Maxwell's Equations

Let E and B be the intensities of an electric and magnetic field, respectively. Maxwell's equations in free space in SI units are then

∇·E = ρ/ε₀
∇·B = 0
∇×E = −(∂B/∂t)
∇×B = μ₀(J + ε₀(∂E/∂t))

where ρ Is the electric charge density, J is electric current density, ε₀ is the permittivity of free space, t is time and μ₀ is the permeability of free space. The parameters ε₀ and ε₀ depend upon the dimensional units used to measure E, B, ρ and J.

The second and third equations are based upon the fact that there are no magnetic charges and hence no currents of them.

The great deduction of James Clerk Maxwell is that the speed of propagation of electromagnetic waves is 1/(μ ₀ε₀)^½. Since this was equal to the speed of light it meant that light was electromagnetic radiation.

Analogues for Nuclear Forces

It is easy enough to conjecture the existence of a corresponding force field for nuclear forces analogous to the magnetic field for the electric field. The crucial matter is finding any evidence of the existence of such a field, The exclusive spin pairing of nucleons may involve such a force.

It is also possible that the principles of special relativity may dictate that any force field has an analogue of magnetism. This would be because the motion of charges has an effect and motion is relative.

Electrodynamics and Special Relativity

Consider a planar loop of wire L encompassing an area S and situated in a magnetic field B(p). The flux Φ through the loop is defined as

Φ = ∫_SB·dS

Suppose the loop is being carried on a train whose tracks pass between the poles of a giant magnet. And suppose the train is traveling at a velocity v between the poles of the magnet. The passage of the electric fields of the free electrons in the metal of the wire loop causes a rate of change in the electric field which creates a change in the magnetic field within the loop. The net result is that there is a rate of change in the flux which generates an electromotive force (emf) E in the loop causing an electrical flow in the loop. The emf is given by

E = −(∂Φ/∂t)

From the view point of an observer on the ground the magnet held still and the loop moved past it. On the other hand, from the view point of an observer on the train the loop held still and the poles of the magnet barreled past it. The time variation in the magnetic field would generate a current in the loop. The emf for that current would be equal to

E = −(∂Φ/∂t)

Thus from the emf of the current in the loop there is no way to tell whether the magnet was still and the loop moved or the loop was still and the magnet moved. This is required by the principles of special relativity.

Magnetism as Required by Special Relativity

Consider two long, thin adjacent wires; one with a string of positive charges moving to the right with a velocity v and the other with a string of negative charges moving to the left at the same speed v. Let λ₀ be the positive charge density. For the positive charged wire its charge density is λ₀ but for the negative wire the positive charge density is −λ₀. The rate at which positive charge is being moved to the right in the positive wire is λ₀v, but in the negative wire it is −(−λ₀v). Therefore the current in the two-wire combination is

J = 2λ₀v

Now consider a point charge Q located a distance r from the wires and traveling to the right at a speed of u. Because the two-wire combination is electrically neutral there is no electrical force imposed upon the moving particle. This is however for a stationary observer.

Consider an observer moving along with the particle. The observed motion of the positive charges is then

v₊ = (v−u)α₊

where α₊ is, according to special relativity, equal to 1/(1−uv/c²).

On the other hand, the observed motion of the negative charges is

v₋ = (v+u)α₋

where α₋ is equal to 1/(1+uv/c²).

Note that v₋ is great than v₊ and thus the Lorentz contraction of the spacing between the charges is greater for the negative charges than it is for the positive charges. Thus the two-wire combination appears to carry a net negative charge to an observer moving with the particle and hence there would be an electrical force on the particle. But if there is a force on the particle for the observer moving with the particle then there must be a force on the particle for a stationary observer. That force cannot be electrical because the two-wire combination is electrically neutral. It must be another force. That force is called magnetic. Thus in order for the system to satisfy special relativity the two-wire system carrying a net current must generate a magnet force for the stationary observer and an electrical force for the moving observer. All of this is independent of the parameters v, u, r and λ₀.

The Extension of the Argument
to Nucleonic Charged Particles

For the above argument to apply there must be particles so the spacing between particles is subject to Lorentz contraction. There must be particles of opposing charges so a system of net charge zero can be constructed. Nothing else special is required.

The nucleons and their interaction force satisfy those requirements. If the nucleonic charge of a proton is taken as +1 then that of a neutron is −2/3. Thus for a stream of neutrons an adjacent stream of protons could be created moving in the opposite direction to the neutrons and having two-thirds the density of the neutron stream. The two-stream combination would be neutral with respect to nucleonic charge for a stationary observer. For an observer traveling along the direction of the streams would find that a nucleon is subject to a net force from the two-stream combination. Special relativity requires that a stationary nucleon outside of the streams must also experience a net force but that force cannot be a nucleonic force because the two-stream combination is nucleonic charge neutral. It must be a magnetism-like force associated with the nucleonic force.

Therefore there exists a set of equations for the nucleonic force and its associated magnetism-like force analogous to Maxwell's equations. The equation parameters howeverMaxwell's equations. may be radically different from those of Maxwell's equations.

Sources:

David Griffiths, Introduction to Electrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1981.

Edward M. Purcell, Electricity and Magnetism, 2nd ed., Harvard University Press, Cambridge, Massachusetts, 1985.