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.