In far-field electromagnetic wave propagation, the electric field (E) and the magnetic field (B) are in phase with each other. This phase relationship is a consequence of the wave equations and the fundamental properties of electromagnetic waves.
The wave equations for electromagnetic waves describe the behavior of electric and magnetic fields as they propagate through space. These equations show that changes in the electric field induce changes in the magnetic field, and vice versa. The fields are interconnected and mutually dependent on each other.
In far-field conditions, where the distance from the source of the wave is large compared to the wavelength, the wavefronts become approximately planar. This allows us to simplify the analysis of the wave propagation.
The speed of propagation for electromagnetic waves is determined by the ratio of the electric permittivity (ε) to the magnetic permeability (μ) of the medium in which they are traveling. In vacuum or free space, this speed is equal to the speed of light (c). Therefore, we can express the speed of propagation as:
c = 1 / sqrt(ε * μ)
Since the speed of propagation is constant, it means that changes in the electric field will occur at the same rate as changes in the magnetic field as the wave travels through space.
Considering this constant speed of propagation, when the electric field reaches its maximum or minimum value, the magnetic field will also reach its corresponding maximum or minimum value at the same location. This alignment of the peaks and troughs of the electric and magnetic fields leads to their being in phase with each other.
In summary, in far-field electromagnetic wave propagation, the electric field and the magnetic field are in phase due to the fundamental properties of electromagnetic waves, as described by the wave equations and the constant speed of propagation for these waves.