Maxwell's Equations

Are Maxwell's equations fundamentally flawed. What if we considered currents free of fields. That is to ask do vector fields really help us understand electromagnetism or do they cover the truth with nice looking math equations that don't explain the physics completely?

Let's start with the Maxwell-Gauss electrical equation. The divergence of the electric field is equal to the charge contained within a volume. This describes electric fields. What if you don't like vector calculus or you want to understand the Maxwell-Gauss concept free of field theory. In this case we can say charges accelerate towards or away from other charges. The field doesn't exist but it forms a convenient representation of the truth.

Second, we have the Maxwell-Gauss magnetism equation. This equation states that there is no divergence to a magnetic field. Magnetism lovers know that the magnetic field is always elliptical in nature. But what if you don't like field theory or magnetic fields. I mean what if magnetic fields don't exist? Magnetism could well be an interaction between a lattice and curling electrons. More on that topic later (and in previous posts). It is most important to examine this equation. If a magnetic field is just the curl of current then the Maxwell-Gauss equation becomes the common vector identity: the divergence of a curl always equals zero. The Maxwell-Gauss magnetism equation is just a vector calculus identity.

The Maxwell-Ampere equation follows the previous equation well. The curl of the magnetic field is equal to the sum of the current and the displacement current. Again, if we realize that the so-called magnetic field is a curl in the group electron movement then we see that the curl of the curl is equal to the current. Imagine a wire with electrons curling off of the wire. They eddy out around the surrounding molecules and atoms in the dielectric material. The curl puts you on the magnetic field lines and the curl of the curl puts you right back on the wire.

It is rarely described in the Maxwell-Ampere equation that the curl of the current or the displacement current produces a straight magnetic field. This is the theory behind an electromagnet but the Maxwell-Ampere equation doesn't describe it specifically.

Finally we consider the Maxwell-Faraday equation. The curl of an electric field is equal to the negative first derivative of the magnetic field. Change the magnetic field and the electric field must curl. A curling electric field gives rise to a changing magnetic field. This equation is really saying that when negative charge is accelerating the curl (derivative) of the current will change. This is a circular equation because it looks at the acceleration of curling charge in terms of the acceleration of curling charge. This equation looks at the dynamics of how the charge moves. Acceleration of charge is very important in electric motors. As some of my previous posts have noted, there are better ways of understanding motors than the Maxwell-Faraday equation.

Fundamental Question in Electromagnetism: Voltage

What is voltage? Text books give a non-answer; something to do with electromotive force. Fuck off. I mean please explain to me what's going on with voltage. What is it and what are it's properties in physics. The fact that we don't have an easy time answering this question leads me to believe we have rather poor understandings of inductance and capacitance too. But that has been the subject of other posts.

Voltage is always given as a potential difference. There is always a system with some amount of energy that will be compared to produce a potential difference. I'll repeat my question: what is this energy difference? What causes it? Let's start with the units. Volts are Joules per Coulomb. This means that six times ten to the power of eighteen electrons in two different places will have an energy difference of one Joule. But what causes this energy difference?

I know all these questions are annoying but it annoys me even more that we can't just pick quick answers off of Wikipedia. Shouldn't Stanford and the DoE's X-Ray lab be looking at this stuff? Let's describe what we do understand. Voltage has been related to tension and pressure in the past as analogies. Those analogies may or may not hold. I'm not as interested in analogies as I am in the physical truth.

Previous posts have shown electric generators to whip electrons onto a return wire sucking electrons from the hot wire creating what we call positive charge. There is now a deficit of electrons on the hot wire that is being back-filled by the load. Electrons move at about 1% the speed of light so they can back-fill fast but they tend not to. The drift velocity of an electric circuit is almost always radically lower than the root mean squared velocity of the electron. So an electron surplus or deficit may contribute to a negative or positive voltage respectively.

That may not be all that is going on. I'll leave out parasitics related to the telegrapher's equations. What about the root mean squared speed of the hot side of the circuit related to the root mean squared speed of the electrons on the return side of the circuit. These quantities are hard to measure. Electrons move quickly and I'm not aware of any laboratories working on characterizing them.

Finally, a contributor to voltage may be hot carriers. Very few, fast moving electrons that can devastate transistors and provide a visible spark show.

Scientists studying the upper atmosphere talk about density and temperature of charge to characterize particles. That helps us only a bit with the characteristics of low voltage and high voltage electric circuits here on Earth.

A deficit or surplus of electrons relative to the nuclei in wires may cause an electric voltage relative to other wires. The root mean squared speed of electrons as it differs from another point in a circuit may cause a voltage. The drift velocity may be a measure that correlates to voltage. Hot carriers may also be a cause of voltage.

Electrostatic Acceleration and Newton's Third Law

Gravity and electromagnetism have not always been linked. It does seem hard to imagine that with so many atoms, molecules and especially electrons that something must be working to keep things down. Specifically objects or mass is attracted to other mass. Often we only look at gravity in planetary terms. The Moon pulls the tides the Sun pulls on the Earth.

So what causes the attraction? It may be that the mysterious dance of electrons and the nuclei they surround can cause an attractive force. Imagine particles in towards the center of mass. The electrons are going to be in abundance and we know from electrostatics that they repel each other. Often enough this repulsion from the center of mass will be enough to accelerate the electron for a while. The acceleration can be seen all the way from the center of mass to the periphery of the mass in a sort of electron buoyancy.

Electrons will be accelerated from the center of mass and they will also be attracted back to maintain charge balance conditions. The attraction back to the center of mass may happen with a drift velocity as the electron bangs from atom to atom - molecule to molecule on the way back down.

I've tried to present this in a diagram shown below. At the center of the circular mass electrons, marked e, accelerate away from the center of mass with a centripetal force. The equal and opposite reaction acts on the molecules around the accelerating electron. The force, F, is dense and acts in a uniform manner. The force at the periphery of the mass turns the electron back towards the center of mass. In reality many particles will fly off into space. The force at the periphery pushes some matter away but this force is diffuse compared with the force towards the center of mass at any point.

The diagram shows X3 at the center of mass so that the number of electrons at the center and at the periphery balances out.

Coils as Electron Structures

The previous post didn't address coils as a part of electron structures. Coils are certainly important as they factor into generators, motors and inductance of magnetic fields. These fields add up as the coil turns we find the measured inductance increase.

In the beginning a long straight wire requires a magnetic field to be set up. The telegraphers' equation show a series inductance as one of the parasitic parameters impeding signal and power propagation along any transmission medium. As the post yesterday pointed out long thin structures tend to have more inductance.

The telegraphers' equations point to four parasitic parameters. Conductance and inductance factor into coils. Conductance tells us that without a doubt electrons leak out from conductors. Electrons are small and extremely fast so this should be no surprise. Electrons have a high propensity to spin around nuclei outside the wire in the dielectric. This forms the inductance signal and power engineers seek to avoid.

As the number of turns in an inductor increases so does the interaction and the additive curl of this spinning electron field. If the curl of the electron field is substituted for the magnetic field we have a clearer picture of what is really going on. These facts are complicated by topics from other posts. Notably, the Ampere and Faraday equations that Maxwell used complicate things with differing eddy current directions.

This happens due to Lenz's effect. More on that later.

Electron Structures

Mechanical structures of conductors with their surrounding dielectrics have a profound influence on electromagnetic signal and power propagation. This post explores some of these mechanical conductor structures and the influence on electron flow.

Microstrip capacitors occur when, relative to the wavelength of the propagating wave, the microstrip widens considerably. Often microwaves are manipulated using microstrip capacitors. Electrons will be trapped momentarily causing the capacitive effect. Electrons will slow down their drift velocity in the wider capacitive microstrip though the group velocity of the electromagnetic wave may not slow down at all.

Microstrip inductor is created to, relative to the wavelength of the propagating wave, the microstrip narrows considerably. Microwaves can be manipulated with microstrip inductors. Electrons will accelerate and move faster through the narrow space. Some of the electrons will jump off the conductor and eddy out. This phenomenon is known as inductance and is used as a linear circuit element in microwave circuits.

Brushes are an old electron structure. They have been used to build high voltages. Excited electrons will find themselves traveling down a conducting brush towards the end. Electrons have been observed to fire very well off the end of conductive brushes no doubt creating some amount of inductance along the way (Ampere-Maxwell's law). Plasma discharge is observed from the ends of brushes as large amounts of electrons fire off the ends of the bristles.

Spike or electron wick is used for electrostatic purposes. When a surface builds up too many excess electrons such as on the wing of an airplane or on the international space station it has to find a way to shed the excess charge. If the charge is not neutralized to the surroundings communication noise or arcing could be a dangerous result. Spikes or wicks can be a tuned electron structure that allows for some inductance or eddy electrons that keep the flow of electrons from the aircraft or the spacecraft. The spikes will act as an electron trap as the conductivity and inductance will propel electrons from the craft but it is far less likely that an electron will spontaneously jump back onto the spike.

Xe charge balance mechanism allows the fuselage of the international space station to reach charge balance with the surrounding orbital plasma if the station finds itself short on electron or positively charged. The station might accelerate Xe nuclei into the surrounding plasma. The electrons that were orbiting the Xe nuclei before they were accelerated will diffuse through the chassis of the station causing a relative match in charge balance with the surrounding plasma.

Resonant cavities are a topic I should attack some other time. These structures are used on microwave frequency radiation to provide bandpass linear effects. Someone has even proposed harvesting thrust from an electromagnetic resonant cavity. More on this some other time.

Circuits Within a Planet or Mass

The force field due to gravity is characterized by a function in i, j and k that has negative components. The gravitational constant, G, is multiplied by the mass of the two objects being attracted to each other. This is divided by the distance between the two masses squared. The result is a force field that points towards the center of mass.

This blog has been exploring what likely happens from an electromagnetic perspective to create the force field known as gravity. We know that electrons move so fast (ten to the sixth power m/s or faster) that the root mean squared speed of protons and neutrons looks like it is effectively zero. Electrons in a dense space will likely experience acceleration due to an effect like buoyancy. The forces on the electrons closer to the center of mass will be greater.

If electrons are propelled out from the center of mass by a force that diminishes with respect to distance (r) squared they will accelerate at a lower rate.These electrons can only continue to accelerate as long as they don't interact with an atom. That means on the way back to the center of mass the electrons might well jump through the shadows of the ions in the mass to balance charge at the center of mass or wherever the negative charge is needed. It is likely that the trip from the atmosphere to the center of mass is a velocity compared with an acceleration on the way from the center of mass outwards.

The acceleration shall be less and less as the electrons move away from the center of mass. A relative velocity (zero acceleration for most of the trip) is likely on the return. The gravity circuit electrons accelerating on the way out and pushing inwards on the way back.

Turning of an Electron Field

Electrons move so quickly they have to be looked at as a vector field or fluid that flows. This field turns and has many statistical properties. The reason it is important to understand the turning of the field of electrons is that turning of a field causes lift. That is to say that every action has an equal and opposite reaction. So every turning field has a centripetal force caused by an object or another field. The equal and opposite reaction is lift. This is known as the Newtonian explanation for lift.

This lift is useful in understanding electrons and their involvement in gravity. Relative to the speed of electrons protons and neutrons can be seen to be standing still. Under dense circumstances electrons in the presence of other electrons will acquire kinetic energy and head for areas of less electron density with some stochastic frequency. At some point the electron will turn around and may make the trip back towards the center of mass again due to statistical interactions with other particles.

Notice that electrons will turn at the center of mass and head outwards away from the center of mass. It is also require for charge balance that electrons be at least modeled as turning around and heading back towards the center of mass. It is important to note that these two turns happen in different places in a gravitational mass. One turn happens in a high density place and one in a lower density place. A model for this could be one electron headed out from the center of mass and two electrons headed back at half the velocity of the electron heading out.

If we look at the dense set of electrons turning at the center of mass we can draw the electrons' centripetal force vector pointed away from the center of mass. Slower particles are causing the 'modeled' 180 degree turn and thus a force is exerted on them with a downwards (towards the center of mass) vector. Every action has an equal and opposite reaction. The opposite is happening far away from the center of mass the electrons have a centripetal force vector pointed downwards. These vectors are less dense meaning that gravity will always prevail as the counter-force to the turning of electrons under the extreme density at the center of a mass.

High density low volume turning vs. low density high volume turning leading to a force imbalance we call gravity. Lift and buoyancy can be analogies that help us understand how matter works in large quantities.

Electrons and Gravity

Electrons at high density near the center of mass are going to acquire kinetic energy and head towards a lower density in much the same way as buoyancy works. The force of an abundance of electrons will seek to distribute the charge. The electrons headed away from the center of mass will seek charge balance equilibrium and will head back towards the center of mass. Electrons travel so quickly they may be modeled as a flow rather than an abundance of particles.

The electrons far away from the center of mass will seek charge balance and will bump down towards the center of mass at a lower group rate compared with the rate they moved out at. There will be more electrons moving slowly towards the center of mass pushing and pulling matter along with it. The turnaround point at the center of mass where a particle has no choice but to head outwards needs a net force acting from the center of mass outwards. For every action there is an equal and opposite reaction and matter will be pulled inwards towards the center of mass. As well, a sphere or most shapes mass that mass takes will see more electrons away from the center of mass converging inwards adding to an electron ‘traffic jam’ as these particles seek the charge balance condition.

Imagine a three lane road with cars moving in one direction and another road with one lane with cars moving in the opposite direction. A set number of cars move along these roads and no car enters or exits the system. The single lane will jam and move slowly while the three lane road will move quickly.

The road system in the previous paragraph is analogous to the path electrons take on our planet or about any mass. The three lane road is the fast path out from the center of mass. The single lane is like the more crowded path back towards the center of mass.  

Reverse Buoyancy Key to Understanding Gravity

Buoyancy is an interesting concept. Today I might suggest that it applies to more than just helium balloons and Styrofoam floating on water. There are a lot of concepts of density and many of them may be additive. That is to say the more atoms and molecules we have the more of this force we observe. We know that the buoyancy of very large objects such as cargo freighters is so solid it resembles the ground itself.

But what of electrons and ions? They have relative densities that correlate with the density of the matter itself at any point. The sheer number of particles can be impossible for the human mind to comprehend. Ten to the twenty particles is an amount of matter that is hard to think of on a particle by particle basis.

So if we have electrons that are bond but not always rigidly bound to an internal nucleus then what do we find as a part of a larger mass? The electrons will float up and even bring a small number of light ions with them. The heavy will sink - gravity - therefore.

When we consider a helium balloon rising we like to focus on the helium - less dense - rising. We don't focus on the fact that the Nitrogen in the air is falling. This is not gravity but it may make up a small part.

Electromagnetism fills in the gaps. There is a circuit like nature to a large mass such as our planet. Electrons rise quickly and fall back towards the center of mass in a step by step manner.