Final answer:
To demonstrate the constancy of the instantaneous Poynting vector for a circularly polarized plane wave, one must understand that the electric and magnetic fields rotate in phase with constant magnitude in a lossless medium, leading to a Poynting vector that is independent of time and distance.
Step-by-step explanation:
To show that the instantaneous Poynting vector of a circularly polarized plane wave propagating in a lossless medium is a constant that is independent of time and distance, let's consider the wave's electric field (ℓ) and magnetic field (ℛ) vectors. For a circularly polarized wave, the fields rotate with the angular frequency (ω), maintaining a constant magnitude, hence the amplitude remains constant.
The Poynting vector (ℙ) is given by the cross product of the electric and magnetic fields: ℙ = ℓ × ℛ. Given that both ℓ and ℛ rotate in phase and have constant magnitudes in a lossless medium, their cross product, which represents the energy flux, will yield a constant vector as well. The constancy over time is due to the cyclical nature of the fields canceling out any time dependency after the cross product is calculated.
Since ℓ and ℛ fields for a circularly polarized wave do not change in magnitude with distance (assuming ideal conditions without any boundary interference or dissipative effects), the Poynting vector magnitude also does not depend on distance, only on the amplitudes of ℓ and ℛ, which are constants for a given wave.