Final answer:
Deriving the expressions for transverse electric and magnetic fields, as well as the Poynting vector, involves using Maxwell's equations and understanding that these fields are both perpendicular to the direction of wave propagation and to each other in a transverse wave.
Step-by-step explanation:
The question is about deriving the expressions related to electromagnetic waves, particularly the electric field E(t) and magnetic field B(t) in a transverse wave, along with understanding the time dependence of these fields and the Poynting vector's magnitude. From Maxwell's equations, we infer that both the electric and magnetic fields are perpendicular to the direction of wave propagation forming a transverse wave. Moreover, the electric field relates to the voltage and separation distance between the plates as E = V/d, and the magnetic field inside the plates has to follow the symmetry argument, producing a constant magnitude along circular paths as ∫B · dı = B(2πr).
The time-varying Poynting vector S illustrates the flow of electromagnetic energy and is related to the electric and magnetic fields by S = E x B, implying that the energy flow is perpendicular to both fields. An expression for the rate at which electromagnetic field energy enters the region between the plates can be found by relating the voltage V(t) and its time derivative to the Poynting vector. The energy density of the electromagnetic fields is given by u(x, t) = ε0 E2 + (1/(2μ0)) B2, and both energy densities of the electric and magnetic fields contribute to the total energy flux.