Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The same four steps with the roles of the two fields exchanged give the equation for $H$ (Figure 5.5.2, right column). Take the curl of the generalised Ampère law, bring $ε$ out and exchange the curl with $\partial / \partial t$ , then substitute Faraday's law $\nabla \times E = - μ \partial H / \partial t$ :

\nabla \times \nabla \times H = ε \frac{\partial}{\partial t} (\nabla \times E) = - μ ε \frac{\partial ^{2} H}{\partial t ^{2}} . (5.5.11)

Expand the double curl with Eq. (5.5.6) and use $\nabla \cdot H = 0$ , which holds in every region since $\nabla \cdot B = 0$ :

\nabla (0 \nabla \cdot H) - \nabla^{2} H = - μ ε \frac{\partial ^{2} H}{\partial t ^{2}} . (5.5.12)

The wave equations for E and H, and the scalar wave equation

In a source-free, simple, lossless medium both fields obey the same equation,

\nabla^{2} E - μ ε \frac{\partial ^{2} E}{\partial t ^{2}} \nabla^{2} H - μ ε \frac{\partial ^{2} H}{\partial t ^{2}} = 0, = 0, (5.5.13)

and so does every Cartesian component $u$ ( $E_{x}$ , $E_{y}$ , $E_{z}$ , $H_{x}$ , $H_{y}$ or $H_{z}$ ) of either field:

\nabla^{2} u - \frac{1}{v ^{2}} \frac{\partial ^{2} u}{\partial t ^{2}} = 0, v = \frac{1}{μ ε} . (5.5.14)

In plain words

One equation governs all six Cartesian field components, each on its own. Whatever is true of waves in general, for instance their speed $v$ , is true of each component of $E$ and of $H$ .

Written out with Eq. (5.5.9), the vector equation for $E$ is three scalar equations,

\frac{\partial ^{2} E _{i}}{\partial x ^{2}} + \frac{\partial ^{2} E _{i}}{\partial y ^{2}} + \frac{\partial ^{2} E _{i}}{\partial z ^{2}} - μ ε \frac{\partial ^{2} E _{i}}{\partial t ^{2}} = 0, i = x, y, z, (5.5.15)

and the same three hold for $H_{x}$ , $H_{y}$ , $H_{z}$ . They are uncoupled: $E_{x}$ appears in the first one only. The coupling between the components, and between $E$ and $H$ , has not been lost, however. It has been moved out of the wave equation into the conditions discussed below.

Common mistake · Applying the scalar equation to cylindrical or spherical components

In cylindrical or spherical coordinates the unit vectors $a_{ρ}$ , $a_{ϕ}$ , $a_{r}$ , $a_{θ}$ turn from point to point, so the Laplacian of a vector is not the Laplacian of its components: $(\nabla^{2} E)_{ρ} \neq = \nabla^{2} E_{ρ}$ . The scalar equation Eq. (5.5.14) holds for $E_{x}$ , $E_{y}$ , $E_{z}$ , and therefore for the axial component $E_{z}$ of a field in a circular waveguide (Section 7.4), but not for $E_{ρ}$ or $E_{ϕ}$ . Exercise 5.5.5 shows a radial field of a coaxial line that satisfies the vector wave equation while its $E_{ρ}$ does not satisfy the scalar one.

Not yet a solution

The wave equations are necessary conditions, not sufficient ones. Taking the curl lost information: the link between $E$ and $H$ and the divergence conditions. A field $E$ that solves Eq. (5.5.8) is an electromagnetic field only if it also satisfies $\nabla \cdot E = 0$ , and then $H$ is not free to choose but follows from Faraday's law $\nabla \times E = - μ \partial H / \partial t$ by one integration in time. Problems 5.5.2 and 5.5.3 use the wave equation and these two conditions together. Section 6.1 shows what they imply for plane waves: $E$ and $H$ perpendicular to each other and to the direction of travel.

Quick check

Does $E = E_{0} sin (k z - ω t) a_{z}$ with $ω / k = v$ describe a field in a source-free region?

No. Its single Cartesian component satisfies the scalar wave equation Eq. (5.5.14), since $\partial^{2} E_{z} / \partial z^{2} = - k^{2} E_{z}$ and $v^{- 2} \partial^{2} E_{z} / \partial t^{2} = - (ω / v)^{2} E_{z} = - k^{2} E_{z}$ . But $\nabla \cdot E = \partial E_{z} / \partial z = k E_{0} cos (k z - ω t) \neq = 0$ , so Gauss's law would require a charge density. A wave in a source-free region cannot have its field along its own direction of travel when the field does not vary across that direction.

Recipe · Testing a given field against the wave equation

Divergence. Check $\nabla \cdot E = 0$ (or $\nabla \cdot H = 0$ ). If it fails, the field is not a source-free field, whatever the wave equation says.
Cartesian components. Write the field in Cartesian components, converting from cylindrical or spherical ones if necessary.
Derivatives. For each component compute $\nabla^{2} u$ and $\partial^{2} u / \partial t^{2}$ . For sines, cosines and exponentials each derivative returns the function times a constant.
Condition. Substitute in $\nabla^{2} u - μ ε \partial^{2} u / \partial t^{2} = 0$ . The result is an algebraic relation between $ω$ , the wave numbers and $μ ε$ , from which the unknown follows.
The other field. Find $H$ from Faraday's law by integrating in time (no static part), and check the generalised Ampère law with it.

5.5.3 The wave equation for H and the scalar form