Question

Determine the values of P[infinity] and E[infinity] for each of the following signals:

(b) x2(t)=ej(2t+π/4)
(c) x3(t)=cos(t)
(e) x2[n]=ej(π/2n+π/8)
(f) x3[n]=cos(π/4n)

Answer 1

The electromagnetic fields are first zero at t = 0 and at multiples of the period. They reach their most negative value at a quarter of the period. The superposition of two waves creates a standing wave, with calculations for amplitude, frequency, and period corresponding to the equations provided for sinusoidal electric fields.

Step-by-step explanation:

To analyze the questions related to electromagnetic waves and wave functions, we break them down into each given scenario. Firstly, for electric and magnetic fields that vary sinusoidally with time and have a frequency of 1.00 GHz, the field strengths will be first zero at t = 0 and then they will be zero whenever the sine function completes a full cycle, that is at multiples of the period. The fields reach their most negative value at a quarter of the period, where the sine function reaches -1. The time needed to complete one cycle, the period (T), can be calculated using the formula T = 1/f, where f is the frequency.

Secondly, the superposition of two waves with a phase difference results in a standing wave pattern, where nodes and antinodes are formed. The resultant wave function can be represented as YR = 2A sin(kx + p/2) cos(wt + p/2), where A is the amplitude, k is the wave number, w is the angular frequency, and p is the phase difference.

For the electromagnetic wave described by E = (30) sin (4.0×106t), the amplitude is the coefficient of the sinusoidal function (30), the frequency (f) is determined by the angular frequency (w) such that f = w/(2π), and the period (T) is the inverse of the frequency (1/f). The graph for this function would be a sinusoidal wave oscillating between ±30 with a frequency and period corresponding to the angular frequency of 4.0×106 rad/s.