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.