Maxwell's Equations

Are Maxwell's equations fundamentally flawed. What if we considered currents free of fields. That is to ask do vector fields really help us understand electromagnetism or do they cover the truth with nice looking math equations that don't explain the physics completely?

Let's start with the Maxwell-Gauss electrical equation. The divergence of the electric field is equal to the charge contained within a volume. This describes electric fields. What if you don't like vector calculus or you want to understand the Maxwell-Gauss concept free of field theory. In this case we can say charges accelerate towards or away from other charges. The field doesn't exist but it forms a convenient representation of the truth.

Second, we have the Maxwell-Gauss magnetism equation. This equation states that there is no divergence to a magnetic field. Magnetism lovers know that the magnetic field is always elliptical in nature. But what if you don't like field theory or magnetic fields. I mean what if magnetic fields don't exist? Magnetism could well be an interaction between a lattice and curling electrons. More on that topic later (and in previous posts). It is most important to examine this equation. If a magnetic field is just the curl of current then the Maxwell-Gauss equation becomes the common vector identity: the divergence of a curl always equals zero. The Maxwell-Gauss magnetism equation is just a vector calculus identity.

The Maxwell-Ampere equation follows the previous equation well. The curl of the magnetic field is equal to the sum of the current and the displacement current. Again, if we realize that the so-called magnetic field is a curl in the group electron movement then we see that the curl of the curl is equal to the current. Imagine a wire with electrons curling off of the wire. They eddy out around the surrounding molecules and atoms in the dielectric material. The curl puts you on the magnetic field lines and the curl of the curl puts you right back on the wire.

It is rarely described in the Maxwell-Ampere equation that the curl of the current or the displacement current produces a straight magnetic field. This is the theory behind an electromagnet but the Maxwell-Ampere equation doesn't describe it specifically.

Finally we consider the Maxwell-Faraday equation. The curl of an electric field is equal to the negative first derivative of the magnetic field. Change the magnetic field and the electric field must curl. A curling electric field gives rise to a changing magnetic field. This equation is really saying that when negative charge is accelerating the curl (derivative) of the current will change. This is a circular equation because it looks at the acceleration of curling charge in terms of the acceleration of curling charge. This equation looks at the dynamics of how the charge moves. Acceleration of charge is very important in electric motors. As some of my previous posts have noted, there are better ways of understanding motors than the Maxwell-Faraday equation.