This realization happened when I was supposed to simulate - model and visualize in 3D - the simplest electromagnetic interaction there is, between electric and magnetic fields from only two electrons/positrons, including the tracing of their individual shape/intensity. Everyone referred me to Maxwell's equations, but - to cut the story short - after much trouble, search and experimenting I had to conclude Maxwell's equations are simply useless for this. They seemed far too specific in relation to experimental setups with wires and permanent magnets, and so they seemed to lack much of the information required to solve any such general case with individual free charges.

I found a solution via Coulomb's law, Biot-Savart law and Lorentz force, I also found I have complete information of these EM field potentials in those equations, including their field "density" or magnitude distribution and geometry, and as well as description of force vectors and not only field vectors, I found that these "older" equations incorporate far more general and complete information about electromagnetic fields and forces.

Well, that is my surprising conclusion anyway, so of course I'm looking for some explanation and hopefully someone here is very familiar with Maxwell's equations so can explain my misunderstandings and say something about their actual implications - what they mean, what they describe, how was it derived and how to apply them to practical situations.

TO BE MORE SPECIFIC:

1. Gauss's law: divE= p/e0

- When do we use this equation? How to get rid of the divergence operator so to solve for just E, and would that be a vector or scalar quantity? Divergence of E field according to Coulomb's law is zero, it has uniform magnitude gradient dropping off with inverse square law, does that not mean divE=0?

2. Gauss's law for magnetism: divB= 0

- According to Biot-Savart law which actually describes this magnetic field potential for point charges, *not wires*, this field is toroidal, its magnitude falls off with inverse square law in perpendicular plane to velocity vector (current direction), but it also falls with the angle according to vector cross product, so at the end it looks like doughnut (toroid/torus) and not like a "ball" of an electric field. This actually means that divergence of this particular magnetic field 'due to moving charge' (this is not intrinsic magnetic dipole moment), has non zero divergence and non zero rotation (curl).

Yes, if you take an infinite wire then divB=0, but that does not say anything about individual fields, it is very specific case that does not reveal anything about how individual magnetic fields look in front and behind that 90 degree plane, it is very crude approximation and hence lacks information. -- Let's say divB=0, then what is just B equal to? I do not see any information about B field here, so where and when do we ever use this equation?

3. Maxwell–Faraday equation: rotE= - dB/dt

- According to Coulomb's law E field has no rotation (curl), it is more of a "radial" kind of thing, so what in the world can this mean if we get rid of the curl operator and solve for just E? How can 'curl of E' tell us anything if 'curl of E' is always supposed to be constant and zero?

What does "dB" refer to?

a.) to second equation: rotE= - (divB= 0)/dt ?

b.) to fourth equation: rotE= - (rotB= J + dE/dt)/dt ?

c.) to Biot-Savart law: rotE= - (B= k qv x 1/r^2)/dt ?

d.) something else?

4. Ampère's circuital law: rotB= J + dE/dt

- What do we get when we get rid of the curl, how to do it, and what just B then equals to? What J equals to? What "dE" refers to, 1st equation, 3rd equation, Coulomb's law? Are these equations for just one field or do they require at least two like Coulomb's law has Q1 and Q2 and Newton's gravity has M1 and M2?

Does rotB, J and dE refer to the fields (potential difference) of one and the same particle, or rotB refers to one field and dE to separate another field? And also, 3rd and 4th equations appear to be kind of 'circular definition' and self-referencing, but hopefully answers to previous question will explain this.

In addition, what is the full and exact meaning of "changing electric field causes... B" or "varying magnetic field produces... E". How can E or B field vary if you look at only one charge (electron) or two? Can E potential of individual charges actually change and can there be a creation of any new magnetic or electric potential (new fields)?

Thank you.

**Edited by ambros, 19 March 2010 - 06:58 AM.**