Relativistic Effects of Electricity---Is Magnetism ``Non-existent''?

One might think that in relativity, there is a factor \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} Therefore, relativistic effects only become prominent in high-speed situations, i.e., when v is relatively close to c. This is correct in general, but not entirely. This is because there is a quite obvious relativistic effect at speeds lower than 1 mm/s—that is the almost universally known “magnetism.”

As previously mentioned, the magnetic field can be explained as a relativistic effect of the electric field; therefore, all electromagnetic phenomena can be attributed to the electric field and relativity. In fact, this is correct, though textbooks do not always state it explicitly. Thus, it is not difficult to understand questions such as “why Maxwell’s equations of electromagnetism are consistent with relativity” and “why the behaviors of electric and magnetic fields are so similar.” It is because their investigation itself lies within the framework of relativity; both the magnetic field and the electric field are results of the same thing.

The following content is primarily based on “Relativity Visualized,” a small popular science book that focuses on providing an intuitive understanding of relativity. It can be described as a prose version of relativity (including general relativity) rather than a formula-heavy one (though it does contain some very accessible formulas).

Let us consider a straight wire with a constant current, and assume there is no resistance in the wire (this is a very ideal case, but having resistance would not significantly change the result). We know that the generation of current is actually the directional movement of electrons carrying a negative charge, while the positive ions in the wire do not move. Let the velocity of the electrons be v, where v is no greater than 1 mm/s, as shown in the figure below:

Imagine there is an electron outside the wire. If it is stationary, it will not be subjected to any force. However, if it moves in the same way as the electrons inside the wire (same speed, same direction), then knowledge of electromagnetism tells us that the electron will be attracted toward the wire, subjected to a force proportional to v^2. This force is what we call the effect of magnetism generated by electricity. Below, we use relativity to explain this effect.

Imagine again that we are “sitting” on an electron; then from our perspective, all electrons are stationary, while the positive ions and the wire itself move in the opposite direction at speed v. Due to relativistic effects, the length of the moving objects will contract, and thus the volume of the wire will change to \sqrt{1-\frac{v^2}{c^2}} times its original value.

Therefore, the positive charge density in the wire becomes \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \approx 1+\frac{v^2}{2c^2} times the original density.

However, the electron density remains unchanged. Therefore, the wire effectively has a net positive charge, the amount of which is proportional to \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1 \approx \frac{v^2}{2c^2}

The wire contains a positive charge proportional to \frac{v^2}{2c^2}, and there is a negative electron outside, so the two naturally attract each other. According to Coulomb’s law, the force between them is proportional to the product of their charges, and thus it is naturally proportional to \frac{v^2}{2c^2}, i.e., proportional to v^2. This explains the phenomenon of magnetism generated by electricity. In other words, even at such low speeds, there are significant relativistic effects.