Question

The electric field of an electromagnetic wave that propagates in a vacuum is given by: ?⃗ (?, ?) = 2 cos (α? - ω?) ? + 2 sin (α? - ω?) ?, ?? (?/?); ???? α ? ω ??? ??????????

a) What is the amplitude E0 of the electric field, its angular frequency, period, frequency f, wave number | ?⃗ |, wavelength, its speed u and its phase constant φ?

b) I) Determine the electric field in the x-y plane (z = 0) for the following time values t = 0, T / 4, T / 2, 3T / 4 and T. II) Then draw these vectors on the same Cartesian plane x-y.

c) Based on item (a), explain what happens with the electric field in the xy plane during the considered time interval.

d) Apply the wave equation to determine the condition that the field must satisfy.

e) I) Deduce the magnetic field ?⃗ (?, ?). II) What direction does the field ?⃗ have in relation to the field ?⃗? Test your statement mathematically. III) Then, for each vector ?⃗ drawn in item (b) draw its respective pair ?⃗.

f) Calculate I) energy density II) the flow of energy density III) their temporary averages <?> and <?>. Tip: use the definition of temporary average .

g) Explain if the wave propagates and if so, in what direction?

Answer 1

​​​​​​

c) During considered time interval, the electric field in x-direction decreases in interval 0 to T/2 and then starts increasing and acquires same values as t=0 at t=T. The whole cycle is completed. In y-direction, exact opposite takes place. The field increases from 0 to T/4, then decreases from T/4 tp 3T/4 and then increases again to the value at t=0.