger than atoms crucial in batteries and nerve cells.
Jinn Liang Liu and I have built a molecular field theory PNPB Poisson Nernst Planck Bikerman that deals with water as molecules and describes local interactions with a steric potential that depends on the volume fraction of molecules and voids between them. The correlations of electrodynamics are described by a fourth-order differential operator that gives (as outputs) ion-ion and ion-water correlations; the dielectric response (permittivity) of ionic solutions; and the polarization of water molecules, all using a single correlation length parameter. The theory fits experimental data on activity and differential capacitance in ionic solutions of varying composition and content, including mixtures. Potassium channels, Gramicidin, L-type calcium channels, and the Na/Ca transporter are computed in three dimensions from structures in the Protein Data Bank.
Numerical analysis faces challenges
Geometric singularities of molecular surfaces
strong electric fields (100 mV/nm) and resulting exponential nonlinearities, and the
enormous concentrations (> 10 M) often found where ions are important, for example, near electrodes in batteries, in ion channels, and in active sites of proteins.
Wide ranging concentrations of Ca^(2+) in (> 10M) and near (10^(-2) to 10^(-8)M) almost every protein in biological cells make matters worse.
Challenges have been overcome using methods developed over many decades by the large community that works on the computational electronics of semiconductors.
Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Biker...Bob Eisenberg
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We have developed a molecular mean-field theory โ fourth-order Poisson-
Nernst-Planck-Bikerman theory โ for modeling ionic and water flows in biological ion channels
by treating ions and water molecules of any volume and shape with interstitial voids,
polarization of water, and ion-ion and ion-water correlations. The theory can also be used to
study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel
cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc.
The theory can compute electric and steric energies from all atoms in a protein and all ions
and water molecules in a channel pore while keeping electrolyte solutions in the extra- and
intracellular baths as a continuum dielectric medium with complex properties that mimic
experimental data. The theory has been verified with experiments and molecular dynamics
data from the gramicidin A channel, L-type calcium channel, potassium channel, and
sodium/calcium exchanger with real structures from the Protein Data Bank. It was also
verified with the experimental or Monte Carlo data of electric double-layer differential capacitance
and ion activities in aqueous electrolyte solutions. We give an in-depth review of
the literature about the most novel properties of the theory, namely, Fermi distributions of
water and ions as classical particles with excluded volumes and dynamic correlations that
depend on salt concentration, composition, temperature, pressure, far-field boundary conditions
etc. in a complex and complicated way as reported in a wide range of experiments.
The dynamic correlations are self-consistent output functions from a fourth-order differential
operator that describes ion-ion and ion-water correlations, the dielectric response (permit2
tivity) of ionic solutions, and the polarization of water molecules with a single correlation
length parameter.
1. INTRODUCTION
Water and ions give life. Their electrostatic and kinetic interactions play essential roles
in biological and chemical systems such as DNA, proteins, ion channels, cell membranes,
Hsinchu maxwell talk january 7 1 2020 for uploadBob Eisenberg
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Applying the Maxwell equations to mitochondria seems a hopeless task: there is so much complexity. But computers and their chips are nearly as complicated. Design of circuits is done every day by uncounted numbers of engineers and scientists, thanks to Kirchoff's Current Law, which is a conservation law in one dimensional (branched) systems of devices. Kirchoff's current law conserves flux, not current in its usual derivation. But Maxwell's equations do not conserve flux; they conserve total current. Total current J equals flux plus displacement current J+ eps_0 partial E/partial t . Maxwell's definition of current allows circuit laws to be applied to complex systems of devices, over a wide range of times and conditions. Channels and enzymes are devices because they localize current flow. Channels and enzymes can be analyzed by the methods of circuit theory, for that reason.
Lens as an Osmotic Pump, a Bidomain Model. DOI: 10.13140/RG.2.2.25046.80966Bob Eisenberg
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The lens of the eye has no blood vessels to interfere with vision. The lens is far too large for diffusion to provide food and clear wastes.
Experimental, theoretical and computational work has shown that the lens supports its own microcirculation. It is an osmotic pump that implements what physiologists have long believed โConvection provides what diffusion cannot.โ
We introduce a general (non-electro-neutral) model that describes the steady-state relationships among ion fluxes, water flow and electric field inside cells, and in the narrow extracellular spaces within the lens.
Using asymptotic analysis, we derive a simplified model exploiting the numerical values of physiological parameters. The model reduces to first generation โcircuitโ models and shows the basis of computer simulations too large to easily understand. The full model helps resolve paradoxes that have perplexed molecular biologists: crucial physiological properties do not depend as expected on the permeability of the lens interior (to water flow).
The name PNP was introduced by Eisenberg and Chen because it has important physical meaning beyond being the first letters of Poisson-Nernst-Planck. PNP also means Positive-Negative-Positive, the signs of majority current carriers in different regions of a PNP bipolar transistor. PNP transistors are two diodes in series PN + NP that rectify by changing the shape of the electric field. Transistors can function as quite different types of nonlinear devices by changing the shape of the electric field. Those realities motivated Eisenberg and Chen to introduce the name PNP.
The pun โPNP = Poisson-Nernst-Planck = Positive-Negative-Positiveโ has physical content. It suggests that Poisson-Nernst-Planck systems like open ionic channels should not be assumed to have constant electric fields. The electric field should be studied and computed because its change of shape is likely to be important in the function of biological systems, as it is in semiconductor systems.
Ions in solutions liquid plasma of chemistry & biology february 27 2019Bob Eisenberg
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This document summarizes a presentation on ions in solutions. It discusses how most of chemistry and all of biology occurs in ionic solutions, which are concentrated mixtures of ions like sodium, potassium, and calcium. However, classical theories like Debye-Hรผckel and Poisson-Boltzmann fail to adequately describe screening effects in real solutions, especially at biological concentrations and for asymmetrical electrolytes. The document calls for new theories and simulations to better model ionic solutions as complex fluids where everything interacts with everything, given the finite size of ions and their interactions are not weak perturbations. It presents work using a nonlocal Poisson-Fermi model as a promising approach.
1) The document discusses the Hodgkin-Huxley equations, which model the nerve impulse.
2) The equations separate the membrane current into individual ionic currents through sodium, potassium, and leak channels, as well as a displacement current governed by the membrane capacitance.
3) The conductances (gNa, gK) that determine the ionic currents depend on variables that can be controlled by genes and evolution, and their dependence is explained by structural biology and molecular biology.
What is different about life? it is inherited oberwolfach march 7 1 2018Bob Eisenberg
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What is different about life? Why do life sciences require different science and mathematics? I address these issues starting from the obvious: all of life is inherited from genes. Twenty thousand genes of say 30 atoms each control an animal of ~1e25 atoms. How is that possible? Answer: the structures of life form a hierarchy of devices that allow handfuls of atoms to control everything. A nerve signal involves meters of nerve but is controlled by a few atoms. Indeed, potassium and sodium differ only in the diameter of the atoms. Life depends on this difference in diameter. Sodium and potassium are otherwise identical. The task of the biological scientist is first to identify the hierarchy of devices and what they do. Then we want to know how the devices work. We want to understand life well enough to improve its devices, in disease and technology.
The action potential signal of nerve and muscle is produced by voltage sensitive channels that include a specialized device to sense voltage. Gating currents of the voltage sensor are now known to depend on the back-and-forth movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential, further drag the S4 segment and induce opening/closing of ion conduction pore by moving the S4-S5 linker. The ion conduction pore is a separate device from the voltage sensor, linked (in an unknown way) by the mechanical motion and electric field changes of the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor so everything must interact with everything else in its mathematical model, as everything does in the whole protein. A PNP-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all movements of charge and mass satisfy conservation laws of current and mass. The resulting 1D continuum model is used to compute gating currents under a wide range of conditions, corresponding to experimental situations. Chemical-reaction-type models based on ordinary differential equations cannot capture such interactions with one set of parameters. Indeed, they may inadvertently violate conservation of current. Conservation of current is particularly important since small violations (<0.01%) quickly (<< 10-6 seconds) produce forces that destroy molecules. Our model reproduces signature properties of gating current: (1) equality of on and off charge in gating current (2) saturating voltage dependence in QV curve and (3) many (but not all) details of the shape of gating current as a function of voltage.
Electricity plays a special role in our lives and life. Equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term the displacement current (of vacuum). Displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as Maxwell did, as the entire source of the curl of the magnetic field. We show how the Bohm formulation of quantum mechanics allows easy definition of current. We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. Matter does not behave the way physicists of the 1800's thought it does with a single dielectric constant, a real positive number independent of everything. Charge moves in enormously complicated ways that cannot be described in that way, when studied on time scales important today for electronic technology and molecular biology. Life occurs in ionic solutions in which charge moves in response to forces not mentioned or described in the Maxwell equations, like convection and diffusion. Classical derivations of conservation of current involve classical treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave that way, classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved exactly in any material no matter how complex the dielectric, polarization or conduction currents are. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.
Electricity is different august 16 1 2016 with doiBob Eisenberg
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Charges are everywhere because most atoms are charged. Chemical bonds are formed by electrons with their charge. Charges move and interact according to Maxwell's equations in space and in atoms where the equations of electrodynamics are embedded in Schrรถdinger's equation as the potential. Maxwell's equations are universal, valid inside atoms and between stars from times much shorter than those of atomic motion (0.1 femtoseconds) to years (32 mega-seconds). Maxwell's equations enforce the conservation of current. Analysis shows that the electric field can take on whatever value is needed to ensure conservation of current. The properties of matter rearrange themselves to satisfy Maxwell's equations and conservation of current. Conservation of current is as universal as Maxwell's equations themselves. Yet equations of electrodynamics find little place in the literature of material physics, chemistry, or biochemistry. Kinetic models of chemistry and Markov treatments of atomic motion are ordinary differential equations in time and do not satisfy conservation of current unless modified significantly. Systems at equilibrium, without macroscopic flow, have thermal fluctuating currents that are conserved according to the Maxwell equations although their macroscopic averages are zero. The macroscopic consequences of atomic scale fluctuating thermal currents are not known but are likely to be substantial because of the nonlinear interactions in systems like these, in which 'everything interacts with everything else'.
Device approach to biology and engineeringBob Eisenberg
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Device Approach to Biology (and Engineering)
The goal of biological research is often more to control than to understand. Devices in biology (like ion channels) control individual functions just as they do in our technology. Study of control requires a multiscale approach because a handful of atoms, moving in 10-15 sec, control biological functions extending meters and taking seconds. Structural biology and molecular dynamics are essential (and beautiful!) parts of this hierarchy, but so are the functions themselves, and the electric field equations that link structure and function on all scales from atoms to nerve cells. Analyzing biological systems as devices is usually successful, and almost always productive.
Talk device approach to biology march 29 1 2015Bob Eisenberg
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Device Approach to Biology (and Engineering)
The goal of biological research is often more to control than to understand. Devices in biology (like ion channels) control individual functions just as they do in our technology. Study of control requires a multiscale approach because a handful of atoms, moving in 10-15 sec, control biological functions extending meters and taking seconds. Structural biology and molecular dynamics are essential (and beautiful!) parts of this hierarchy, but so are the functions themselves, and the electric field equations that link structure and function on all scales from atoms to nerve cells. Analyzing biological systems as devices is usually successful, and almost always productive.
Talk two electro dynamics and ions in chemistryBob Eisenberg
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The universal nature of electrodynamics and the strength of the electric field needs to be incoporated into chemical theory and thinking. The law of mass action for example is incompatible with continuity of current and thus with Maxwell's equations. Maxwell's equations are accurate from unimaginably small to unimaginably large dimensions with immeasurable accuracy. Violations of Maxwell's equations have large effects because the electric field is so large. Methods are shown to incorporate these realities into chemical descriptions.
Talk multiscale analysis of ionic solutions is unavoidableBob Eisenberg
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Ions in channels and solutions control most living functions. Analysis in atomic detail is needed, but so is prediction of functions on the macroscopic scale. Computational electronics has solved similar issues and we all benefit from the computational devices it provides us. These slides show how a similar approach can be used, and is necessary in my view, for ions solutions and biological systems, most notably in ion channels
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
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Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and โ70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
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Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niลก, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
The binding of cosmological structures by massless topological defectsSรฉrgio Sacani
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Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
When I was asked to give a companion lecture in support of โThe Philosophy of Scienceโ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers โ hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters โ are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposรฉ invitรฉ Journรฉes Nationales du GDR GPL 2024
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sรฉrgio Sacani
ย
Context. With a mass exceeding several 104 Mโ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 ร 10โ8 photons cmโ2
s
โ1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
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Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease.ย Plants have an innate immune system that allows them to recognize pathogens and provide resistance.ย However, breeding for long-lasting resistance often involves combining multiple resistance genes
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luรญsa Pinho
ย
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Maxwell equations without a polarization field august 15 1 2020
1. Maxwell Equations without a Polarization Field
using a paradigm from biophysics
Robert S. Eisenberg
Department of Applied Mathematics
Illinois Institute of Technology;
Department of Physiology and Biophysics
Rush University Medical Center
Chicago IL
Bob.Eisenberg@gmail.com
โข DOI: 10.13140/RG.2.2.32228.60805
August 15, 2020
Keywords: electrodynamics, Polarization, Maxwell equations, Gating Current
Document file is
Maxwell Equations without a Polarization Field August 15-1 2020.docx
2. 1
ABSTRACT
Electrodynamics is almost always written using a polarization vector field to describe the
response of matter to an electric field, or more specifically, to describe the change in distribution
of charges as an electric field is applied or changed. This approach does not allow unique
specification of a polarization field from measurements of the electric and magnetic fields and
electrical current.
Many polarization fields will produce the same electric and magnetic fields, and current, because
only the divergence of the polarization enters Maxwellโs first equation, relating charge and
electric field. The curl of any function can be added to a polarization field without changing the
electric field at all. The divergence of the curl is always zero. Models of structures that produce
polarization cannot be uniquely determined from electrical measurements for the same reason.
Models must describe charge distribution not just distribution of polarization to be unique.
I propose a different approach, using a different paradigm to describe field dependent charge,
i.e., to describe the phenomena of polarization. I propose an operational definition of
polarization that has worked well in biophysics where a field dependent, time dependent
polarization provides the gating current that makes neuronal sodium and potassium channels
respond to voltage. The operational definition has been applied successfully to experiments for
nearly fifty years. Estimates of polarization have been computed from simulations, models, and
theories using this definition and they fit experimental data quite well.
I propose that the same operational definition be used to define polarization charge in
experiments, models, computations, theories, and simulations of other systems. Charge
movement needs to be computed from a combination of electrodynamics and mechanics
because โeverything interacts with everything elseโ.
The classical polarization field need not enter into that treatment at all.
3. 2
INTRODUCTION
Polarization has a central role in electrodynamics. Faraday and Maxwell thought all charge
depends on the electric field. All charge would then be polarization.
Maxwell used the ๐ and ๐ fields as fundamental dependent variables. Charge only
appeared as polarization, usually over-approximated by a dielectric constant ๐ ๐ that is a single
real positive number. Charge independent of the electric field was not included, because the
electron had not been discovered: physicists at Cambridge University (UK) did not think that
charge could be independent of the electric field. The electron was discovered some decades
later, in Cambridge, ironically enough [1, 2]1. It became apparent to all that the permanent charge
of an electron is a fundamental source of the electric field. The electron and permanent charge
must be included in the equations defining the electric field, e.g., eq. (3) & eq. (6).
For physicists today, the fundamental electrical variable is the ๐ field that describes the
electric force on an infinitesimal test charge. ๐ and ๐ fields are auxiliary derived fields that many
textbooks think unnecessary, at best [5-10].
THEORY
Maxwellโs first equation for the composite variable ๐ relates the โfree chargeโ ๐ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก),
units cou/m3, to the sum of the electric field ๐ and polarization ๐. It is usually written as
๐๐ข๐ฏ ๐(๐, ๐, ๐|๐) = ๐ ๐(๐, ๐, ๐|๐) (1)
๐(๐ฅ, ๐ฆ, ๐ง|๐ก) โ ๐0 ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) + ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) (2)
The physical variable ๐ that describes the electric field is not visible in the classical formulation
eq. (1). Maxwell embedded polarization in the very definition of the dependent variable
๐ โ ๐0 ๐ + ๐. ๐0 is the electrical constant, sometimes called the โpermittivity of free spaceโ.
Polarization is described by a vector field ๐ with units of dipole moment per volume, cou-m/m3,
that can be misleadingly simplified to cou-m-2. The charge ๐ ๐ cannot depend on ๐ or ๐ in
traditional formulations and so ๐ ๐ is a permanent charge.
1
Thomsonโs monograph [3] โintended as a sequel to Professor Clerk-Maxwell's Treatise on electricity and magnetismโ does
not mention charge, as far as I can tell. Faradayโs chemical law of electrolysis was not known and so the chemistโs โelectronโ
postulated by Richard Laming and defined by Stoney [4] was not accepted in Cambridge as permanent charge, independent
of the electric field. It is surprising that the physical unit โthe Faradayโ describes a quantity of charged particles unknown to
Michael Faraday. Indeed, he did not anticipate the existence or importance of permanent charge on particles or elsewhere.
4. 3
When Maxwellโs first equation is written with ๐ as the dependent variable, the source
terms are ๐ ๐ and the divergence of ๐. Maxwellโs first equation for ๐ is
๐0 ๐๐ข๐ฏ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) = ๐ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) โ ๐๐ข๐ฏ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) (3)
๐ itself does not enter the equation. Only the divergence of ๐ appears on the right hand side of
the Maxwell equation for ๐(๐ฅ, ๐ฆ, ๐ง|๐ก), eq. (3). ๐ does not have the units of charge and should
not be called the โpolarization chargeโ.
๐(๐ฅ, ๐ฆ, ๐ง|๐ก) and the polarization ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) are customarily over-approximated in
classical presentations of Maxwellโs equations: the polarization is assumed to be proportional to
the electric field, independent of time.
๐(๐ฅ, ๐ฆ, ๐ง|๐ก) โ (๐ ๐ โ 1)๐0 ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) (4)
๐(๐ฅ, ๐ฆ, ๐ง|๐ก) โ ๐ ๐ ๐0 ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) (5)
The proportionality constant (๐ ๐ โ 1)๐0 involves the dielectric constant ๐ ๐ which must be a single
real positive number if the classical form of the Maxwell equations is taken as an exact
mathematical statement of a system of partial differential equations. If ๐ ๐ is generalized to
depend on time, or frequency, or the electric field, the form of the Maxwell equations change. If
๐ ๐ is generalized, traditional equations cannot be taken literally as a mathematical statement of
a boundary value problem. They must be changed to accommodate the generalization.
1) Polarization and thus ๐ ๐โhowever definedโdepend on time or frequency in complex
ways in all matter [11-16]. Many of the most interesting applications of electrodynamics
arise from the dependence of polarization and ๐ ๐ on field strength.
2) ๐ ๐ should be taken as a constant only when experimental estimates, or theoretical
models are not available, in my view.
It is difficult to imagine a physical system in which the electric field produces a change in charge
distribution independent of time (see examples shown on p.12). The time range in which
Maxwellโs equations are used in the technology of our computers, smartphones, and video
displays starts around 10โ10
sec. The time range in which Maxwellโs equations are used in biology
start around 10โ15
sec in simulations of the atoms that control protein function. The time range
in which Maxwellโs equations are used to design and operate the synchrotrons that generate
x-rays (time scale ~10โ19
sec) to determine protein structure is very much faster than that,
something like 10โ23
sec.
A dielectric constant ๐ ๐, independent of time is an inadequate over-approximation in
many cases of practical interest today.
Despite these difficulties, Maxwellโs first equation for ๐
๐ ๐ ๐0 ๐๐ข๐ฏ ๐( ๐ฅ, ๐ฆ, ๐ง|๐ก) = ๐ ๐( ๐ฅ, ๐ฆ, ๐ง|๐ก) (6)
is often written using the dielectric constant ๐ ๐ to describe polarization, without mention of the
over-approximation involved.
5. 4
RESULTS
Ambiguities in the traditional formulation. Equations (6) and (3) are ambiguous in an important
way. They do not mention the shape or boundaries of the regions in question. In fact, if ๐ varies
from region to region, but is constant within each region, charge accumulates at the boundaries,
and is absent within the region: when ๐ is constant, ๐๐ข๐ฏ ๐ = 0.
Such boundary charges are important in almost any system of dielectrics. Dielectric boundary
charges have a particular role in biological systems, see Appendix on Proteins p.14 and [17].
Most of the properties of dielectric rods studied by Faradayโand predecessors going
back to Benjamin Franklin, if not earlierโarise from the dielectric boundary charges. Textbooks
typically spend much effort teaching why polarization charge appears on dielectric boundaries in
systems with constant ๐ where ๐๐ข๐ฏ ๐ = 0 (e.g., Ch. 6 of [7]). Students wonder why regions
without polarization charge have polarization charge on boundaries.
A general principle is at work here: a field equation in itselfโlike eq. (3) and (6) that are
partial differential equations without boundary conditionsโis altogether insufficient to specify
an electric field. A model is needed that has boundary conditions, an explicit structure and
describes the spatial variation of ๐. Without specifying boundary conditions (defined explicitly in
specific structures), using ๐ in eq. (3) is ambiguous and confusing.
Indeed, using ๐ without boundary conditions is so incomplete that it might be called incorrect.
๐ is not unique. The general nature of the ambiguity becomes clear once one realizes that:
Adding ๐๐ฎ๐ซ๐ฅ โฬ(๐ฅ, ๐ฆ, ๐ง|๐ก) (7)
to ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) in Maxwellโs first equation, eq. (3)
changes nothing
because
๐๐ข๐ฏ ๐๐ฎ๐ซ๐ฅ โฬ(๐ฅ, ๐ฆ, ๐ง|๐ก) โก 0 ;
see [18, 19]
(8)
The ambiguity in ๐ means that any model ๐ ๐๐๐ ๐๐(๐ฅ, ๐ฆ, ๐ง|๐ก) of polarization can have
๐๐ฎ๐ซ๐ฅ โฬ( ๐ฅ, ๐ฆ, ๐ง|๐ก) added to it, without making any change in the ๐๐ข๐ฏ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) in Maxwellโs first
equation eq.(3), or its over-approximated version eq. (6). In other words, the polarization
๐๐ข๐ฏ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) in Maxwellโs first equation eq. (3) or (6) does not provide a unique structural
model of polarization ๐ ๐๐๐ ๐๐(๐ฅ, ๐ฆ, ๐ง|๐ก).
Any structural model can be modified by adding a polarization ๐ฬ(๐ฅ, ๐ฆ, ๐ง|๐ก) โ ๐๐ฎ๐ซ๐ฅ โฬ(๐ฅ, ๐ฆ, ๐ง|๐ก)
to its representation of polarization without changing electrical properties at all.
Models of the polarization ๐ ๐๐๐ ๐๐
๐
and ๐ ๐๐๐ ๐๐
๐
of the same structure written by different
authors may be strikingly different but they can give the same electrical results. If the models
differ by the curl of a vector field, they will give the same result in the Maxwell equations even
though the models can appear to be very different. The ๐๐ฎ๐ซ๐ฅ โฬ(๐ฅ, ๐ฆ, ๐ง|๐ก) field can be quite
complex and hard to recognize in a model. The two models ๐ ๐๐๐ ๐๐
๐
and ๐ ๐๐๐ ๐๐
๐
produce the same
6. 5
charge distribution ๐๐ข๐ฏ ๐ ๐๐๐ ๐๐
๐
and div ๐ ๐๐๐ ๐๐
๐
in Maxwellโs first equation eq. (3) and so they
cannot be distinguished by measurements of electric field ๐(๐ฅ, ๐ฆ, ๐ง|๐ก), magnetic fields
๐(๐ฅ, ๐ฆ, ๐ง|๐ก), and electrical current ๐ ๐๐๐๐๐ (see eq. (11).
The P field is arbitrary. It is not surprising then that the structural models analyzed in detail by
Purcell and Morin [5] are not unique, see p. 500 โ 507.
Purcell and Morin are not guilty of hyperboleโindeed they may be guilty of
understatementโwhen they say โThe concept of polarization density ๐ is more or less arbitraryโ
(slight paraphrase of [5], p. 507) and the ๐ field is โis an artifice that is not, on the whole, very
helpfulโ [5], p. 500.
The classical approach criticized by Purcell and Morin [5] does not allow unique
specification of a polarization field ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) from measurements of the electric ๐(๐ฅ, ๐ฆ, ๐ง|๐ก)
and magnetic fields ๐(๐ฅ, ๐ฆ, ๐ง|๐ก), and electrical current ๐ ๐๐๐๐๐ (see eq. (11).
It seems clear that most formulations of electrodynamics of dielectrics in classical
textbooks are โmore or less arbitraryโ. An arbitrary formulation is not a firm foundation on which
to build a theory of electrodynamics. See the Appendix p.14 for a discussion of polarization in
proteins in particular.
Maxwell First Equation, updated
๐๐ข๐ฏ ๐0 ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) = ๐ ๐
( ๐ฅ, ๐ฆ, ๐ง|๐ก; ๐) (9)
Here ๐ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก; ๐) describes all charge whatsoever, no matter how small or fast or transient,
including what is usually called dielectric charge and permanent charge, as well as charges driven
by other fields, like convection, diffusion or temperature. Updated formulations of the Maxwell
equations [13, 20] avoid the problems produced by ambiguous ๐ and over-simplified ๐ ๐.
Flow. Most applications of electrodynamics involve flow. We are interested in the flux of charges
๐ ๐ as well as their density and so I turn to Maxwellโs second equation that describes current. It is
understandable that Maxwellโand his Cambridge contemporaries and followersโhad difficulty
understanding current flow when their models did not include permanent charge, electrons or
their motions.
Maxwellโs extension of Ampereโs law describes the special properties of current flow ๐ ๐ก๐๐ก๐๐
(eq. (11) that make it so different from the flux of matter. Maxwellโs field equations include the
ethereal current ๐0 ๐๐ ๐๐กโ that makes the equations resemble those of a perfectly incompressible
fluid. Maxwellโs field equations describe the incompressible flow ๐ ๐ก๐๐ก๐๐ over a dynamic range of
something like 1016
, safely accessible within laboratories. The dynamic range is much larger if one
includes the interior of stars, and the core of galaxies in which light is known to follow the same
equations of electrodynamics as in our laboratories.
Maxwellโs field equations are different from material field equations (like the Navier
Stokes equations) because they are meaningful and valid universally [21], both in a vacuum devoid
of mass and matter and within and between the atoms of matter [13]. The ethereal current
๐0 ๐๐ ๐๐กโ responsible for the special properties of Maxwellโs equations arises from the Lorentz
(un)transformation of charge. Charge does not vary with velocity, unlike mass, length, and time,
all of which change dramatically as velocities approach the speed of light, strange as that seems.
7. 6
This topic is explained in any textbook of electrodynamics that includes special relativity.
Feynmanโs discussion of โThe Relativity of Electric and Magnetic Fieldsโ was an unforgettable
revelation to me as a student [6], Section 13-6: an observer moving at the same speed as a stream
of electrons sees zero current, but the forces measured by that observer are the same as the
forces measured by an observed who is not moving at all. The moving observer describes the force
as an electric field ๐(๐ฅ, ๐ฆ, ๐ง|๐ก). The unmoving observer describes the force as a magnetic field
๐(๐ฅ, ๐ฆ, ๐ง|๐ก). The observable forces are the same, whatever they are called, according to the
principal and theory of relativity.2
The ethereal current reveals itself in magnetic forces which have no counterpart in
material fields. The ethereal current is apparent in the daylight from the sun, that fuels life on
earth, and in the night light from stars that fuels our dreams as it decorates the sky. The ethereal
current is the term in the Maxwell equations that produces propagating waves in a perfect
vacuum like space.
Magnetism ๐
defined by
Maxwellโs Ampere Law
Maxwellโs Second Equation
1
๐0
๐๐ฎ๐ซ๐ฅ ๐ = ๐ ๐ + ๐0
๐ ๐
๐๐ก
(10)
๐ ๐ก๐๐ก๐๐ โ ๐ ๐ + ๐0
๐ ๐
๐๐ก
(11)
1
๐0
๐๐ฎ๐ซ๐ฅ ๐ = ๐๐ก๐๐ก๐๐ (12)
Note that ๐ ๐ includes the movement of all charge with mass, no matter how small, rapid or
transient. It includes the movements of charge classically approximated as the properties of an
ideal dielectric. It describes all movements of the charge described by ๐ ๐(๐ฅ, ๐ฆ, ๐ง|๐ก; ๐); ๐ ๐ is one
of the components of ๐ ๐.
Indeed, ๐ ๐ can be written in terms of ๐ฏ ๐ the velocity of mass with charge. In simple cases, such
as a plasma of ions each with charge ๐ ๐
๐ ๐ = ๐ฏ ๐ ๐ ๐ ๐ ๐ (13)
where ๐ ๐ is the charge per particle and ๐ ๐ is the number density of particles. Note
that sets of fluxes ๐ ๐
๐
, velocities ๐ฏ ๐
๐
, charges ๐ ๐,
๐
number densities ๐ ๐
๐
, and charge
densities ๐ ๐
๐
are needed to keep track of each elemental species ๐ of particles in a
mixture. Plasmas are always mixtures because they must contain both positive and
negative particles to keep electrical forces within safe bounds, as determined by
(approximate) global electroneutrality.
2
The principal and theory of relativity are confirmed to many significant figures every day in the GPS (global positioning
systems) software of the map apps on our smartphones, and in the advanced photon sources (synchrotrons) that produce
x-rays to determine the structure of proteins.
8. 7
In other cases, more complex than plasmas, ๐ ๐ and ๐ ๐ are related to material properties in more
subtle ways. For example, the Maxwell equations do not describe charge and current driven by
other fields, like convection, diffusion, or temperature. They do not describe constraints imposed
by boundary conditions and mechanical structures. Those must be specified separately. If the
other fields, structures, or boundary conditions involve matter with charge, they will respond to
changes in the electric field. The other fields and constraints thus contribute to the phenomena
of polarization and must be included in a description of it, as we shall discuss further below in
the examples shown on p. 12. The theory of complex fluids can provide examples that deal with
such cases, spanning scales, connecting micro (even atomic) structures with macro phenomena.
The charge density ๐ ๐ธ and current ๐ ๐๐๐๐๐ can be parsed into components in many ways,
some helpful in one historical context, some in another. Ref. [12, 13, 20, 22-27] define and
explore those representations in tedious detail. Simplifying those representations led to the
treatment in this paper.
Conservation of Mass with Charge and Conservation of Current
Maxwellโs Ampereโs law implies two equations of great importance and generality.
First, it implies a continuity equation that describes the conservation of charge with mass. The
continuity equation is the relation between the flux of charge with mass and density of charge
with mass.
Second, it implies conservation of total current.
Derivation: Take the divergence of both sides of eq. (10), use ๐๐ข๐ฏ ๐๐ฎ๐ซ๐ฅ = ๐ [18, 19], and get
๐๐ข๐ฏ ๐ ๐ = ๐๐ข๐ฏ (โ ๐0
๐๐
๐๐ก
) = โ ๐0
๐
๐๐ก
๐๐ข๐ฏ ๐ (14)
if we interchange time and spatial differentiation
But we have a relation between ๐๐ข๐ฏ ๐ and charge ๐ ๐ from Maxwellโs first equation, eq. (9), so
the result is the
Maxwell Continuity Equation
that describes the conservation of mass with charge
๐๐ข๐ฏ ๐ ๐ = โ ๐0
๐ ๐ ๐
๐๐ก
(15)
๐๐ข๐ฏ (๐ฏ ๐ ๐ ๐ ๐ ๐) = โ ๐0
๐ ๐ ๐
๐๐ก
(16)
Note that sets of fluxes ๐ ๐
๐
and sets of charge densities ๐ ๐
๐
are needed to keep
track of each elemental species ๐ of particles in a mixture, along with sets of
velocities ๐ฏ ๐
๐
, charges ๐ ๐,
๐
and number densities ๐ ๐
๐
, as described near eq. (13).
9. 8
Derivation: Taking the divergence of both sides of Maxwellโs Second law eq. (10) yields
Conservation of Total Current
๐๐ข๐ฏ ๐ ๐ก๐๐ก๐๐ โ ๐๐ข๐ฏ (๐ ๐ + ๐0
๐๐
๐๐ก
) = 0 (17)
๐๐ข๐ฏ ๐ ๐ก๐๐ก๐๐ = 0 (18)
or
๐๐ข๐ฏ ๐ ๐ก๐๐ก๐๐ โ ๐๐ข๐ฏ (๐ฏ ๐ ๐ ๐ ๐ ๐ ๐ ๐ + ๐0
๐๐
๐๐ก
) = 0 (19)
Conservation of current in one dimensional systems. One dimensional systems are of great
importance despite, or because of their simplicity. The design of one dimensional systems is
relatively easy. Their dimensionality rules out spatial singularities. Systems are more robust when
steep slopes and infinities are not present to create severe sensitivity.
Branched one dimensional systems describe the electronic networks and circuits of our
technology. Branched one dimensional systems describe the metabolic pathways of biological cells
that make life possible. Branched one dimensional systems can be described accurately by a simple
generalization of Kirchhoffโs law: all the ๐ ๐ก๐๐ก๐๐ that flows into a node must flow out [20, 24-26].
Unbranched one dimensional systems include the diodes of electronic technology and the
ion channels of biological cells which are crucial components of technology and life.
Conservation of current in series systems. Unbranched one dimensional systems have
components in series, each with its own current voltage relation arising from its microphysics. In
a series one dimensional system, the total current ๐ ๐ก๐๐ก๐๐ is equal everywhere at any time in every
location no matter what the microphysics of the flux ๐ ๐ of charge with mass. Maxwellโs equations
ensure that ๐0 ๐๐ ๐๐ก,โ etc., take on the values at every location and every time needed to make
the total currents ๐ ๐ก๐๐ก๐๐ equal everywhere. An example is described in detail near Fig. 2 of [25].
There is no spatial dependence of total current in a series one dimensional system. No
spatial variable or derivative is needed to describe total current in such a system [27], although of
course spatial variables are needed to describe the flux ๐ ๐ of charge with mass, or the electrical
current ๐ ๐
๐
๐ ๐ก๐๐ก๐๐
๐
of individual elemental species, or to describe the velocities, charge, and number
densities ๐ฏ ๐, ๐ ๐
, ๐ ๐
, and ๐ ๐.
It is important to realize that the flux of charge with mass ๐ ๐ is not conserved. In
fact, ๐๐ข๐ฏ ๐ ๐ = ๐๐ข๐ฏ (๐ฏ ๐ ๐ ๐ ๐ ๐) supplies the flow of charge that is the current ๐๐ ๐ ๐๐กโ necessary
to change ๐๐ข๐ฏ (๐0 ๐๐ ๐๐กโ ) as described by the following continuity equation. Charges can
10. 9
accumulate. ๐ ๐ can accumulate as ๐ ๐
. Total current ๐ ๐ก๐๐ก๐๐ cannot, not al all, not anywhere ornot
any time, when or where Maxwellโs equations are valid.
๐๐ข๐ฏ ๐ ๐ = ๐๐ข๐ฏ ๐0
๐๐
๐๐ก
=
๐
๐๐ก
๐๐ข๐ฏ (๐0 ๐) =
๐๐ ๐
๐๐ก
(20)
Because conservation of total current applies on every time and space scale, including those of
thermal motion, the properties of ๐ ๐ differ a great deal from the properties of ๐ ๐ก๐๐ก๐๐. For
example, in one dimensional channels, the material flux ๐ ๐ can exhibit all the complexities of a
function of infinite variation, without a time derivative, like a trajectory of a Brownian stochastic
process, that reverses direction an uncountably infinite number of times in any interval, however
small, while the electrical current ๐ ๐ก๐๐ก๐๐ will be spatially uniform [27], strange as that seems. The
fluctuations of ๐0 ๐๐ ๐๐กโ and other variables are exactly what is needed to smooth the bizarrely
infinite fluctuations of ๐ ๐ into the spatially uniform ๐ ๐ก๐๐ก๐๐.
DISCUSSION
How is the phenomenon of polarization included in the updated version of the Maxwell
equations eq. (9) & eq. (12)? First, we need a general paradigm to define polarization, even when
dielectrics are far from ideal, time and frequency dependent, and voltage dependent as well.
Biophysical Paradigm. I propose adopting the operational definition of โgating currentโ used to
define nonlinear time and voltage dependent polarization by biophysicists since 1972 [28-30]. A
community of scholars has studied the nonlinear currents that control the opening of voltage
sensitive protein channels for nearly fifty years , inspired by [31]. They have developed protocols
that may be as useful in other systems, as they have been in biophysics.
The basic idea is to apply a set of step functions of potential and observe the currents that
flow. The currents observed are transients that decline to a steady value, often to near zero after
a reasonable (biologically relevant) time. The measured currents are perfectly reproducible. If a
pulse is applied, the charge moved (the integral of the current) can be measured when the
voltage step is applied. The integration goes on until ๐ก1 when the current ๐๐๐๐๐ is nearly
independent of time, often nearly zero. That integral is called the ON charge ๐ ๐๐.
When the voltage is returned to its initial value (the value that was present before the ON
pulse), another current is observed that often has quite different time course [28-30]. The
integral of that current is the OFF charge ๐ ๐๐ ๐ .
This gating current depends on the voltage before the step. It also depends separately on
the voltage after the step, although Fig. 1 does not illustrate the dependence documented in the
literature [28-30]. The voltage and time dependence defines the molecular motions underlying
the gating current [32-34]. If the ON charge is found experimentally to equal the OFF charge, for
11. 10
a variety of pulse sizes and range of experimental conditions, the current is said to arise in a
nonlinear (i.e., voltage dependent) polarization capacitance and is interpreted as the movement
of charged groups in the electric field that move to one location after the ON pulse, and return
to their original location following the OFF pulse. The charge is called โgating chargeโ and the
current that carries the charge is called โgating currentโ.
The current observed in the set up is equal to the current carried by the charged groups
remote from the boundaries of the setup because the setup is designed to be an unbranched one
dimensional circuit with everything in series. In a one dimensional series setup the total current
is equal everywhere in the series system at any one time, even though the total current varies
significantly with time. The equality of current can easily be checked in the experiment. The
spatial equality of current needs to be checked in simulations as in [32-34].
If the currents reach a steady value independent of time, but not equal to zero, as in Fig. 1,
the steady current ๐๐๐๐๐ is considered to flow in a resistive path that is time independent, but
Figure 1
Fig. 1 shows the response to a step function change in potential and the charges
measured that are proposed as an operational definition of polarization.
12. 11
perhaps voltage dependent, in parallel with the device in which the gating charges ๐ ๐๐ and ๐ ๐๐ ๐
flow. If the current does not reach a steady value, or if the areas are not equal, the currents are
not considered โcapacitiveโ and are interpreted as those through a time and voltage dependent
โresistorโ. It is important to check the currents through the resistive path by independent
methods to see if they are time independent, by blocking the path with drugs, or one way or the
other. If the resistive currents are not time independent, the definition of ๐ ๐๐ and ๐ ๐๐ ๐ in Fig. 1
needs to be changed.
Clearly, this approach will only work if step functions can reveal all the properties of the
system. If the underlying mechanisms depend on the rate of change of velocity, step functions
are clearly insufficient.
Much work has been done showing that step functions are enough to understand the
voltage dependent mechanisms in the classical action potential of the squid axon [35-37], starting
with [38], Fig. 10 and eq. 11. Hodgkin kindly explained the significance of this issue to colleagues,
including the author (around 1970). He explained the possible incompleteness of step function
measurements: if sodium conductance had a significant dependence on ๐๐ ๐๐ก,โ the action
potential computed from voltage clamp data would differ from experimental measurements. He
mentioned that this possibility was an important motivation for Huxleyโs heroic hand integration
[31] of the Hodgkin Huxley differential equations. Huxley confirmed this in a separate personal
communication, Huxley to Eisenberg. Those computations and many papers since [35-37] have
shown that voltage clamp data (in response to steps) is enough to predict the shape and
propagation of the action potential in nerve and skeletal muscle. It should be clearly understood
that such a result is not available for biological systems in which the influx of Ca++ drives the action
potential and its propagation [39].
The conductance of the voltage activated calcium channel has complex dependence on
the current through the channel because the concentration of Ca++ in the cytoplasm is so low
(~10-8M at rest) that the current almost always changes the local concentration near the
cytoplasmic side of the channel. Current through the channel changes the local concentration.
The local concentration changes the gating and selectivity characteristics of the channel protein,
as calcium ions are prone to do at many physical and biological interfaces. It seems unlikely that
the resulting properties can be described by the same formalism [31] used for voltage controlled
sodium and potassium channels of nerve and skeletal muscle. That formalism uses variables that
depend on membrane potential and not membrane current because it was evident to Cole [40]
and Hodgkin [41-43] that neuronal action potentials were essentially voltage dependent, not
current dependent. There may of course be other reasons the formalism [31] is inadequate.
Experiments are needed to show that responses to steps of voltage allow computation of a
calcium driven action potential.
Polarization defined in theories and simulations. The polarization protocol described here can
be applied to models and simulations of polarization as well as experiments. Indeed, the
13. 12
operational definition of polarization has been applied even when theories [32] or simulations
are enormously complicated by atomic detail including the individual motions of thousands of
atoms [33, 34].
Does the estimated polarization equal ๐? The question of general interest is how does the
polarization defined this way correspond to the polarization ๐ in the classical formulation of the
Maxwell equations eq. (3)?
Polarization cannot be defined in general. The variety of possible responses of matter to a step
of potential prevents a general answer.
Indeed, a main point of this paper is that polarization must be defined by a protocol in a
specific setting. It cannot be defined in general. The possible motions of mass and mass with
charge are too many to permit a general definition of polarization. Every possible motion of mass
(with charge, including rotations and translations and changes of shape) would produce a
polarization. Polarization currents can be as complicated as the motions of matter.
Insight can be developed into various kinds of polarization by constructing โtoyโ models
of classical systems, applying the operational definition of polarization, and understanding the
resulting estimates of polarization currents (that are called โgating currentsโ in the biophysical
literature).
Those models must specify the mechanical variables ๐ฏ ๐, ๐ ๐, ๐ ๐ and ๐ ๐ (or their equivalent) and
solve the field equations of mechanics along with the Maxwell equations. Indeed, toy problems
are examples that may help develop insight:
(1) simple electro-mechanical models, like a charged mass on a spring with damping.
(2) ideal gases of permanently charged particles, i.e., biological and physical plasmas.
(3) ideal gases of dipoles (point and macroscopic), quadrupoles, and mixtures of dipoles and
quadrupoles, that rotate and translate while some are attached by bonds that vibrate
(see (1)). These mixtures should provide decent representations of liquid water in ionic
solutions, if they include a background dielectric, even if the dielectric is over-
approximated with a single dielectric constant ๐ ๐.
(4) molecular models of ionic solutions that include water as a molecule. It is best to use
models that are successful in predicting the activity of solutions of diverse composition
and content and include water and ions as molecules of unequal nonzero size [44].
(5) classical models of impedance, dielectric .and molecular spectroscopy [16, 45-50].
(6) well studied systems of complex fluids, spanning scales, connecting micro (even atomic)
structures with macroscopic functions.
These examples, taken together, will form a handbook of practical examples closely related to
the classical approximations of dielectrics.
14. 13
These problems have time dependent solutions except in degenerate, uninteresting
cases. As stated in [20] on p. 13
โIt is necessary also to reiterate that ๐ ๐ is a single, real positive constant in
Maxwellโs equations as he wrote them and as they have been stated in many
textbooks since then, following [51-53]. If one wishes to generalize ๐ ๐ so that it
more realistically describes the properties of matter, one must actually change
the differential equation (6) and the set of Maxwellโs equations as a whole. If, to
cite a common (but not universal) example, ๐ ๐ is to be generalized to a time
dependent function (because polarization current in this case is a time dependent
solution of a linear, often constant coefficient, differential equation that depends
only on the local electric field), the mathematical structure of Maxwellโs
equations changes.
[Perhaps it is tempting to take a short cut by simply converting ๐ ๐ into a
function of time ๐ ๐(๐ก) in Maxwellโs equations, as classically written.] Solving the
equations with a constant ๐ ๐ and then letting ๐ ๐ become a function of time
creates a mathematical chimera that is not correct. The chimera is not a solution
of the equations. [The full functional form, or differential equation for ๐ ๐(๐ก) must
be written and solved together with he Maxwell equations. This is a formidable
task in any case, but becomes a formidable challenge if convection or
electrodiffusion modify polarization, as well as the electric field.]
Even if one confines oneself to sinusoidal systems (as in classical impedance
or dielectric spectroscopy [11, 45, 54, 55]), one should explicitly introduce the
sinusoids into the equations and not just assume that the simplified treatment of
sinusoids in elementary circuit theory [56-60] is correct: it is not at all clear that
Maxwellโs equationsโ combined with other field equations (like Navier Stokes
[61-78] or PNP = drift diffusion [66, 79-94]); [joined] with constitutive equations;
and boundary conditionsโalways have steady state solutions in the sinusoidal
case. They certainly do not always have solutions that are linear functions of just
the electric field [95-98].โ
It seems clear that the classical Maxwell equations with the over-approximated dielectric
coefficient ๐ ๐ cannot emerge in the time dependent case. Of course, the classical Maxwell
equations cannot emerge when polarization has a nonlinear dependence on the electric field, or
depends on the global (not local) electric field, or depends on convection or electrodiffusion.
Indeed, in my opinion, when confronted with these models of polarization, the classical
Maxwell equations will be seen as useful only when knowledge of the actual properties of
polarization is missing.
A generalization of Maxwellโs ๐ may emerge from these examples that describes a wide enough
range of systems to be useful, one imagines that Maxwell hoped ๐ and ๐ would.
15. 14
Until then, one is left with
(1) bewilderingly complete measurements, over an enormous range of frequencies (e.g.,
[14]) of the dielectric properties and conductance of ionic solutions of varying
composition and content. These measurements embarrass the theoretician with their
diversity and complexity. They have not yet been captured in any formulas or programs
less complicated than a look up table of all the results.
(2) computations of the motion of all charges on the atomic scale [33, 34], described by the
field equations of mechanics and electrodynamics [32].
What should be done when little is known? Sadly, the actual properties of polarization are often
unknown. Then, one is left with the over-approximated eq. (6) or nothing at all. Eq. (6) is certainly
better than nothing: it is almost never useful to assume polarization effects contribute nothing
to phenomena. Eq. (6) can be useful if it is used gingerly: toy models can usefully represent an
idealized view of part of the real world.
In some cases, the toy models can be enormously helpful. They allow the design of circuits
in our analog and digital electronic technology [99-102]. They allow the understanding of
selectivity [44, 103-105] and current voltage relations of several important biological channel
proteins in a wide range of solutions [44, 106-108]. In other casesโfor example the description
of ionic solutions with many componentsโthey can be too unrealistic to be useful. Experiments
and experience can tell how useful the toy model actually is in a particular case: pure thought
usually cannot.
APPENDIX
๐(๐ฅ, ๐ฆ, ๐ง|๐ก) in Proteins
Ambiguities in the meaning of the polarization field ๐(๐ฅ, ๐ฆ, ๐ง|๐ก) can cause serious
difficulties in the understanding of protein function. The protein data bank contains 167,327
structures in atomic detail today (August 11, 2020) and the number is growing rapidly as cryo-
electron microscopy is used more and more.
Protein structures are usually analyzed with molecular dynamics programs that assume
periodic boundary conditions. Proteins are not periodic in their natural setting. It seems obvious
that periodic systems cannot conserve total current ๐ ๐๐๐๐๐ in generalโor perhaps even in
particularโas required by the Maxwell equations, see eq. (17). In other words, it is likely that
molecular dynamics analyses of periodic structures do not satisfy the Maxwell equations,
although almost all known physics does satisfy those equations.
It is also unlikely that standard programs of molecular dynamics compute electrodynamics
of nonperiodic systems correctly, despite their use of Ewald sums, with various conventions, and
force fields (tailored to fit macroscopic, not quantum mechanical) data. Compare the exhaustive
methods used to validate results in computational electronics [86] with those in the computation
of electric fields in proteins.
The electrostatic and electrodynamic properties of proteins are of great importance.
Many of the atoms in a protein are assigned permanent charge greater than 0.2๐ in the force
fields used in molecular dynamics, where ๐ is the elementary charge, and these charges tend to
16. 15
cluster in locations most important for biological function. Enormous densities of charge
(> 10M, sometimes much larger) are found in and near channels of proteins [44, 109-111] and
in the โcatalytic active sitesโ [112] of enzymes. Such densities are also found near nucleic acids,
DNA and all types of RNA and binding sites of proteins in general.
A feel for the size of electrostatic energies can be found from Coulombโs law between
isolated charges (in an infinite domain without boundary conditions).
๐ธ๐๐๐ข =
560
๐ ๐
๐๐ ๐๐
๐๐๐
(21)
Here ๐ธ๐๐๐ข is in units of the thermal energy ๐ ๐, with gas constant ๐ and absolute temperature
๐; charge ๐๐ or ๐๐ are in units of the elementary charge ๐; and ๐๐๐ is in units of Angstroms
= 10โ10
meters.
For water, with ๐ ๐ โ 80, this becomes
๐ธ๐๐๐ข(water) = 7
๐ ๐ ๐ ๐
๐ ๐๐
; units RT (22)
For the pore of a channel, one can guess ๐ ๐ โ 10 and then
๐ธ๐๐๐ข(channel: ๐ ๐ โ 10 ) = 56
๐ ๐ ๐ ๐
๐ ๐๐
; units RT (23)
Electrostatic energy has to be computed very accurately indeed to predict current voltage
relations. Gillespie ([108] particularly in โSupplemental Dataโ) found that errors of energy of
0.05 RT produced significant changes in current voltage relations of the ryanodine receptor
protein. Similar sensitivity is expected for any model of an ion channel or the active site of an
enzyme or the binding site of a protein in which the underlying energetics are similar.
Accuracy of this sort is not claimed in most calculations of molecular or quantum dynamics
involving ionic solutions. Typical accuracies claimed for quantum chemical calculations start
around 2 RT and for molecular dynamics calculations around 0.5 RT, in my (limited) experience.
However, reduced models of proteins allow calculations of the precision needed to deal with
Gillespieโs results, as much work demonstrates [106, 108, 113-116]. Reduced models use lower
resolution representations drawn from the full detail atomic structure. Typically the accuracy of
the reduced model itself cannot be calculated from first principles. But if the model fits a wide
range of data, measured in solutions of different composition and contents with a single set of
parameters, as does Gillespieโs, the precision of the model is assured by the fit to the data itself
[108].
Polarization must be accurately calculated to understand protein function. Polarization
phenomena are as important as permanent charges. Charges that depend on the electric field
are as important as the permanent charges analyzed in the last paragraphs (see [117] and
references cited there). The enormous effort and investment in developing polarizable force fields
[118] is an eloquent testimonial to the importance of polarization in biological and chemical
applications.
The ambiguous nature of the ๐ field means that ๐ provides a poor guide to the importance
of polarization in a particular protein. Identical electrodynamics will arise from structures that
appear different, but only differ by ๐๐ฎ๐ซ๐ฅ โฬ (๐ฅ, ๐ฆ, ๐ง|๐ก). Situations like this can produce confusion
17. 16
and unproductive argument, because not all scientistsโparticularly structural biologistsโrealize
the inherent unavoidable ambiguity of ๐๐ข๐ฏ ๐ or eq. (3).
What is needed is a model of both the dynamics of mass and the electrodynamics of charge, that
allows the unambiguous calculation of the response of the protein to an applied electric field. The
combined dynamics of mass and electrodynamics of charge (that depends on the electric field)
are the appropriate model of the polarization phenomena, not the classical ๐ field.
Low resolution models may do surprisingly well provided they actually satisfy the Maxwell
equations and conservation of current, avoiding periodic boundary conditions for the electric
field in nonperiodic systems. Compare [32] and [33, 119].
ACKNOWLEDGMENT
It is a particular pleasure to thank my friend and teacher Chun Liu for his continual
encouragement, advice, and for correcting mistakes in my mathematics as I developed these
ideas over many years.
18. 17
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