Final answer:
The student's question deals with the characteristics of an electromagnetic wave based on a given magnetic field and calculating the Poynting vector and average power. It involves using Maxwell's equations, the relationship between the speed of light and the fields, and algebraic manipulation to derive necessary information.
Step-by-step explanation:
The student is dealing with an electromagnetic wave problem where they are given a magnetic field equation and must find various characteristics of the wave including the wavelength (λ), propagation constant (β), and frequency (f), as well as derive the electric field (E) and calculate the Poynting vector (P) and the average power.
Finding β, λ, and f
The magnetic field equation H(z, t) = 20 sin(π x 108 t + βz) A/m suggests that the angular frequency ω is π x 108 radians per second. The frequency f can be found by dividing ω by 2π, leading to f = 50 MHz. If c is the speed of light in free space, λ can be found using f = c/λ.
Finding the Electric Field E(z, t)
Using Maxwell's equations, one can find the electric field E associated with the magnetic field H. Since the speed of light c is considered to be 3 x 108 m/s, E can be derived from H using the relationship c = E/H.
Calculating the Poynting Vector P
Once E and H are known, the Poynting vector P, which is the cross product of E and H, represents the power per unit area. It points in the direction of the wave propagation and varies with both space z and time t.
Finding the Average Power
To find the average power transmitted by the electromagnetic wave, one would square the absolute value of the Poynting vector and average it over a period. This is related to the time-averaged intensity of the wave, which can be further explored through algebraic manipulation of relevant equations tying together field strengths and the constant c. The mention of the correct option in the final answer requires a comprehensive understanding of electromagnetic theory and the ability to perform complex calculations.