The Maxwell-Heaviside Equations

This blog has been alive for two years and I wanted to reflect on the basic tenants of electromagnetics. Namely, the four Maxwell-Heaviside equations. These four equations and the Lorentz force equation backbone electromagnetics for all practicing electrical engineers.

Gauss' law of electricity is useful as it shows the direction that a  charge will travel in given the presence of an electric field. The electric field is a useful construct because its magnitude gives us insight into the behaviour of electric phenomena.

Gauss' law of magnetism is somewhat less useful. A magnetic field is the normal vector in the curl of an electron field. Gauss' law of magnetism points out that the divergence of a curl is zero. That is the magnetic field is electrons curling and the divergence of that curling field is zero. This fact is also a vector calculus identity.

The Maxwell-Ampere equation can be read two ways. The curl in a magnetic field gives a current and the curl of electrons gives a magnetic field which is the normal to the field of curling electrons. The curl of a curling field does add up to a current. This is backwards from the way we should be thinking about electricity. The current in a wire generates a magnetic field which surrounds the wire. The current from the wire spins off a type of leakage current which spins. The telegraphers equations dictate this type of behaviour.

The Maxwell-Faraday equation involves Lenz's law and the principle of electromagnetic induction. The equation, in differential form, states that the change in a magnetic field will be a curl in the electric field. Understanding what is really going on takes closer consideration. When a loop of current sees an increase in the curl of an electron field one has to consider the nature of the curl. The curl in the electron field is very tight as it was generated by a permanent or electromagnet. The magnetic domains or curl in the electron field of the coil is random.

Due to particle interactions the coil starts to see a tight curl in its electron fields. The law of conservation of angular momentum causes a Lenz' phenomenon curling in the opposite direction. This phenomenon is harnessed as current to drive a load in an electric generator.

When the Maxwell-Heaviside equations were developed the developers had jar batteries, wires and coils at their disposal. Reconciling their world with a modern electronic world takes understanding and patience.

On Capacitance

How capacitance works is poorly defined. Some texts will point out that a dielectric is polarized such that energy is stored to counter the prevailing electric field. But what are the electrons doing? How are these electrons moving? I have explored this topic previously and will revisit it again because capacitance is complicated and so many texts make it sound simple.

The energy in a capacitor is proportional to the voltage squared. Voltage is the excited energy of an electron. Energy is also proportional to the dielectric constant of the dielectric material. The surface area is also proportional to the energy stored but the surface area is not always equal on both plates of a capacitor but we will get to some of the subtleties of capacitance later. Lastly, the energy stored between two plates of a capacitor is inversely proportional to the distance between the two plates or poles.

So what are the electrons doing in a capacitor to store energy? First they move. Electrons move at a fraction of the speed of light. Estimates of how fast an electron move vary but electrons don't move at the speed of light. Electrons don't move at speeds a regular person could understand. Electrons move at a fraction of the speed of light that is to say a speed that is meaningless in kilometers per hour.

We know that electron take in a large number of electrons before they begin to excite at the voltage levels of the conductor charging the capacitor. Electrons flow in and the voltage or excitement of the electrons in the conductor don't immediately rise. The electrons flow through the dielectric to the return and are immediately back-filled by electrons in the dielectric. As more energized or higher voltage electrons enter the dielectric the dielectric becomes more energetic. The electrons that are back-filling the incoming electrons have a higher and higher voltage until their voltage matches the incoming electrons. The capacitor is fully charged.

When a capacitor is fully charged there is an excitement at both plates. The capacitor has a lot of statistical properties that may well have to do with the exponential distribution or the Poisson distribution. Electrons will move into the dielectric with a high relative energy and they will keep moving towards the return. Eventually the electron will return towards the energized plate. It is the continuous dance between the energetic plate and the return that constitutes capacitance. Electrons moving quickly towards the opposing plate only to be back-filled by electrons seeming to polarize the dielectric.

The statistics of electrons in a capacitor has yet to be fully understood. Understanding that things are not fully understood is the first step in understanding the capacitor and eventually the diode and transistor.