SINUSOIDAL ELECTROMAGNETIC WAVES
Sandal electromagnetic waves are directly analogous to sinusoidal transverse aves on a stretched string, which we studied in Section 19-4. In a sinue ectromagnetic wave, E and i at any point in space are sinusoidal functions of “””””‘- lnJU at any instant of time the spatial variation of the fields is also sinusoidal. sinusoidal electromagnetic waves are plane waves; they share with the waves /%:lobed in Section 33-3 the property that at any instant the fields are uniform over any perpendicular to the direction ‘of propagation. The entire pattern travels in the c:::ealOll of propagation with speed c. The directions of E and i are perpendicular to the o:i:eaioo of propagation (and to each other), so the wave is transverse. Electromagnetic produced by an oscillating point charge, shown in Fig. 33-1, are an example of idal waves that are not plane waves. But if we restrict our observations to a rela-
; small region of space at a sufficiently great distance from the source, even these are well approximated by plane waves; in the same way, the curved surface of the y) spherical earth appears flat to us because of our small size relative to the earth’s. In this section we’ll restrict our discussion to plane waves. Tbe frequency f, the wavelength A, and the speed of propagation c of any periodic are related by the usual wavelength-frequency relation c = Af. If the frequency fis power-line frequency of 60 Hz, the wavelength is and a moderate distance can include many complete waves.
Figure 33-8 shows a sinusoidal electromagnetic wave traveling in the +x-direction. The E and i vectors are shown for only a few points on the positive x-axis. Imagine a plane perpendicular to the x-axis at a particular point, at a particular time; the fields have
the same values at all points in that plane. The values are different on different planes. In planes where E is in the +y-direction, i is in the +z-direction; where E is in the -y-direction, i is in the -z-direction. Note that in all planes the vector product Ex i is in. the direction in which. the wave is tlrotlagating (the +x-direction). We can describe electromagnetic waves by means of wave functions, just as we did z in Section 19-4 for waves on a string. One form of the wave function for a transverse wave traveling in the +x-direction along a stretched string is Eq. (19-7):
where y(x, t) is the transverse displacement from its equilibrium position at time t of a point with coordinate x on the string. The quantity A is the maximum displacement, or amplitude, of the wave; co is its angular frequency, equal to 2n times the frequency t.and k is the wave number; equal to 2nlA, where A is the wavelength. Let E(x, t) and B(x, t) represent the instantaneous values of the y-component of E and the z-component of i,respectively, in Fig. 33-8, and let and represent the maximum values, or amplitudes, of these fields. The wave functions for the wave are then.
CAUTION. Note the two different k’s, the unit vector f in the z-direction and the wave number k. Don’t get these confused. The sine curves in Fig. 33-8 represent instantaneous values of the electric and magnetic fields as functions of x at time t = 0, that is, E(x, t = 0) and 8(x, t = 0). As time goes by, the wave travels to the right with speed c. Equations (33-16) and (33-17) show that at any point the sinusoidal oscillations of E and 8are in phase. From Eq. (33-4) the amplitudes must be related by.
These amplitude and phase relations are also required for E(x, t) and B(x,t) to satisfy
Eqs. (33-12) and (33-14), which came from Faraday’s law and Ampere’s law respectively.
Can you verify this statement? (See Problem 33-32.) Figure 33-9 shows the electric and magnetic fields of a wave traveling in the negative x-direction. AtjlOints where E is in the positio,!!y-direction, 8 is in the negative z-direction; where E is in the negative y-direction, B is in the positive z-direction. We note that the direction of propagation is the direction of Ex 8, as mentioned in Section 33-3 in the list of characteristics of electromagnetic waves. The wave functions for this wave.
As with the wave traveling in the +x-direction, at any point that the sinusoidal oscillations of the E and 8 fields are in phase, and the vector product E X 8points in the direction of propagation.
The sinusoidal waves shown in Figs. 33-8 and 33-9 are both linearly polarized in the y-direction; the E field is always parallel to the y-axis. Example 33-1 concerns a wave that is linearly polarized in the z-direction.
SOLUTION Equations (33-19) describe a wave traveling in the negative x-direction with H along the y-axis; that is, a wave that is linearly polarized along the y-axis. By contrast, the wave in this example is linearly polarized along the z-axis. At points where H is in the positive z-direction, B must be in the positive
y-direction in order for the vector product H X B to be in the negative x-direction (the direction of propagation). A possible pair of wave functions that atisfies these requirements.