Validity of Maxwell's laws

Question

I know Lorentz's force changes, according to Wikipedia, when the charges are moving nearly light speed but what about Maxwell's laws? Are they still valid when particles move at light speed? I ask this because, according to Halliday et al., it is said that the Coulomb's law is valid to describe interactions between electrons and protons within an atom or molecular interactions but when particles move nearly light speed (e.g. within a particles accelerator), one must use Maxwell's laws. So the question is the following: can they change at light speed or not?

Answer 1

Maxwell's equations describe the electric and magnetic fields of a charge moving at any velocity, so long as you account for the fact that a moving charge acts like a current, and that changing electric & magnetic fields also can act like a source of magnetic and electric fields (through the $\partial \mathbf{B}/\partial t$ and $\partial \mathbf{E}/\partial t$ terms in Faraday's and Ampere's Laws, respectively.)

Actually solving these equations in general for a charge moving with arbitrary motion is usually reserved for upper-level undergraduate and/or graduate-level courses in electrodynamics. The "easiest" route is via what are called the Lienard-Wiechert potentials, which are written in terms of the scalar potential $\phi$ and the vector potential $\vec{A}$.

However, if you take relativity as a given, you can actually make relatively simple arguments about the electric and magnetic fields for a charge moving at constant velocity. I can't do justice to this argument in this space; if you want to see these arguments, I highly recommend Purcell & Morin's Electricity and Magnetism for a discussion at the introductory level.

The net result of these arguments, though, is that the electric field of a moving charge is not uniform in all directions (as predicted by Coulomb's Law), but is instead stronger in the directions perpendicular to the direction of motion. It is often said that the field lines of a moving charge are "compressed" along the direction of motion, since closer-spaced field lines correspond to a stronger electric field. In addition, there is a magnetic field that circulates around the line of travel of the charge, and this magnetic field is proportional to the electric field (i.e., stronger at points perpendicular to the direction of motion.)

As far as the Lorentz force goes, it turns out that it remains superficially the same: $d \mathbf{p}/dt = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})$, regardless of the magnitude of $\bf{v}$. One does, however, have to be careful of what one means by $\mathbf{p}$ here; it has to be defined as the relativistic momentum $\mathbf{p} = \gamma m \mathbf{v}$ for this equation to hold.

Answer 2

Maxwell's Equations in their 19th C form are Lorentz covariant, which means they are valid after being boosted into a new frame of any boost-able velocity when transformed by a Lorentz transformation. (Note: $v=c$ is not a boostable speed).

It does take some effort to show they are unchanged, unless you put them in their "manifestly covariant" form. So for instance, you make a 4-vector potential:

$$ A^{\mu} = (\phi/c, \vec A)$$

from the usual electric potential and magnetic vector potential. The electric and magnetic fields are contained in an antisymmetric 4-tensor:

$$ F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} $$

(It's a good exercise to work out the derivates to just how nicely electric and magnetism are related in relativity).

The current 4-vector comprises charge and 3-current via:

$$ J^{\mu} = (c\rho, \vec j)$$

With that, Maxwell's Equation (with sources) can be written in a manifestly covariant way:

$$\partial_{\mu}F^{\mu\nu} = \mu_0J^{\nu}$$

$$ \partial_{\mu}\epsilon^{\mu\nu\sigma\lambda}F_{\sigma\lambda} = 0$$

These equations are guaranteed to be valid after a Lorentz transformation.