A SIMPLE PLANE ELECTROMAGNETIC WAVE
Using an zyz-coordinate system (Fig. 33-2), we imagine that all space is divided into two regions by a plane perpendicular to the x-axis (parallel to the yz-plane). At every point to the left of this plane there are a uniform electric field E in the +y-direction and a uniform magnetic field jj in the +z-direction, as shown. Furthermore, we suppose that the boundary plane, which we call the wave front, moves to the right in the +x-direction
with a constant speed c, as yet unknown. Thus the E and jj fields travel to the right intopreviously field-free regions with a definite speed. The situation, in short, describes a rudimentary electromagnetic wave. A wave such as this, in which at any instant the fields are uniform over any plane perpendicular to the direction of propagation, is called a plane wave. In the case shown in Fig. 33-2, the fields are zero for planes to the right of
the wave front and have the same values on all planes to the left of the wave front; later we will consider more complex plane waves.
We won’t concern ourselves with the problem of actually producing such a field configuration. Instead, we simply ask whether it is consistent with the laws of electromagnetism, that is, with Maxwell’s equations. We’ll consider each of these four equations in turn. Let us first verify that our wave satisfies Maxwell’s first and second equations, that is, Gauss’s laws for electric and magnetic fields. To do this, we take as our Gaussian surface a rectangular box with sides parallel to the xy, zz, and yz coordinate planes (Fig. 33-3). The box encloses no electric charge, and you should be able to show that the total electric flux and magnetic flux through the box are both zero; this is true even if part of the box is in the region where E = B = O. This would not be the case·if E or jj had an x-component, parallel to the direction of propagation. We leave the proof as a problem (see Problem 33-30). Thus in order to satisfy Maxwell’s first and second equations, the electric and magnetic fields must be perpendicular to the direction of propagation; that is, the wave must be transverse.
The next of Maxwell’s equations to be considered is Faraday’s law:
To test whether our wave satisfies Faraday’s law, we apply this law to a rectangle efgh that is parallel to the xy-plane (Fig. 33-4a). As shown in Fig. 33-4b, a cross section in the xy-plane, this rectangle has height a and width ~x. At the time shown, the wave front
has progressed partway through the rectangle, and E is zero along the side ef In applying Faraday’s law we take the vector area dA of rectangle efgh to be in the +z-direction, With this choice the right-hand rule requires that we integrate E . dl counter clock wise.
Hence the left-hand side of Am..rere’s law, Eq. (33-5), is nonzero; ~e right-~and side must be nonzero as well. Thus E must have a y-component (perpendicular to B) so that the electric flux <l>Ethrough the rectangle and the time derivative d<l>idt can be nonzero. We come to the same conclusion that we inferred from Faraday’s law: In an electromagnetic wave, E and jj must be mutually perpendicular. In a time interval dt the electric flux <l>Ethrough the rectangle increases by d<l>E= E(ac dt). Since we-chose ciA to be in the +y-direction, this flux change is positive;
Our assumed wave is consistent with all of Maxwell’s equations, provided that the wave front moves with the speed given above, which you should recognize as the speed of light! Note that the exact value of c is defined to be 299,792,458 m/s; for our purposes, c = 3.00 X 108 mls is sufficiently accurate.
We chose a simple wave for our study in order to avoid mathematical complications, but this special case illustrates several important features of all electromagnetic waves: 1. The wave is transverse; both E and B are perpendicular to the direction of propagation of the wave. The electric and magnetic fields are also perpendicular to each other. The direction of propagation is the direction of the vector product E XB.
2. There is a definite ratio between the magnitudes of E and B: E = cB. 3. The wave travels in vacuum with a definite and unchanging speed. 4. Unlike mechanical waves, which need the oscillating particles of a medium such as water or air to transmit a wave, electromagnetic waves require no medium. What’s “waving” in an electromagnetic wave are the electric and magnetic fields. We can generalize this discussion to a more realistic situation. Suppose we have several wave fronts in the form of parallel planes perpendicular to the x-axis, all of which are moving to the right with speed c. Suppose that the E and B fields are the same at all points within a single region between two planes, but that the fields differ from region to region. The overall wave is a plane wave, but one in which the fields vary in steps along the x-axis. Such a wave could be constructed by superposing several of the simple step waves we have just discussed (shown in Fig. 33-2). This is possible because the E _ and B fields obey the superposition principle in waves just as in static situations: When
two waves are superposed, the total E field at each point is the vector sum of the E fields of the individual waves, and similarly for the total B field. We can extend the above development to show that a wave with fields that vary in steps is also consistent with Ampere’s and Faraday’s laws, provided that the wave fronts all move with the speed c given by Eq. (33-9). In the limit that we make the individual steps infinitesimally small, we have a wave in which the E and B fields at any instant vary continuously along the x-axis. The entire field pattern moves to the right with speed c. In Section 33-4 we will consider waves in which E and B are sinusoidal functions of x and t. Because at each point the magnitudes of E and B are related by E = cB, the peri variations of the two fields in any periodic traveling wave must be in phase.
Electromagnetic waves have the property of polarization. In the above discussion ice of the y-direction for E was arbitrary. We could just as well have specified the . for E; then i would have been in the -y-direction. A wave in which E is always el to a certain axis is said to be linearly polarized along that axis. More generally, wave traveling in the x-direction can be represented as a superposition of waves linearly polarized in the y- and z-directions. We will study polarization in greater detail, . special emphasis on polarization of light, in Chapter 34.