Propagation of Electromagnetic Waves in Media & Plasmas

Derivation of the Vector Wave Equation

Consider source-free vacuum where charge density $\rho = 0$ and conduction current density $\mathbf{J} = 0$. Maxwell's equations reduce to:

1. $\nabla \cdot \mathbf{E} = 0$
2. $\nabla \cdot \mathbf{B} = 0$
3. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
4. $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

To decouple these coupled first-order partial differential equations, take the curl of Faraday's law:

Using the fundamental vector identity $\nabla \times (\nabla \times \mathbf{A}) \equiv \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ on the left-hand side:

Since $\nabla \cdot \mathbf{E} = 0$ in vacuum, and substituting Ampère-Maxwell's law for $\nabla \times \mathbf{B}$:

Rearranging gives the celebrated homogeneous vector wave equation for the electric field:

Taking the curl of Ampère-Maxwell's law in identical fashion yields the wave equation for the magnetic field:

The Speed of Light and Maxwell's Synthesis

Comparing these to the standard 3D scalar wave equation $\nabla^2 \psi - \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} = 0$, the wave propagation speed $v$ is determined strictly by electromagnetic constants:

Substituting experimental values:

This exact match between the derived velocity of electromagnetic waves and the measured speed of light led James Clerk Maxwell to declare: "Light is an electromagnetic disturbance in the form of waves propagating through the electromagnetic field according to electromagnetic laws."

Monochromatic Plane Wave Solutions

A plane wave traveling in direction $\hat{\mathbf{k}}$ with wavevector $\mathbf{k} = k \hat{\mathbf{k}}$ and angular frequency $\omega$ is described in complex exponential notation by:

where physical fields are the real parts $\text{Re}\{\mathbf{E}\}$.

Substituting into the wave equation gives the vacuum dispersion relation:

Rigorous Proof of Transverse Nature (No Longitudinal Component)

Apply Gauss's law $\nabla \cdot \mathbf{E} = 0$:

Similarly, applying $\nabla \cdot \mathbf{B} = 0$:

Conclusion: Both $\mathbf{E}$ and $\mathbf{B}$ are strictly perpendicular to the propagation vector $\mathbf{k}$. Electromagnetic waves in unbounded homogeneous media are purely transverse waves ($E_k = 0, B_k = 0$).

Orthogonality and Phase Relationship

Applying Faraday's law $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$:

This fundamental vector relation establishes that:

1. $\mathbf{B}$ is perpendicular to $\mathbf{E}$ ($\mathbf{E} \cdot \mathbf{B} = 0$).
2. $\mathbf{E}$, $\mathbf{B}$, and $\hat{\mathbf{k}}$ form an orthogonal right-handed triad:

1. The magnitudes are related by $B_0 = \frac{E_0}{c}$.
2. In vacuum, $\mathbf{E}$ and $\mathbf{B}$ oscillate exactly in phase, reaching crests and nodes at identical positions and times.

Intrinsic Wave Impedance of Free Space

The ratio of the transverse electric field to the transverse magnetic intensity $H = B/\mu_0$ defines the wave impedance:

Evaluating numerically:

The wave impedance of free space represents the resistance of vacuum to the generation of electric and magnetic flux. In a linear medium with permittivity $\epsilon$ and permeability $\mu$, the intrinsic impedance is $\eta = \sqrt{\mu / \epsilon}$.

Time-Averaged Energy Flux (Intensity)

For real sinusoidal fields traveling along $+z$:

The instantaneous Poynting vector is:

Since the time average of $\cos^2(\theta)$ over a full cycle is $\frac{1}{2}$, the time-averaged Poynting vector (wave intensity $I$) is:

Energy Density Distribution

The time-averaged electric energy density is:

The time-averaged magnetic energy density is:

Thus, $\langle u_E \rangle = \langle u_B \rangle$: electromagnetic energy is partitioned equally between electric and magnetic fields at all times in a plane wave.

In a linear, homogeneous, isotropic dielectric medium with permittivity $\epsilon = \epsilon_r \epsilon_0$, permeability $\mu = \mu_r \mu_0$, and conductivity $\sigma = 0$:

Maxwell's equations yield the modified wave velocity:

where $n \equiv \sqrt{\epsilon_r \mu_r}$ is the index of refraction of the medium. For non-magnetic optical materials ($\mu_r \approx 1$), Maxwell's relation gives:

Wavelength and Wavevector in Matter

Because frequency $\omega$ is fixed by the driving source, the wavelength in the dielectric shrinks:

The wavevector increases:

The wave impedance becomes:

The time-averaged intensity in the dielectric is:

The Telegrapher-Type Wave Equation in Conductors

In an ohmic conducting medium with electric conductivity $\sigma$, free conduction currents flow according to Ohm's Law: $\mathbf{J}_f = \sigma \mathbf{E}$. There are no static free charges ($\rho_f = 0$).

Maxwell's curl equations become:

Taking the curl of Faraday's law:

Since $\nabla \cdot \mathbf{E} = 0$:

The first-order time derivative term $\mu \sigma \frac{\partial \mathbf{E}}{\partial t}$ acts as a dissipative frictional damping term that extracts energy from the wave and converts it into Joule heat.

Complex Wavevector Formulation

Substitute a monochromatic plane wave $\mathbf{E} = \mathbf{E}_0 e^{i(\tilde{k} z - \omega t)}$:

The ratio $\tan \delta \equiv \frac{\sigma}{\omega \epsilon}$ is known as the loss tangent. It quantifies the relative magnitude of conduction current compared to displacement current:

Let the complex wavevector be $\tilde{k} \equiv \beta + i \alpha$:

Equating real and imaginary parts:

Solving this system yields the exact analytical expressions:

The spatial electric field in the conductor is therefore:

The wave amplitude decays exponentially with distance into the conductor.

The Good Conductor Limit ($\sigma \gg \omega \epsilon$)

For metals and good conductors (such as copper, silver, aluminum, and sea water at radio frequencies), conduction current dwarfs displacement current:

Under this approximation, the term $\sqrt{1 + (\sigma/\omega\epsilon)^2} \approx \frac{\sigma}{\omega \epsilon}$, and the formulas for $\alpha$ and $\beta$ simplify dramatically:

Definition of Skin Depth ($\delta$)

The skin depth $\delta$ (also called penetration depth) is defined as the distance over which the wave amplitude drops by a factor of $1/e \approx 0.3679$ (36.8% of its surface value):

Over a depth of $z = 5\delta$, the wave amplitude decays to $e^{-5} \approx 0.0067$ (under 0.7%), meaning high-frequency currents are confined almost entirely to a microscopically thin outer shell of a conductor.

Table: Practical Skin Depths Across Frequencies

For pure copper ($\sigma = 5.8 \times 10^7 \text{ S/m}$, $\mu = \mu_0$):

| Frequency ($f$) | Skin Depth $\delta$ in Copper | Application / Impact | | :--- | :--- | :--- | | 50 Hz / 60 Hz | $9.38 \text{ mm}$ | AC mains power transmission (large cables require hollow or stranded conductors). | | 10 kHz | $0.66 \text{ mm}$ | Audio and induction heating frequencies. | | 1 MHz | $66 \, \mu\text{m}$ | AM radio broadcast frequencies. | | 100 MHz | $6.6 \, \mu\text{m}$ | FM radio and VHF communications. | | 10 GHz | $0.66 \, \mu\text{m}$ | X-band radar and microwave waveguides (requires surface silver plating). |

Phase Delay Between $\mathbf{E}$ and $\mathbf{B}$

From Faraday's law in a good conductor:

Since $\tilde{k} = \beta + i \alpha = \alpha(1 + i) = \alpha \sqrt{2} e^{i\pi/4}$:

Physical Result: In a good conductor, the magnetic field lags the electric field by a phase angle of $45^\circ$ ($\pi/4$ radians), and the magnetic energy density dramatically exceeds the electric energy density:

The Cold Plasma Dielectric Function

Consider an ionized gas (such as the Earth's ionosphere or interstellar plasma) consisting of free electrons (mass $m_e$, charge $-e$, density $n_e$) and heavy, immobile positive ions. In the presence of a monochromatic electric field $\mathbf{E}(t) = \mathbf{E}_0 e^{-i\omega t}$, the equation of motion for a conduction electron (neglecting damping collisions) is:

The induced macroscopic dipole polarization density is:

The electric displacement is:

The Plasma Frequency ($\omega_p$)

We define the characteristic electron plasma frequency:

In terms of frequency in Hertz:

where $n_e$ is in $\text{electrons/m}^3$.

The relative permittivity of the plasma is:

Dispersion Relation and Propagation Regimes

The wavevector in the plasma satisfies:

1. High Frequency Regime ($\omega > \omega_p$):

$k$ is purely real:

The wave propagates freely without attenuation. The phase velocity exceeds the speed of light:

The group velocity (signal energy velocity) is strictly less than $c$:

Notice that $v_p \cdot v_g = c^2$, satisfying special relativity.

2. Low Frequency Cutoff Regime ($\omega < \omega_p$):

$k$ becomes purely imaginary:

The fields decay exponentially: $\mathbf{E}(z, t) = \mathbf{E}_0 e^{-\kappa z} e^{-i\omega t}$. No real energy is propagated; instead, the wave undergoes total reflection at the plasma boundary.

Application to Radio Communications: The Earth's ionosphere has peak electron density $n_e \approx 10^{12} \text{ m}^{-3}$, giving a plasma critical frequency $f_p \approx 9 \text{ MHz}$. Shortwave radio signals ($3 - 30 \text{ MHz}$) below the critical frequency are totally reflected back to Earth, enabling global intercontinental communication without satellites. Satellite transmissions (GPS, 1.5 GHz) easily exceed $f_p$ and pass through the ionosphere unhindered.