The electromagnetic field equations are indeed invariant under Lorentz transformations, which means they have the same form in all inertial reference frames. However, the reason we sometimes use different equations to describe electromagnetic fields in different situations is because of the different symmetries and conditions that may be present.
Maxwell's equations, which form the foundation of classical electrodynamics, are covariant under Lorentz transformations. They describe the behavior of electromagnetic fields in the presence of charges and currents. These equations consist of Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of electromagnetic induction, and Ampere's law with Maxwell's addition.
However, in certain situations or under certain conditions, it is more convenient or necessary to use different equations that are derived from Maxwell's equations or have additional terms. Here are a few examples:
Electrostatics: In the absence of time-varying fields and currents, the electromagnetic fields are described by electrostatics. In this case, Gauss's law for electric fields and the absence of magnetic monopoles (Gauss's law for magnetic fields) are sufficient to describe the behavior of the fields. The equations simplify, and we no longer need to consider the time dependence and magnetic effects.
Magnetostatics: Similarly, in the absence of time-varying fields and charges, the electromagnetic fields are described by magnetostatics. Here, Ampere's law with Maxwell's addition and the absence of electric fields due to magnetic monopoles are sufficient. Again, the equations simplify, and we focus on the magnetic effects.
Wave equations: When studying the propagation of electromagnetic waves, such as light, it is common to use wave equations derived from Maxwell's equations. These wave equations explicitly describe the time-varying behavior of the fields as they propagate through space. They allow us to analyze properties like wave interference, diffraction, and polarization.
Boundary conditions: When electromagnetic fields encounter interfaces or boundaries between different media, additional equations known as boundary conditions come into play. These conditions ensure the continuity of the fields and their derivatives across the interface and allow us to determine the reflection, transmission, and scattering of electromagnetic waves.
In summary, while the fundamental equations of electromagnetism, Maxwell's equations, are invariant under Lorentz transformations, different equations or variations of them are used in specific contexts or under certain conditions. These equations capture the essential features and simplify the analysis of electromagnetic phenomena in those situations, taking into account specific symmetries and constraints.