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.

Sharp Charge vs. Dull Charge in an Atom

On the left side of the periodic table we have atoms with what are termed a positive charge. On the right side of the periodic table we see negative charges. All, in reality, are balanced. Each proton has an associated electron to balance it.

Positive atoms on the left side of the periodic table have an extra electron above the previous filled complement of electrons. This electron peels off easily leading to negatively charged ion. These two configurations can be seen as sharp. When the electrons balance the protons the negative charge can be seen as sharp. Where the electron resided the charge is sharp and very negative. When the electron peels off the result will be a dull positive charge. The imbalance caused by more protons than electrons will be felt through the Maxwell-Gauss electric equation.

Negative atoms on the right side of the periodic table have too few electrons to make up a full complement of electrons for a shell. The extra positive charge that presents itself around the atom will tend to draw electrons from farther away. This positive charge can be seen as sharp. When the atom is successful in drawing in another electron the valence shell will be complete. This atom now has a dull negative charge which is spread around the outside of the electron and not quite balanced by the protons' charge in the nucleus of the atom.

Sharp charge vs dull charge can be important in understanding why chemical reactions happen.

Molecule Attraction in Masses

Masses have an attractive force that pulls large things down. Towards the center of mass and less often towards the periphery of mass electrons will tend to jump. This sharp negative charge will move quickly over a distance between molecules.

Charge balance will seek to reassert itself as the electron moves from the center of mass towards the periphery of mass. Electric dipoles in the larger mass will be dragged negative side down. This drags large mass down while small mass (electrons) often move in the opposite direction.

Electrons don't just move straight out to the periphery of mass. They may move at angles to perfect radial movement. With ten to the twenty molecules in a mass the averaging effect happens quickly.

There is sharp charge and dull charge at play in a gravitational force. The sharp negative charge comes from excess electrons on the inside of any sphere or mass. The dull and moving dipoles in otherwise balanced molecules provides the dull negative charge that makes its way inwards giving us a gravitational force.

Little things, electrons, are always moving up sharply while both big and little things are attracted towards the center of a mass in a process we call gravity. The attraction back towards the center of mass serves a charge balance function as well as replacing the mass that bolted sharply and quickly from within the mass.