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.