Solutions of Plane Electromagnetic Waves and Their Physical Parameters

In the study of electromagnetics, the plane electromagnetic wave serves as the most fundamental and widely utilized solution to Maxwell's equations. Beyond its theoretical importance, the plane wave model acts as a critical approximation for real-world engineering applications, including antenna radiation patterns, fiber optic propagation, and microwave transmission lines.

To derive the analytical solutions for these waves, we typically operate under a set of idealized assumptions regarding the medium:

• Homogeneous and Isotropic Medium: The permittivity ($\varepsilon$) and permeability ($\mu$) are constant throughout space and do not depend on the direction of the field.
• Source-Free Environment: We assume the absence of free charge densities ($\rho_f = 0$) and free current densities ($\mathbf{J}_f = 0$).
• Planar Geometry: The phase of the wave changes only along a single fixed direction, denoted by the unit vector $\hat{\mathbf{k}}$, while the field components perpendicular to this direction remain constant.

Under these conditions, the electric field $\mathbf{E}$ and magnetic field $\mathbf{H}$ can be expressed in their phasor form as:

[
\mathbf{E}(\mathbf{r},t)=\mathbf{E}_0,e^{j(\mathbf{k}\cdot\mathbf{r}-\omega t)},\qquad
\mathbf{H}(\mathbf{r},t)=\mathbf{H}_0,e^{j(\mathbf{k}\cdot\mathbf{r}-\omega t)}
]

Here, $\mathbf{k} = k\hat{\mathbf{k}}$ represents the wave vector, $\omega$ is the angular frequency, and $j$ is the imaginary unit.

Maxwell's Equations in the Time-Harmonic Domain

When working with time-harmonic fields (assuming a time dependence of $e^{-j\omega t}$), Maxwell's equations simplify into a set of coupled algebraic equations. In a source-free, homogeneous medium, they are expressed as:

[
\begin{aligned}
\nabla\times\mathbf{E} &= -j\omega\mu,\mathbf{H} \
\nabla\times\mathbf{H} &= j\omega\varepsilon,\mathbf{E} \
\nabla\cdot\mathbf{E} &= 0 \
\nabla\cdot\mathbf{H} &= 0
\end{aligned}
]

By substituting the exponential plane wave form into these equations, the differential operator $\nabla$ is effectively replaced by $j\mathbf{k}$. This transformation yields:

[
\begin{aligned}
\mathbf{k}\times\mathbf{E}_0 &= \omega\mu,\mathbf{H}_0 \
\mathbf{k}\times\mathbf{H}_0 &= -\omega\varepsilon,\mathbf{E}_0 \
\mathbf{k}\cdot\mathbf{E}_0 &= 0 \
\mathbf{k}\cdot\mathbf{H}_0 &= 0
\end{aligned}
]

These results are profound: they dictate that the electric field, the magnetic field, and the direction of propagation are all mutually orthogonal. This confirms that plane waves in such media are transverse waves.

Derivation of Key Wave Parameters

1. Wave Number and Phase Velocity

To find the relationship between the spatial and temporal components, we take the cross product of the first two equations. Using the vector identity $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$ and the fact that $\mathbf{k} \cdot \mathbf{E}_0 = 0$, we arrive at the dispersion relation:

[
k^2 = \omega^2\mu\varepsilon \implies k = \omega\sqrt{\mu\varepsilon}
]

From this, we define the phase velocity ($v_p$), which is the speed at which the phase of the wave propagates through the medium:

[
v_p = \frac{\omega}{k} = \frac{1}{\sqrt{\mu\varepsilon}}
]

2. Intrinsic Impedance

The relationship between the magnitudes of the electric and magnetic fields is governed by the medium's properties. From the equation $\mathbf{k}\times\mathbf{E}_0 = \omega\mu,\mathbf{H}_0$, we can derive the intrinsic impedance ($\eta$):

[
\mathbf{H}_0 = \frac{1}{\eta},\hat{\mathbf{k}}\times\mathbf{E}_0, \quad \text{where} \quad \eta = \sqrt{\frac{\mu}{\varepsilon}}
]

The intrinsic impedance represents the "resistance" the medium offers to the formation of the electromagnetic wave and is independent of the wave's frequency in a lossless medium.

3. Polarization States

The polarization of a plane wave is determined by the orientation of the electric field vector $\mathbf{E}_0$ as it oscillates in time:

• Linear Polarization: The $\mathbf{E}$ field oscillates along a single fixed line in space.
• Circular Polarization: The $\mathbf{E}$ field vector rotates in a circle, occurring when two orthogonal components have equal magnitudes and a phase difference of $\pm\pi/2$.
• Elliptical Polarization: The most general case, where the $\mathbf{E}$ field traces an ellipse, occurring when the components have unequal magnitudes or a phase difference other than $\pm\pi/2$.

Summary of Physical Parameters

The following table summarizes the essential parameters used to characterize a plane wave:

Note on Lossy Media: In conducting or lossy media, the permittivity becomes complex ($\varepsilon = \varepsilon' - j\varepsilon''$). Consequently, $k$ and $\eta$ also become complex, leading to a wave that both oscillates and decays exponentially as it travels (attenuation).

Practical Example: A 10 GHz Wave in Free Space

To illustrate these concepts, consider a plane wave propagating in free space ($\varepsilon_0, \mu_0$) with a frequency of $f = 10\text{ GHz}$ in the $+z$ direction, with linear polarization along the $x$-axis.

1. Fundamental Constants & Wave Properties:

• Angular Frequency: $\omega = 2\pi f = 2\pi \times 10^{10}\text{ rad/s}$.
• Phase Velocity: $v_p = c \approx 3 \times 10^8\text{ m/s}$.
• Wavelength: $\lambda_0 = c/f = 0.03\text{ m}$ (or $3\text{ cm}$).
• Wave Number: $k_0 = 2\pi/\lambda_0 \approx 209.44\text{ rad/m}$.

2. Impedance and Field Distribution:

• Intrinsic Impedance: $\eta_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377\ \Omega$.
• Field Expressions:
  [
  \mathbf{E}(z,t) = E_0 \hat{\mathbf{x}} e^{j(k_0z - \omega t)}
  ]
  [
  \mathbf{H}(z,t) = \frac{E_0}{377} \hat{\mathbf{y}} e^{j(k_0z - \omega t)}
  ]

3. Power Density:
If the electric field amplitude $E_0 = 1\text{ V/m}$, the time-averaged power density (magnitude of the Poynting vector) is:
[
\langle S \rangle = \frac{|E_0|^2}{2\eta_0} \approx \frac{1}{754} \approx 1.33 \times 10^{-3}\text{ W/m}^2
]

Conceptual Insights and Common Considerations

1. Why is the wave transverse? The requirement that $\nabla \cdot \mathbf{E} = 0$ in a source-free medium necessitates that the electric field has no component in the direction of propagation ($\mathbf{k} \cdot \mathbf{E} = 0$). This is a fundamental consequence of Gauss's Law in a vacuum or homogeneous medium.
2. Plane Waves vs. Real Waves: While plane waves are mathematically "infinite," real-world waves (like those from a dipole antenna) are spherical. However, in the far-field region, these spherical waves locally behave like plane waves, which is why the plane wave model remains so effective in engineering.
3. The Role of Complex Impedance: In lossy materials, the complex nature of $\eta$ implies a phase shift between $\mathbf{E}$ and $\mathbf{H}$. This phase lag is a direct indicator of the energy being dissipated as heat within the medium.