Faraday discovered that a changing magnetic field creates an induced electric field, and Maxwell’s postulated displacement current says that a changing electric field creates an induced magnetic field. The idea behind electromagnetic waves, illustrated in FIGURE 31.18, is that the fields can exist in a self-sustaining mode if a changing magnetic field creates an electric field that, in turn, happens to change in just the right way to recreate the original magnetic field. Notice that it has to be an electromagnetic wave, with changing electric and magnetic fields. A purely electric or purely magnetic wave cannot exist.

FIGURE 31.18 Induced fields can be self-sustaining.

You saw in Section 30.6 that an induced electric field, which can drive an induced current around a conducting loop, is perpendicular to the changing magnetic field. And earlier in this chapter, when we introduced the displacement current, the induced magnetic field in a charging capacitor was perpendicular to the changing electric field. Thus we’ll make the assumption that $\vec{\mathit{E}}$ and $\vec{\mathit{B}}$ are perpendicular to each other in an electromagnetic wave. Furthermore—we’ll justify this shortly—$\vec{\mathit{E}}$ and $\vec{\mathit{B}}$ are each perpendicular to the direction of travel. Thus an electromagnetic wave is a transverse wave, analogous to a wave on a string, rather than a sound-like longitudinal wave.

We will also assume, to keep the mathematics as simple as possible, that an electromagnetic wave can travel as a plane wave, which you will recall from Chapter 16 is a wave for which the fields are the same everywhere in a plane perpendicular to the direction of travel. FIGURE 31.19a shows an electromagnetic plane wave propagating at speed $v_{\text{em}}$ along the x-axis. $\vec{E}$ and $\vec{B}$ are perpendicular to each other, as we’ve assumed, and to the direction of travel. We’ve defined the y- and z-axes to be, respectively, parallel to $\vec{E}$ and $\vec{B}$. Notice how the fields are the same at every point in a yz-plane slicing the x-axis.

FIGURE 31.19 An electromagnetic plane wave.

Because a wave is a traveling disturbance, FIGURE 31.19b shows that the fields—at one instant of time—do change along the x-axis. These changing fields are the disturbance that is moving down the x-axis at speed $v_{\text{em}}$, so $\vec{E}$ and $\vec{B}$ of a plane wave are functions of the two variables x and t. We’re not assuming that the wave has any particular shape—the shape of the wave is what we want to predict from Maxwell’s equations—simply that it’s a transverse wave moving along the x-axis.

Now that we know something about the structure of the wave, we can start to check its consistency with Maxwell’s equations. FIGURE 31.20 shows an imaginary box, a Gaussian surface, centered on the x-axis. Both electric and magnetic field vectors exist at each point in space, but the figure shows them separately for clarity. $\vec{E}$ oscillates along the y-axis, so all electric field lines enter and leave the box through the top and bottom surfaces; no electric field lines pass through the sides of the box.

FIGURE 31.20 A closed surface can be used to check Gauss’s law for the electric and magnetic fields.

Because this is a plane wave, the magnitude of each electric field vector entering the bottom of the box is exactly matched by an electric field vector leaving the top. The electric flux through the top of the box is equal in magnitude but opposite in sign to the flux through the bottom, and the flux through the sides is zero. Thus the net electric flux is ${\text{Φ}}_{\text{e}}=0$. There is no charge inside the box, because there are no sources in this region of space, so we also have $Q_{\text{in}}=0$. Hence the electric field of a plane wave is consistent with the first of the source-free Maxwell’s equations, Gauss’s law.

The exact same argument applies to the magnetic field. The net magnetic flux is ${\text{Φ}}_{\text{m}}=0;$ thus the magnetic field is consistent with the second of Maxwell’s equations.

Suppose that $\vec{E}$ or $\vec{B}$ had a component along the x-axis, the direction of travel. The fields change along the x-axis—that’s what a traveling wave is—so it would not be possible for the flux through the right face to exactly cancel the flux through the left face at every instant of time. An x-component of either field would violate Gauss’s law by creating a net flux when there are no enclosed sources. Thus our claim that an electromagnetic wave must be a transverse wave, with the fields perpendicular to the direction of travel, is a requirement of Gauss’s law.