Were We Foolish To Follow Maxwell?

Heaviside interpreted Maxwell's equations as four equations. Let's consider what these four equations really mean.

Maxwell's Gauss equation for electricity states that the charge contained in a volume is equal to the electric field from that volume. Electrons moving at a fraction of the speed of light move far faster than protons and neutrons moving at a fraction of this speed. This root mean squared speed differential leads to Gauss' equation for electricity.

Maxwell Gauss' equation for magnetism states that the net magnetic flux from a volume will always equal zero. This mean that magnetic fields will look elliptical at all times. A magnetic field is a curling field of electrons. If we put a curling electron or current field into the Gauss magnetism equation we get the divergence of a curl is equal to zero. This is a vector calculus identity.

The Maxwell Ampere equation has the curl of a curl. The magnetic field is a curling field of electrons. Those electrons curl from a current of electrons due to the electron's tendency to orbit surrounding nuclei. The magnetic field is the curl vector from the field of rotating electrons. These magnetic fields bend elliptically around currents.

The Maxwell Faraday equation states the change in magnetic field causes a curl in the electric field. Understanding how electrons curl is instrumental to understanding this equation. When a conductive curled wire enters or turns in a magnetic field the field induces tight curls of electrons in the wire. At a higher level the electrons oppose the tight curls due to conservation of angular momentum. As the curls of the electrons change there is a brief counter curl at the macroscopic level in the curled conductor.

Conductivity: How Does It Work

How does conductivity work. Normally we talk about 'free' electrons carrying charge through a lattice. An electron puts electrostatic pressure on the lattice by moving into one end of the lattice. Another electron moves out of the lattice near the 'load'.

Free electrons and lattices with holes have been observed to conduct. More well balanced material does not. I want to explore why. When an insulator is injected with a hot electron it gives up an electron quickly.

Electrons propagate their influence in a conductor at very high speeds. The wave of electrical influence propagates at a fraction of the speed of light.

Atoms and molecules that are electrically balanced, bonded and are not prone to ionization make poor conductors. These electrons may slow the drift of charge such that it gets reflected back to the source or the generator.

Sharp Charge - Dull Charge

Look at both sides of the periodic table. Ignore the balanced noble gases. Chlorine and the halogens have almost full valence bands they will attract negative charge now and then with statistically spurious positive charge from the nucleus of the atom. If a spurious electron briefly fills the valance band of the Chlorine atom the atom is said to be negatively charged.

The forces of charged particles pulling at a small distance will be far from uniform. Sharp charge pulls far and statistically seldom (Poisson). Dull charge pulls a short distance with a more constant force.

The act of an electron jumping quickly from a surface or a molecule leads to a net force in the opposing direction as the electrons backfill the electron that has jumped. The electrons that jump seem to move through thin air unopposed while the backfill electrons bump into matter and drag this matter with them. 

Electrons tend to move in an orbiting manner. There are statistical orbitals that have been deemed likely places to find electrons. These orbitals rotate quickly as electrons are moving at a real fraction of the speed of light. These orbitals will tend towards spherical or elliptical orbits. These stable configurations for electron movement become the building blocks for atoms and molecules alike.

Mechanism for Magnetic Force

Magnetism is caused by a curl in the field of moving electrons. How these curling fields interact is key to understanding the magnetic force. When a magnetic South pole is inverted and approaches a magnetic North pole the inversion causes the fields to line up with the same curl. This curl attracts density and as a result the two poles come together.

When a magnetic North pole is inverted and moved towards another North pole there is a repelling effect. The spinning electrons cannot simply begin spinning in the opposite direction. The spinning electrons wrap back towards the nearest South pole. This spinning field near both North poles attracts gaseous matter in the way of air and keeps the two poles apart.

When two wires are conducting current in opposite directions they are pushed apart. The current on each wire eddies off of the wire creating a curling magnetic field. When this curl is additive the air in between the wires increases. This causes the wires to be pushed apart.

When two wires are conducting current in the same direction these wires are pulled towards each other. The current on each wire eddies off the wire creating a curling electron field. When this curl cancels the electron fields from the other wire the electrons will disperse. In this electron dead environment matter does not have an easy time sticking around. The less matter there is between the two wires the closer they will come to each other.

Maxwell's Equations

Maxwell-Gauss equation for electricity shows the path of ballistic or drift of hot charge carriers.

Maxwell-Gauss equation for magnetism shows the opposite of the normal vector to the path of curling electrons.

The Maxwell-Ampere equation describes curling electrons around a current. The current eddies out around surrounding molecules giving rise to this effect. The curl of the curl of the electron field can be seen as a current in some instances.

The Maxwell-Faraday equation the change in a curling electron field will create a looser electron circulation in the opposite direction to the electron field. This opposite circulation is due to conservation of angular momentum.

Asymmetric Capacitors and Ion Movement

NASA published the paper Asymmetrical Capacitors for Propulsion by Canning et al. and the American Institute of Physics published High Efficiency Lifter Based on the Biefeld-Brown Effect by Einat and Kalderon. The Army Research Laboratory published Force on an Asymmetric Capacitor by Bahder and Fazi.

An asymmetric capacitor is constructed with a large grounded electrode and a positive small electrode. A cloud of electrons is emitted from the large electrode. These electrons are accelerated towards the positive electrode. Some of the electrons hit the electrode but many miss and fly in a ballistic manner upwards past the positive electrode into the dielectric.

Electrons drift down to back-fill the electrons that have moved up. More electrons are emitted from the large electrode.The electrons emitted from the large electrode are accelerated. Momentum is distributed in a sparse manner above the capacitor because of the high velocity of the electrons. Momentum is zero in the region of acceleration. Momentum is downwards and highly concentrated below the large electrode of the capacitor.

Asymmetric Capacitors and Force

NASA published the paper Asymmetrical Capacitors for Propulsion by Canning et al. and the American Institute of Physics published High Efficiency Lifter Based on the Biefeld-Brown Effect by Einat and Kalderon. The Army Research Laboratory published Force on an Asymmetric Capacitor by Bahder and Fazi.

Asymmetric capacitors most likely cause electrons or ions to go ballistic in the space between the two electrodes. Many of the ballistic electrons likely overshoot the small electrode causing a net momentum increase above the capacitor. Momentum is conserved and larger ions and even molecules drift towards the larger electrode.

The momentum is very concentrated at the bottom of the capacitor because the velocity of the particles is far less than those ions moving in a ballistic manner in the opposite direction. The drift ions most likely undergo many collisions with other molecules and move in a slower manner causing pressure at the far end of the larger electrode.

Ballistic movement of small particles in one direction towards the small electrode. This causes a drift movement of molecules in the other direction, towards the large electrode, causing a pressure differential. This will cause a net force pointing from the large electrode towards the small electrode.

Asymmetric Momentum Distribution

When electrons are accelerated away from the center of mass there is an equal and opposite force in the other direction which conserves momentum. Electrons may be accelerated to high velocities. The matter moving in the other direction will be more massive and will move with less velocity.

The momentum distributions for the electrons span more space than for the matter moving towards the center of mass.

Asymmetric Acceleration

NASA published the paper Asymmetrical Capacitors for Propulsion by Canning et al. and the American Institute of Physics published High Efficiency Lifter Based on the Biefeld-Brown Effect by Einat and Kalderon.

The acceleration in an asymmetrical capacitor happens between the electrodes which are kept at many kilo-volts of potential difference. In between the electrodes of the asymmetrical capacitor is an electric field measured in V/m. Charge will accelerate at a fantastic rate in such an electric field. Electrons will take off at a large fraction of the speed of light. The momentum from this kinetic electron will be spread out over kilometers if it misses the top, small electrode of the capacitor as well as the atoms of the atmosphere around the asymmetrical capacitor.

Momentum is conserved. When electrons are accelerated there is an equal and opposite reaction in the opposite direction with equal momentum. Mass travels downwards. In fact so much air travels downwards that it pushes the asymmetrical capacitor upwards.

A sphere of any size contains mass. This mass will concentrate negative charge towards the center of mass and the negative charge will be relatively more sparse towards the periphery of mass. Negative charge will accelerate towards the periphery of mass. Mass around the accelerated charge will be pulled towards the center of mass due to conservation of momentum.

The acceleration, therefore, is asymmetrical. The acceleration of ions towards the small side of the asymmetric capacitor causes an acceleration downwards of more mass. This causes the capacitor to lift (the capacitor can push downwards if pointed that way).

Ions and Large Bodies of Mass

A sphere has more surface area on its outer perimeter than near the center. While that sounds like an obvious statement it is important in understanding how ions travel with respect to a large mass. A mass of any size exhibits these principles.

Electrons nearer to the center of mass are more crowded and those at the periphery are more spread out. Electrons nearer the center will be accelerated outwards quickly and there will be a return current which is slower.

Electrons generally jump outwards while everything else gets pulled in the opposite direction in a drift force. This is why an apple falls towards the Earth and doesn't just float.

Asymmetric Capacitors

NASA published the paper Asymmetrical Capacitors for Propulsion by Canning et al. and the American Institute of Physics published High Efficiency Lifter Based on the Biefeld-Brown Effect by Einat and Kalderon.

The ions fly towards the small conductor and overshoot. The ions are accelerated in the space between the two electrodes. This creates a momentum distribution of fast moving particles distributing momentum over a large area. The larger ions move in the opposite direction creating low pressure above the lifter and high pressure below the lifter.

For every action there is an equal and opposite reaction.

Asymmetric Capacitors and Propulsion

NASA published the paper Asymmetrical Capacitors for Propulsion by Canning et al. and the American Institute of Physics published High Efficiency Lifter Based on the Biefeld-Brown Effect by Einat and Kalderon.

These lifters or pressers push a mass from the big end of an asymmetrical capacitor to the thin end. The big end of the capacitor becomes a source for electrons. These electrons accelerate and move past the positive small end of the capacitor at a fast speed. For every action there is an equal and opposite reaction.

The space between the electrodes in the asymmetrical capacitor accelerates the electrons. A slow wind moves from the small electrode to the big electrode. More pressure is found underneath the lifter when it is lifting.

Electricity and Return Current

Universal to the concept of electricity is the idea of a return current. No circuit can exist without the return. Electrons under pressure will tend to pop, one at a time, towards an area with less electron pressure or negative potential voltage. In this case the return current may not be a discrete electron returning to where the original kinetic electron left. Instead, a number of electrons will drift in to back-fill where the kinetic electron left.

The electrons will leave according to the probability mass function lambda to the k multiplied by e to the negative lambda divided by k factorial where lambda is the electron ejection rate. The Poisson distribution tells us that for a certain period of time and a certain surface area an average ejection rate will be described by the equation described above. 

The flux in the flow of electrons as energetic electrons move one way and then less energetic electrons back-fill in large numbers moving in the other direction show the capacitive charging of a dielectric medium. Electron kinetic energy begins to move both ways as the medium maximizes the energy stored in the kinetic electrons in the medium.

Electron Movement at a PN Junction

Electrons must move faster at a pn junction. The root mean squared speed of electrons must increase as they move from one side to the other side of the junction. There must be a pressure exerted by a static voltage. This pressure must be described by a statistical process.

Electrons entering the p end of the pn junction with extra energy sit on the edge of the junction and become part of the pressure process. This is the reverse bias case. Electrons entering the n side of the pn junction with extra energy are conducted through the depletion region to the p side where they are conducted out of the diode.

Electron Movement in a Capacitive Structure

So just how do electrons move. We know that they move very quickly. Electrons move at a fraction of the speed of light. These electrons tend to move around a more massive nucleus and can be fit into orbitals. Various levels of energy can be attributed to these orbitals.

Do electrons move differently when they are in the presence of a capacitor, inductor or resistor? This blog post seeks to explore the statistical process of electron movement in the presence of a capacitor. Perhaps in capacitors electrons move as the pistons move in a boxer engine.

We know that charge carriers pile into capacitors as the voltage increases. An uncharged capacitor contains no potential energy. When the capacitor is charged we say that the electric field formed charging the capacitors contains energy. But what does this look like from an electron movement point of view. How far are electrons traveling and if they travel far why doesn't electric current just flow?

When a coulomb of electrons approaches a capacitor some charge makes its way through the capacitor and out the other end. This is not like a conductor though. As the charge makes its way through the dielectric the charge is replaced by electrons bonded to the dielectric. These bonded electrons are far less energetic than the electrons they are replacing.

Electrons flow into the dielectric and are replaced by less energetic electrons until all of the charges match and the capacitor is said to be charged. Back to the analogy of the boxer engine where we have pistons and electrons moving one way and almost at the same time we have a counter movement of electrons or pistons moving in the opposite direction. All of this happens at a fraction of the speed of light that is to say very quickly.

The Maxwell-Heaviside Equations

This blog has been alive for two years and I wanted to reflect on the basic tenants of electromagnetics. Namely, the four Maxwell-Heaviside equations. These four equations and the Lorentz force equation backbone electromagnetics for all practicing electrical engineers.

Gauss' law of electricity is useful as it shows the direction that a  charge will travel in given the presence of an electric field. The electric field is a useful construct because its magnitude gives us insight into the behaviour of electric phenomena.

Gauss' law of magnetism is somewhat less useful. A magnetic field is the normal vector in the curl of an electron field. Gauss' law of magnetism points out that the divergence of a curl is zero. That is the magnetic field is electrons curling and the divergence of that curling field is zero. This fact is also a vector calculus identity.

The Maxwell-Ampere equation can be read two ways. The curl in a magnetic field gives a current and the curl of electrons gives a magnetic field which is the normal to the field of curling electrons. The curl of a curling field does add up to a current. This is backwards from the way we should be thinking about electricity. The current in a wire generates a magnetic field which surrounds the wire. The current from the wire spins off a type of leakage current which spins. The telegraphers equations dictate this type of behaviour.

The Maxwell-Faraday equation involves Lenz's law and the principle of electromagnetic induction. The equation, in differential form, states that the change in a magnetic field will be a curl in the electric field. Understanding what is really going on takes closer consideration. When a loop of current sees an increase in the curl of an electron field one has to consider the nature of the curl. The curl in the electron field is very tight as it was generated by a permanent or electromagnet. The magnetic domains or curl in the electron field of the coil is random.

Due to particle interactions the coil starts to see a tight curl in its electron fields. The law of conservation of angular momentum causes a Lenz' phenomenon curling in the opposite direction. This phenomenon is harnessed as current to drive a load in an electric generator.

When the Maxwell-Heaviside equations were developed the developers had jar batteries, wires and coils at their disposal. Reconciling their world with a modern electronic world takes understanding and patience.

On Capacitance

How capacitance works is poorly defined. Some texts will point out that a dielectric is polarized such that energy is stored to counter the prevailing electric field. But what are the electrons doing? How are these electrons moving? I have explored this topic previously and will revisit it again because capacitance is complicated and so many texts make it sound simple.

The energy in a capacitor is proportional to the voltage squared. Voltage is the excited energy of an electron. Energy is also proportional to the dielectric constant of the dielectric material. The surface area is also proportional to the energy stored but the surface area is not always equal on both plates of a capacitor but we will get to some of the subtleties of capacitance later. Lastly, the energy stored between two plates of a capacitor is inversely proportional to the distance between the two plates or poles.

So what are the electrons doing in a capacitor to store energy? First they move. Electrons move at a fraction of the speed of light. Estimates of how fast an electron move vary but electrons don't move at the speed of light. Electrons don't move at speeds a regular person could understand. Electrons move at a fraction of the speed of light that is to say a speed that is meaningless in kilometers per hour.

We know that electron take in a large number of electrons before they begin to excite at the voltage levels of the conductor charging the capacitor. Electrons flow in and the voltage or excitement of the electrons in the conductor don't immediately rise. The electrons flow through the dielectric to the return and are immediately back-filled by electrons in the dielectric. As more energized or higher voltage electrons enter the dielectric the dielectric becomes more energetic. The electrons that are back-filling the incoming electrons have a higher and higher voltage until their voltage matches the incoming electrons. The capacitor is fully charged.

When a capacitor is fully charged there is an excitement at both plates. The capacitor has a lot of statistical properties that may well have to do with the exponential distribution or the Poisson distribution. Electrons will move into the dielectric with a high relative energy and they will keep moving towards the return. Eventually the electron will return towards the energized plate. It is the continuous dance between the energetic plate and the return that constitutes capacitance. Electrons moving quickly towards the opposing plate only to be back-filled by electrons seeming to polarize the dielectric.

The statistics of electrons in a capacitor has yet to be fully understood. Understanding that things are not fully understood is the first step in understanding the capacitor and eventually the diode and transistor.

Electron Field Curl and Force

If an electron flies through a dense medium of particles it will be deflected.

If a group of electrons deflects in a circle or an elipse a magnetic field has been created. The field of electrons has a measurable curl.

If enough electrons are moving in a circular pattern they will interact to cause similar movement in nearby electrons. The curl of the electron field can be seen as the magnetic field.

Electrons ejected from a coil or a permanent magnet will wrap back from one pole to the opposing pole.

When electron curls add, such as when a South pole comes near a North pole, mass will be drawn in causing the two poles to attract. When two wires have additive curls they will attract mass between them causing a repulsive force known as Ampere's force.

When electron curl is opposite then the field of curling electrons will bend back to terminate at its opposing pole. Matter will be drawn into the curl causing a repulsive force.

When common mode wires eject electrons the curl of their electrons cancels each other out causing matter to scatter as the atoms have less curl. The scattered matter causes the wires to attract. This is known as Ampere's force law.

A wire will eject electrons as the telegraphers model states. Some of the ejected electrons will travel in a circle or an ellipse and end up back on the wire. This phenomenon will happen all around the wire.

Ejected electrons tend to curl in a tight circle or elipse. When these tight curls interact with a conductive media the tend to induce a larger curl in the opposite direction. This opposing curl is temporarty and caused by a counter-spin of opposing atoms.

When the curl of ejected electrons influences a nearby wire it will temporarily cause eddy current to flow on the near side or the far side of the wire. This is known as the proximity effect.

When the internal curl of electrons turns towards the core of a conductor as it does; the eddy current opposes the current at the center of the conductor. This is known as the skin effect.

On Inductance

Ampere's equation states that the magnetic field wraps around the current in a steady state. This means that electrons might leave a conductor and spin back onto that same conductor around the atoms and molecules in the surrounding media. The normal of the electron spin is represented by magnetic field lines.

Faraday's law will be the focus of this post. Faraday's law states that a changing magnetic field will induce an electric field according to the left hand rule. This is opposite the right hand rule used in Ampere's law. We must explore why this is.

A magnetic field is a tight curl of electrons spinning in the same direction. Electrons that were spinning in the opposite direction because of the movement of the magnetic field or the circuit within the magnetic field will push outwards and cause a momentary electromotive force through a circuit. Electric motors take advantage of this.

In the figure below the circulating electrons counter-clockwise are due to an external magnetic field. Really the electrons are just lining up due to collisions. In between the spinning due to magnetism exists a counter-spin. In this case clockwise. This clockwise spin is not constrained to a small tight spin so it pushes outwards to a larger and larger Faraday current until it can produce an electromotive force in a circuit.

This is how a changing magnetic field induces an electromotive force in a circuit.

The Missing Maxwell's Equation

Heaviside's  version of Maxwell's equations are missing one equation. Either that or the equation has to be inferred.

Maxwell's first two equation known as the Gauss equations of electricity and magnetism define static electric and magnetic fields. Ampere and Faraday have equations that follow Gauss' to define magnetic fields and currents and currents in the presence of changing magnetic fields.

The missing equation would be based on the simple electromagnet. A curling electron or current field produces a magnetic field. The curl of the volume current density is equal to the magnetic field vector with a proportionality constant. Now Gauss' equation for magnetism breaks down into the divergence of a curl which by vector calculus identities is just an identity.

Faraday's law ends up being a result of Lenz' law. A tight magnetic field (spin of electrons) will cause a larger counter-spin of electrons due to the conservation of angular momentum.