Even Mode Ampere's Law Using Circulating Current

Maxwell's equations 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.

The Maxwell-Faraday equation states that the curl of the electric field is equal to the first derivative of the magnetic flux density with respect to time. 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 before. 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 opposite curl as the electrons ejected by the other conductor. These electrons spin against each other interacting in the middle of the two conductors. A gentle matter stream begins to develop. Th matter flowing out from between the two conductors forces the wires together. This is ampere's force described using Ampere's circulating currents.

The eddy current flows within the conductor and outside the conductor. Even though these electron flows are shown in the diagram to be in a few radial positions with one radius, they occur at all radial positions and with a statistically varying radius. The eddy currents circulate end to end throughout the conductor to some extent. The eddy currents circulate outside the conductor most vigorously near to the conductor and less so as one extends away from the conductor. As in the conductor these eddy currents occur at all radial positions and they circulate throughout the dielectric surrounding the conductor until they are too small to be measured.

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 cancels the 'magnetic' field from the right hand conductor. 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.