The Maxwell-Faraday Equation and the Proximity Effect

Maxwell's equations are woven together like a carbon-boron nano-tube ribbon for a space elevator. They describe electromagnetic fields as far as an electrical engineering student would understand them. Gauss, Ampere, Faraday and Heaviside are all said to be linked to these equations through a history of development of electricity, magnetism and the vector calculus behind the equations.

Of course the concept of 'magnetism' may just be a convenient way of looking at things. Ampere, Maxwell, Faraday and Heaviside were working things out before Bohr laid out his view on electrons, protons and neutrons. In this post, I would like to focus on hot carriers or the acceleration of electrons and how that affects the lattice they travel through.

The Maxwell-Faraday equation states that the curl of the electric field is equal to the first derivative of the magnetic flux density. Deconstructing this relationship shows us that the spacial change in the electric field will affect the acceleration of electrons. The electrons may jerk in a curled field, they may accelerate or they may curl with a group drift velocity. The other side of the equation points to a 'magnetic' field that changes with time. The changing magnetic field is an important component for the rest of this post.

If a 'magnetic' field really points to a drift or acceleration of electrons in a curling motion then we have to relate that to the equation at hand. The Maxwell-Faraday equation indicates, at a very minimum, an acceleration of electrons in a circular motion. The electrons may, in fact, jerk in a circular direction. This jerk and acceleration is an important part of the ejection of electrons so to influence the flux of the electron flow outside the conductor. The parameters of the telegrapher's equations describe all of these electron-dielectric interactions and they have been discussed on previous posts.

I've posted about the proximity effect and the skin effect previously. The proximity effect describes the interaction between one conductor and another when electrons are accelerating in a circular way. The skin effect deals with the interaction of similarly accelerating eddy currents only the skin effect is one conductor's eddy currents acting on itself. The eddy currents can be looked at as a result of the parameter's described by the telegrapher's equations. Alternatively, we can use the fictitious 'magnetic' field to explore the interactions between the two conductors.

First, let's look at the proximity effect using eddy currents alone. Electrons eddy out behind the ions of the atoms they are passing causing a curl. The fastest electrons are moving through the lattice near the surface of the conductor where they can 'leap frog' for a faster speed and less impeded acceleration. Electrons eddy out of the conductor causing a flux in the flow of electrons surrounding the conductor. In this case, the electrons curl.

The diagram below shows the proximity effect with curl in the electron flow in addition to the drift of electrons against the traditional current.

There may be more to the diagram above than meets the eye. The field of curling electrons has the same curl as the electrons ejected by the opposing conductor. These electrons spin together in a generally self-reinforcing way allowing tight curls. The curl keeps the angular momentum of the electrons from quickly leaving the space between the two conductors. This, in turn, draws in more positive charge in the form of atoms or ions. This extra matter forces the wires apart. Furthermore the influence of the curl makes the inside of the conductors a bit of a positive feedback loop meaning that most of the current will travel along the inside of the conductor.

Now lets describe the same phenomenon, the proximity effect, using the fictitious 'magnetic' field. The diagram below illustrates the same electromagnetic effect as the diagram above only with 'magnetic' field lines drawn in in the traditional fashion.

Using this model we see the current in the left hand conductor producing a 'magnetic' field that overlaps the right hand conductor. Using the right-hand rule and Maxwell's equations the 'magnetic' field induces eddy currents in the opposite conductor. These eddy currents oppose the flow of current near the outside of the conductor and reinforce the current on the inside of the conductor.

The so called 'magnetic' field from each of the two conductors adds up between the two conductors. The right hand rule the magnetic field originating from the conductors with opposing current the field lines point the same way between the conductors. Sorting out the directions of the 'magnetic' field can be done using Maxwell-Ampere's 'law'.

Either model amounts to the same effect. The concentration of current due to the proximity effect can be a real menace at high frequencies.


Material Referenced from: Johnson and Graham, High Speed Signal Propagation, 2003.

Electron Structures: How Does the ISS Ground Itself?

The International Space Station is floating through space so the thoughtful people in United States of America thought it just had to be up to code. That means that the ISS has to be properly grounded and can't be 'floating' with respect to surrounding plasma.

On Earth the resistivity of soil may be 7662 ohm -cm, in places, according to the United States' Department of Defence MIL-HDBK-419A. In orbit around the Earth the ion -  electron plasma makes the concept of grounding far different. The fine folks at NASA will seek to minimize hazards by making the space station's net charge the same as the surrounding plasma in space. If the space station has too large a potential difference between the chassis of the space station and the surrounding plasma an arc hazard develops. Nobody wants lightning hitting the ISS. Talk about hot carrier injection!

So, it has come to my attention that the USA and perhaps another nation or two has developed two ways of controlling the potential difference between the chassis of the ISS and the surrounding plasma. The first mechanical-electrical structure to control charge is a spike. The other is a noble 'gas' ion gun. I use the word gas carefully because there is nothing about this element that is gaseous as it is used on the ISS after it has been fired.

The spike has been considered since the brushes of the Van der Graaf generator were invented. Photocopiers have bristles on them to control charge. These bristles or spikes sweep or conduct and launch or conduct electrons in a very particular way. I want to focus on the spike on this post rather than the brushes though they have similarities. Mechanically, the spike looks like the beginning of a conductor. It seems to just terminate in space. Electrons surely travel the length of the conductor at a high rate of speed and then just launch themselves off the proverbial plank.They might accelerate into the dielectric after a relatively consistent velocity in the lattice of the conducting spike. Off the electron flies but what is the probability that it will just fly back to the conductor that launched it? The answer is very low. So the spike sheds negative charge making the chassis more positive providing half of the grounding for an orbiting structure.

Now I'm just making educated estimates at how all of this works but we have a century of research and experimentation to draw on postulate.

The ion cannon is fun speculation. When the crew or sensors of our International Space Station feel they are in electron deficit they fire an ion cannon (particle accelerator) of Xe ions into space. The trick is that moments before the Xe+ are accelerated they had a full complement of electrons. The electrons spread through the mechanical structure of the conductive apparatus into the hull of the space station chassis with a net addition of negative charge to the ISS superstructure.

My post today involves much speculation. The speculation into the grounding of such a complicated electrical system helps resolve problems on Earth such as the grounding of a regenerative braking subway train. Of course there are far more people that ride on subways when compared to a space station so far away in orbit.

The Skin Effect

The proximity effect is the effect of eddy currents from one conductor’s changing current on the current flow distribution of another conductor. The skin effect, then, is an issue for alternating current on the self-same conductor.

The skin effect on AC transmission lines involves all of the telegrapher’s components. Most specifically though the electron eddy currents associated with, L, will be found especially at the boundary between the conductor and the dielectric. Fast moving electrons will eddy into the dielectric starting in the direction of current movement and then curling. Likewise, the current will eddy into the conductor. These eddy currents will add up from all around the conductors to oppose electron flow at the middle of the conductor. This means less current at the center of a conductor and more at the skin of the conductor.

As a surplus of electrons with a surplus of energy over the lattice is exposed to the conductor the charges will begin conduction as long as there is an energy gradient. The electrons will have the easiest time accelerating near the cladding of the conductor as they leap-frog their way to an electrical load or away from this load. Some electrons will escape the cladding boundary into the dielectric creating easier acceleration for subsequent electrons in the flow at the cladding.

The leap-frog effect above may be a capacitive phenomenon. Capacitance fundamentally refers to the capacity of the dielectric to store electrons. The accelerating particles accelerate down the boundary between the dielectric and the conductor. The conductor allows electrons to move quickly but if an electron jumps out of the lattice, into the dielectric and then back into the lattice, this class of electron moves the most quickly out of any governed by the parameters of the telegrapher’s equations.

The acceleration and faster relative velocities near the conductor-dielectric boundary can be seen as a type of efficiency. The electrons of an AC current are eddying in both directions and the non-eddy electron shoots down the middle efficiently. As well, with AC, there is a change of direction necessary and this change in direction is facilitated by the capacitance specified by the telegraphers equations. In reality that capacitance represents the charge that jumps off the conductor, into the dielectric, and then they make their way back into that self-same conductor.

The velocity or inertia of electrons eddying out around a given nucleus can knock other electrons out of their orbitals and into the curl of the flux of the electron flow. Again with more than 10²⁰ electrons traveling in a very small space the electron density means this may be fairly common. It will be less common for an electron to collide with a nucleus. 

Proximity Effect without the Old Idea of Magnetism

The proximity effect is the result of eddy currents going to work on a nearby conductor while the skin effect involves the eddy currents going to work on the conductor that is producing them. Both the proximity effect and the skin effect involve accelerated charge and the idea of drift eddy currents or a drift velocity in the curl of the electron field.

If two parallel wires carry a changing current in a opposite directions electrons will be ejected from each wire and some of them will curl in the dielectric between the two conductors.The curl of the two electron flows are in the same direction. The fact that the eddy currents circulate in the same direction means exponentially more energy can be stored in the eddy currents. This will cause efficiency that enables more electrons to congregate in less space. The curl is maximized as there is a lot of spin. The efficient packing of electrons will mean a surplus of negative charge which is neutralized quickly by positive ions. The extra matter in between the two wires forces the wires apart in what is known as Ampere's force.

The curl has increased between the two conductors with opposing currents. It is a trivial statement to say that the curl is circular but that circulating eddy currents means that more current is going to travel in the side of the conductor nearest the other conductor.This curl current induces stored energy in the form of a curl in the current or energy in a rotating mass. As the electrons crash back into the conductor they will cause this proximate current distribution.This is traditionally known as the fictitious magnetic field.

I'll have to go into more detail about why a changing current causes the proximity effect as opposed to a direct current.

It is interesting to consider the opposite condition where two conductors carry current in the same direction. In this case the current flow that curls between the conductors curls in the opposite direction. The fictitious 'magnetic fields' would be said to be cancelled in the middle. The current flow in the center between the two parallel conductors will be additive causing a current that in opposite in direction to the original current flow. This current will have eddy currents disrupting the curls around the middle of the two conductors. A bit of noise one might say.

Heaviside's Force Law

I give Heaviside more credit than others. He may have published Lorentz's force law first and he had so much to do with the telegraphers equations to say nothing of his influence on the way we right Maxwell's equations today with vector calculus.

The law agrees with Maxwell's equations and I thought I'd work through some educated speculation and some facts.

First, if a positive charge is located a positive distance along the x-axis from a charge carrying wire on the z-axis with the current flowing out of the page. The current flowing in the wire will generate a 'magnetic field' circulating in a counter-clockwise direction around the z-axis.

The electrons flow counter to the current. There are more than ten to the eighteenth power electrons per Coulomb so a more manageable 1 mA current will still have a massive more than ten to the fifteenth power electrons flowing per second. Try for a second to grasp how large that number is. That's a lot of electrons traversing what might be a small wire.

The telegrapher's equations tell us that there is a certain small G, C and especially L that sap some of the ten to the power of fifteen electrons per second from our current. I want to focus on the L which must escape the wire (nothing is really holding it in but a hot conductor). The escaped electron will generally be traveling in the same direction as the electron flow. This negative charge carrier will be a hot carrier injected into the flow causing flux. The electron will eddy out behind one or more positive nuclei causing curl. The cross product of this vector curl is said to be the fictitious magnetic field.

Heaviside's force law will take the cross product of the positive ion traveling out of the page along the z-axis with the 'magnetic field' which is pointed up at that point. The positive ion will move towards the conductor which agrees with three of Maxwell's equations. The 'magnetic field' isn't changing so one of Maxwell's equations is trivial.

Back to the telegrapher's view of electric matter. The flow around the conductor is in flux due to the hot carriers flying off the wire. These hot carriers are almost all electrons and they curl. The hot carriers congregating at the boundary of the new medium may cause the positive carriers to move in to balance the charge.

Are There Issues with Electromagnetism?

Did the fine compass needle lead us astray? Do magnetic field lines really exist? Are they just a construct humanity invented to explain the compass needle?

How about electric field lines? Perhaps, by definition, these lines just describe the acceleration of charge?

Maybe humanity has the cart before the horse.

To answer the questions above we'll find our best clues in Maxwell's Equations and Heaviside's Force equation. Heaviside's telegraph equations offer some real hints as well.

What about mass that congregates at the equators of so many planets?

Magneto-Attraction

The north pole of a magnet will attract a south pole as this blog has pointed out. Like poles repel through an opposite type of magneto-reaction.

The spinning of the electrons in the North end of the magnet will create a flux in the electron flow in the surrounding fluid. This flux is a curl. This curl will be nested and will permeate a distance from the North pole of the magnet. The South end of a second magnet will cause an upside-down counter spin or negative curl. It should be noted that these are electron spins that will now add up. There is now, at a certain point, enough spin to cause some negative electron induced pressure. This is not conventional pressure because it is really only the light charge carriers that are causing this induction. The two magnets, North and South, attempt to create a vacuum and in so doing create magneto-attraction.

Certain conductors is a different thing. Now suppose we present the North pole of a magnet to a fluid, air, and then we but a small conductor. The magnet will set the electrons in the fluid spinning. The curl of the fluid will induce the same curl in the conductor. This flux in the flow of the air will eventually be of a magnitude that will cause a vacuum-like evacuation of the air forcing the magnet and the conductor together.

It could be that a magnetic movement of electrons in a curl can be seen as a sort of an extension of a chemical bond between atoms. This is true and it isn't a chemical bond is very tight and a magnetic field represents a great number of electrons in a large, curled field between the North and the South pole of a magnetic dipole.

The Proximity Effect

The proximity effect can be explained conventionally by magnetic fields inducing eddy currents in a conductor causing the wires to carry their currents either on the inside of the conductor or on the outside of the conductor. The wires will either repel each other or attract each other.

Alternatively we can look at the dielectric field flux. The escaping electrons from the conducting conductor will eddy out quickly behind surrounding molecules in the dielectric material. Other electrons will continue on eddying out eventually to spin with a much larger radius. If this eddy effect reaches the return conductor, and it will, it will cause an additive current effect on the inside of the conductor. We see the proximity effect due to eddy currents in the dielectric.

At the same time we note the tight circular spinning and the large circular spinning in the dielectric. This may be measured as a decrease in the electric field between the conductors. Really the electrons are efficiently spinning, briefly, without associated positive charge. Very quickly, and some might say instantaneously positive ions rush in to balance the charge. The positive ions, too, will spin in a spatially efficient manner. The effect is more matter spinning between the wires forcing them apart.

When we get talking about inductance and spinning electrons or spinning molecules some people love to get hung up on vortexes. While I am certain vortexes will be shown to play a huge part in inductance and the proximity effect it is most important to focus on the efficiencies of matter organizing itself in a spinning manner. Think about the bottom of a tornado.

Inductance and the Telegrapher's Equations

What is inductance? A circular definition asks us to believe it is the induced magnetic field by a current or a displacement current. But what is it really?

Inductance is the eddy currents experienced when electrons are rushed down a wire as described by Gausses equations described by Maxwell and then Heaviside. The inductance is formed outside a conductor in a dielectric. Eddies within eddies of current rotating around atoms, molecules and groups thereof. This eddy phenomenon that happens around nuclei in a sort of a telegrapher's drift storing energy in angular momentum of electrons and then atomic nuclei themselves. This energy storage ability of a dielectric medium is described by magnetic permeability and electric permeability.

It is interesting to think about how many electrons flow through a wire. One Coulomb is in the order of ten to the eighteen electrons. That is a huge number of charge carriers. The telegrapher's equations point out that some charge bleeds off in the conductance parameter. Mainly the inductance parameter and also the capacitance parameter are important. As the charge heads down the wire certain 'hot carriers' will bleed off into the dielectric and eddy out around a dielectric nucleus. This spin gives the atoms that extra bit of angular momentum which is also known as energy stored in a magnetic field or inductance. It may once have been known as field flux where the field is contained in the dielectric.

Neat to think about how nested these electron eddies can be. More turns in an inductor or electromagnet lead to more feverish eddies or spin. More angular momentum leads to more energy stored in the magnetic field and more inductance.

After a current has been removed from a wire the angular momentum of the electron eddies unwinds. The magnetic field is said to collapse and the spinning electrons have a propensity to collapse into the wire keeping the current going. This is just as the inductor in the telegrapher's equations would have predicted.

There is a reason for those wonky inductance calculations.

Gauss' Laws in Tight

So as any two atoms or molecules come together there has to be a slight attraction. We see this hot side attraction will come about from a divergence. The electrons are so much less massive than the nucleus that they scatter quickly. Some electrons from one nucleus will end up overloading a second nucleus. The result of this electron-proton-proton-electron oscillation is a brief and slight attraction instead of repulsion. The vast majority of the time this attraction does not result in a chemical bond but rather some gravity.

Next it is important to see how these slight 'hot side' attractions build in a structure to form larger objects such as planets or stars. Gauss' laws add up and there is always a propensity for the hot side to reside in closer to the middle of a mass. There is a more dense electron cloud as we approach the center of a sphere of mass.

Relationship Between Chemical Bonds and Gravity

So we have an idea whereby when large numbers of atoms and molecules are in close proximity they tend to stick together. There are a lot of molecules. There may be upwards of ten to the twenty-fifth power molecules in a meter squared. Humans just can't comprehend that type of magnitude.

Gauss' law of electric fields will force electrons in a repulsive manner. These electrons will clear a way towards the nucleus which will attract the electron from the adjacent molecule or atom.

It is interesting to compare a strong Gaussian attraction to a weak one. Notably, a strong diatomic atomic - chemical attraction and a gravitational attraction.

A diatomic molecule has an almost full valence shell. Two atoms together form what looks more like a stable double shell together. Statistically these atoms stay together. We can imagine that the atoms come together the electrons scatter before the nucleus feels any of the push or pull of Maxwell's equations. Because the electrons move so quickly the attractive force of the nucleus is exposed. Electrons from the adjacent atom will be attracted beginning the diatomic attraction. This attraction settles out in a molecule because of the inherent stability of the double shell.

Let's look at the weakest case we can find to see if it fits with our model for the weak force of gravity, helium. We can imagine that the helium atoms come together the electrons scatter before the nucleus feels any of the push or pull of Maxwell's equations. Because the electrons move so quickly the attractive force of the nucleus is exposed. Electrons from the adjacent atom will be briefly attracted before being pushed away. This force is brief and it is statistically valid for only small fractions of time. Larger masses see more statistical attraction building towards the gravitational attraction of large masses.

Small Gravity Large Gravity

It has long been postulated that gravity holds for small masses as it does for large masses.

If two small helium atoms met in the vacuum of space their outer shell would repel exposing their inner nucleus which would also repel. What wouldn't repel is the whole structure which would have a net attraction.

Now if three small atoms met in a vacuum their outer shells would experience a definite repulsion. Electrons which travel at a fraction of the speed of light would quickly move out of the way exposing a line on the nucleus. This balance may precipitate a momentary exchange of electrons even if a proper chemical bond is not formed.

When many atoms and larger molecules get together in a bunch the middle molecules will have a net crowd of electrons which will experience a natural repulsion. The electrons will move towards the outside of the total mass but the nuclei will pull back. This total gravitational attraction is hard to ignore.

Electrostatics and Gravity

There are definite tendencies that exert themselves on a mass that is clumped in a bundle of molecules or atoms. We find gravity which may well be electrostatic. There will be an outward pressure pushing lighter negatively charged electrons out and heavier ions in. The pressure is due to an ever so slight abundance of negative charge which finds itself on the inside of the bundle of molecules or atoms. Some lighter electrons will escape and be drawn outwards at an accelerated pace. This electron will tend outwards until it is accelerated inwards by the imbalanced, remaining, positive charge. The electron will be accelerated inwards until it collides with the heavier ions. Eventually if there is no collision the electron will be drawn out again at an accelerated pace restarting the cycle.

The charge imbalances are so slight and the number of electrons and ions are so great gravity ends up being quite fluid. A type of reverse buoyancy.

Imagine an ion at each face of a cube. Pretending that electrons don’t repel each other then, statistically, we would expect there to be a higher electron density inside the cube when compared with the outside of the cube. Because electrons do repel each other we would expect an accelerating divergence from inside the cube.

At a certain point the electron will be attracted back towards the cube. The inward-most electrons will tend to scatter while the outward-most electrons will exert a pressure towards the center of a large mass.

It is important to remember that the electrons are far lighter and faster than the ions. These little things will go to work around the main mass causing movement. The electron will dictate gravity through an outwards and inwards oscillation described above.

Flux and Electromagnetism

The concept of flux in Elecromagnetics may be more than 150 years old. What does it mean really. The use of electric flux and magnetic flux is pervasive. Has the meaning changed over the last century. It could be that the founders of electrostatics were thinking a lot more about the particulars of what was going on then we do today. Maybe today we take certain sets of equations for granted and plow through formulas with multi-processor computers using method of moments to gather a solved 3D field.

When the idea of vector field flux was applied to electromagnetics chemistry was still nascent. I'm not certain our fore-bearers had a solid concept of the lattice structure of many metals or a statistical model of the electron voltage vs the positron voltage. Maybe these people were doing what they had to to get published. Regardless the concept of flux remains with respect to electric fields and magnetic fields.

Electric fields - magnetic fields? But fields of what. Well we know there are an abundance of elementary particles out there. Electrons just have to dance in the right way and we will see field like effects because there are so many of them and they move so fast.

The fast movement of electrons in a conductor seem to have effects that stretch far from the conductor itself. The dielectric region beyond the conductor exhibits or contains the flux as it changes in the presence of a surplus or deficit of electrons.

In the case of an antenna this alternating surplus or deficit of electrons that propagates kilometers or mega-meters from the antennas location. There are trillions of electrons located in small bits of atmosphere. It's a hard to conceive of just how may electrons surround an antenna. The chemistry of antenna's could be said to be a statistical marvel.

Electrons at boundaries behave in particular manner. Electrons deep inside a conductor behave in another way all together. Perhaps that is a topic best described in another post.

Magnetics and Vacuums

This topic may seem both trivial or deep depending on how one looks at it.

So earlier I posted that magnetism might just be a result of the vector field flux associated with fast moving particles interacting, every once in a while, with particles that are located far from the point of origin of the electric or electron source of the magnetism according to Maxwell's equations. It is hard to conceive of how much faster electrons move when compared with neutrons and protons.

If in a vacuum a solid magnet exists it breaks the vacuum by definition. That fact aside, lets consider the speed of the electrons in the magnet. They are going at a solid fraction of the speed of light depending on temperature. The solid magnet must be leaking electrons like crazy. Consider the magnetic field lines. A spiral along the magnetic field lines is where you will find a trail of electrons in this near vacuum.

A North pole will then attract a South pole, in a vacuum, in a slightly different way then it would in an environment where the air is thick and the air must 'flux'.