Unification of Two Phenomena Well Known

Unification is a key word that invokes some associations: The Grand Unified Theory and Einstein’s unsuccessful quest for it, of course the detection of the Higgs boson and the confirmation of the validity of the Standard Model of Particle Physics, or Kepler’s Harmonices Mundi.

Unification might be driven by the search for elegance and simplicity in the universe. Nevertheless, in retrospect it might be presented as down-to-earth and straight-forward.

Electricity and magnetism have been considered distinct phenomena until they have been “unified” by describing them by Maxwell’s equation that are consistent with the theory of relativity. What does this mean?

I am summarizing the explanation given by Richard Feynman in chapter 13 of volume II of his Physics Lectures:

Consider a wire carrying an electric current and a small test charge near the wire. The test charge moves with constant velocity and follows a path parallel to the wire. The wire comprises positive ions and free electrons, and it is electrically neutral, so it exerts no electrical force on the test charge. However, the motion of the electrons gives rise to a magnetic field. Since the test charge is moving, the magnetic field gives rise to a force (the Lorentz force) that makes the charge move in a direction perpendicular to the wire (The sign depends on the type of charge. A negative charge would be attracted to the wire if it travels in the same direction as the electrons in the wire).

Now imagine you would watch this experiment from the perspective of an observer who moves with a velocity equal to the velocity of the test charge. The charge is now at rest. If the test charge had moved with exactly the same speed as the electrons before (this is an assumptions made for the sake of simplicity), from the travelling observer’s perspective the electrons in the wire would be at rest and the positive ions would be moving. So since some carriers of charge do still move, a magnetic field would also exist in that frame of reference. However, the field would not exert a magnetic force on the test charge that is now standing still.

If the charge would move towards the wire and eventually hit it in one frame of reference, the same effect needs to be observed in the other. What kind of force would be accountable for that in the second frame of reference?

It is the electrical force, and it is due to the fact that the wire is electrically charged in the second system. Electrical charges of particles do not change with switching to different inertial frame, but dimensions parallel to the relative velocity do. And thus does the charge density – the charge per unit volume or per unit length of the wire. If a charge density ρ  is measured by an observer at rest, the observer in motion relative to the charges measures a larger charge density because the volume has shrunk by a factor of √(1 – v²/c²) (This is the infamous factor appearing in all kinds of equations in relativity, c being the speed of light in vacuum). If charge density changes, there is a net overall charge per unit volume.

Why do the swap of the roles of positive and negative charges not compensate for that? The travelling electrons turned to static charges and the static ions turned to moving positive charges. Remember that the wire is electrically neutral in the system considered first. Thus in this system the charge density of electrons is larger than their density measured in the travelling system. Switching to the latter system, the correction factors are applied to each type of charge in a different way – starting from the densities measured in the system “at rest”: The ion density is increased as these are moving now, but the electron charge density is reduced, as we have measured the increased density in the other system.

Actually, the forces turn out be different by a factor equal to the square root mentioned above, the force is smaller for system 2. But this is needed for consistency: The effect of the force is measured by its impact – its momentum. In special relativity the momentum is often illustrated by the penetration depth of a bullet (driven into some material, in a direction perpendicular to the relative velocity). Momentum is force times the interval of time the force is acting on a particle. But time is dilated according to special relativity, that is: time intervals appear longer if the particle is moving (system 1). Thus by calculating the product of force and time interval, the factors cancel out exactly.

In summary, the forces of electricity and magnetism morph into each other – dependent on the frame of reference chosen. They are two aspects of some underlying “unified” force. On the one hand, this changed the way we think about the electromagnetism.

On the other hand – technically it just means that we take the components of electrical and magnetic fields (3 numbers each – these are vectors) and stuff them into a more general mathematical structure consisting of 6 numbers. This is called a tensor. This sounds simple and there is a reason for that: Historically, Maxwell’s equation that govern the spatial and temporal evolution of electrical and magnetic fields have been laid down before Einstein developed the theory of special relativity. Maxwell’s equations had already been consistent with special relativity and they did not need amendment – as Newton’s law. So unification did exist already – mathematically, but the consequences had not been fully understood.

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