Part II: Electromagnetic Waves | Chapter 4

Maxwell's Equations & the Wave Equation

The unification of electricity, magnetism, and light

4.1 Maxwell's Equations

All of classical electromagnetism is contained in four equations. In differential form, for free space (no charges or currents):

Maxwell's Equations (Free Space)

I.$\nabla \cdot \vec{E} = 0$Gauss's law

II.$\nabla \cdot \vec{B} = 0$No magnetic monopoles

III.$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$Faraday's law

IV.$\nabla \times \vec{B} = \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$Ampere-Maxwell law

The crucial addition by Maxwell was the displacement current term$\mu_0\epsilon_0 \partial\vec{E}/\partial t$ in Ampere's law. Without it, the equations would be inconsistent with charge conservation. With it, they predict electromagnetic waves.

With sources: In the presence of charge density $\rho$ and current density $\vec{J}$, Gauss's law becomes$\nabla \cdot \vec{E} = \rho/\epsilon_0$ and the Ampere-Maxwell law becomes$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\partial\vec{E}/\partial t$.

4.2 Deriving the Electromagnetic Wave Equation

Figure. Electromagnetic wave propagating along z. The electric field E (red, y-direction) and magnetic field B (blue, x-direction) oscillate perpendicular to each other and to the propagation direction.

Derivation 1: From Maxwell to Waves

Take the curl of Faraday's law:

$$\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$$

Use the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2\vec{E}$and Gauss's law ($\nabla \cdot \vec{E} = 0$):

$$-\nabla^2\vec{E} = -\frac{\partial}{\partial t}\left(\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}\right)$$

The Electromagnetic Wave Equation

$$\nabla^2\vec{E} = \mu_0\epsilon_0\frac{\partial^2\vec{E}}{\partial t^2} \quad \text{and} \quad \nabla^2\vec{B} = \mu_0\epsilon_0\frac{\partial^2\vec{B}}{\partial t^2}$$

Both $\vec{E}$ and $\vec{B}$ satisfy the wave equation with speed $c = 1/\sqrt{\mu_0\epsilon_0}$.

4.3 The Speed of Light

Derivation 2: c from Electrical Constants

Comparing the wave equation with the standard form $\nabla^2 u = (1/v^2)\partial_t^2 u$, we identify the wave speed:

Speed of Light from Electromagnetic Constants

$$c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = \frac{1}{\sqrt{(4\pi \times 10^{-7})(8.854 \times 10^{-12})}} \approx 2.998 \times 10^8 \text{ m/s}$$

Maxwell recognized that this speed, calculated purely from electrical and magnetic measurements, matched the known speed of light. This led him to propose that light is an electromagnetic wave — one of the greatest unifications in physics.

In a linear, isotropic medium with permittivity $\epsilon$ and permeability$\mu$, the wave speed is:

$$v = \frac{1}{\sqrt{\mu\epsilon}} = \frac{c}{n} \quad \text{where } n = \sqrt{\mu_r\epsilon_r} \text{ is the refractive index}$$

4.4 Plane Wave Solutions

Derivation 3: Structure of EM Waves

The simplest solutions are plane waves. For a wave propagating in the $\hat{z}$direction:

$$\vec{E} = E_0 \cos(kz - \omega t)\,\hat{x}$$$$\vec{B} = \frac{E_0}{c} \cos(kz - \omega t)\,\hat{y}$$

Key properties of electromagnetic plane waves:

Transverse

$\vec{E}$ and $\vec{B}$ are perpendicular to the propagation direction and to each other: $\vec{E} \perp \vec{B} \perp \hat{k}$

In Phase

$\vec{E}$ and $\vec{B}$ oscillate in phase, with a fixed ratio$|\vec{E}| = c|\vec{B}|$

Dispersion Relation

$\omega = ck$

Linear: no dispersion in vacuum

Wavelength-Frequency

$c = \lambda f$

The universal relation

4.5 Energy and the Poynting Vector

Derivation 4: Electromagnetic Energy Flow

The energy density stored in electromagnetic fields is:

$$u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$$

For a plane wave, $B = E/c$ so the electric and magnetic contributions are equal:$u_E = u_B = \frac{1}{2}\epsilon_0 E^2$. The energy flux (power per unit area) is given by the Poynting vector:

The Poynting Vector

$$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$$

The Poynting vector points in the direction of wave propagation, with magnitude$S = u \cdot c$. Its time average gives the intensity:$I = \langle S \rangle = \frac{1}{2}\epsilon_0 c E_0^2$.

Derivation 5: Poynting's Theorem (Energy Conservation)

From Maxwell's equations, one can derive Poynting's theorem:

$$\frac{\partial u}{\partial t} + \nabla \cdot \vec{S} = -\vec{J} \cdot \vec{E}$$

This is a continuity equation for electromagnetic energy: the rate of decrease of field energy equals the power flowing out through the surface plus the power dissipated by currents ($\vec{J} \cdot \vec{E}$ is Ohmic heating).

Radiation Pressure

Electromagnetic waves carry momentum as well as energy. The momentum density is$\vec{g} = \vec{S}/c^2$, and the radiation pressure on a perfectly absorbing surface is:

$$P_{\text{rad}} = \frac{I}{c} \quad (\text{absorption}), \qquad P_{\text{rad}} = \frac{2I}{c} \quad (\text{reflection})$$

4.5b Electromagnetic Waves in Linear Media

In a linear, isotropic, homogeneous medium with permittivity $\epsilon = \epsilon_r\epsilon_0$and permeability $\mu = \mu_r\mu_0$, Maxwell's equations give:

$$\nabla^2\vec{E} = \mu\epsilon\frac{\partial^2\vec{E}}{\partial t^2} \quad \Longrightarrow \quad v = \frac{c}{n} \text{ where } n = \sqrt{\epsilon_r\mu_r}$$

For a conducting medium with conductivity $\sigma$, Ohm's law gives $\vec{J} = \sigma\vec{E}$, and the wave equation becomes:

$$\nabla^2\vec{E} = \mu\epsilon\frac{\partial^2\vec{E}}{\partial t^2} + \mu\sigma\frac{\partial\vec{E}}{\partial t}$$

The extra term causes exponential attenuation. The skin depth is the distance over which the wave amplitude drops by $1/e$:

Skin Depth

$$\delta = \sqrt{\frac{2}{\mu\sigma\omega}}$$

For copper at 60 Hz: $\delta \approx 8.5$ mm. At 1 GHz: $\delta \approx 2$ $\mu$m. This is why high-frequency currents flow only on the surface of conductors.

4.5c The Electromagnetic Spectrum

All electromagnetic waves are fundamentally the same phenomenon — oscillating electric and magnetic fields propagating at the speed of light. They differ only in wavelength and frequency, related by $c = \lambda f$:

Radio Waves ($\lambda > 1$ mm)

Used for communication, radio astronomy, MRI. Generated by oscillating currents in antennas. The cosmic microwave background (CMB) peaks in the microwave range at$\lambda = 1.9$ mm, a relic of the Big Bang.

Infrared ($700$ nm – $1$ mm)

Thermal radiation from room-temperature objects. Used in night vision, remote controls, thermal imaging, and fiber-optic telecom (1.3 and 1.55 $\mu$m bands).

Visible Light (380–700 nm)

The narrow band to which human eyes are sensitive, centered near the peak of solar emission. The Sun's surface temperature of 5778 K gives peak emission at 502 nm (green).

Ultraviolet, X-rays, Gamma Rays

Progressively higher energies. UV causes sunburn; X-rays penetrate tissue for medical imaging; gamma rays from nuclear transitions and cosmic sources carry $E > 100$ keV. At these energies, the photon nature of light becomes dominant over its wave nature.

4.5d Worked Example: Sunlight Radiation Pressure

Problem: The intensity of sunlight at Earth's distance is$I \approx 1370$ W/m$^2$ (the solar constant). Calculate: (a) the electric field amplitude, (b) the radiation pressure on a perfectly reflecting solar sail, and (c) the force on a 100 m $\times$ 100 m sail.

Solution

(a) From $I = \frac{1}{2}\epsilon_0 c E_0^2$:

$$E_0 = \sqrt{\frac{2I}{\epsilon_0 c}} = \sqrt{\frac{2 \times 1370}{8.854 \times 10^{-12} \times 3 \times 10^8}} \approx 1015 \text{ V/m}$$

(b) Radiation pressure for perfect reflection:$P_{\text{rad}} = 2I/c = 2 \times 1370/(3 \times 10^8) \approx 9.1 \times 10^{-6}$ Pa

This tiny force, sustained over months, can accelerate a lightweight spacecraft to significant velocities for interplanetary travel without any fuel.

4.6 Applications

Radio Communication

Maxwell's prediction of EM waves, confirmed by Hertz in 1887, led to radio communication by Marconi (1895). The entire electromagnetic spectrum — from radio waves through microwaves, infrared, visible, UV, X-rays, to gamma rays — is described by Maxwell's equations.

Solar Sails

Radiation pressure from sunlight can propel spacecraft. The IKAROS mission (2010) successfully demonstrated solar sailing. The Poynting vector determines the thrust: $F = 2IA/c$ for a perfectly reflecting sail of area $A$.

Wireless Power Transfer

Near-field electromagnetic coupling enables wireless charging of phones and electric vehicles. The Poynting vector describes energy flow from transmitter to receiver coils, with efficiency depending on coupling and resonance conditions.

Optical Tweezers

Focused laser beams can trap and manipulate microscopic particles using radiation pressure. Arthur Ashkin received the 2018 Nobel Prize for this invention, which enables studying single molecules and cells.

4.6b Maxwell's Equations: Integral Form and Gauge Freedom

The integral forms of Maxwell's equations, obtained by applying Gauss's and Stokes' theorems, are often more useful for problems with high symmetry:

$$\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \quad \text{(Gauss)}$$

$$\oint \vec{B}\cdot d\vec{A} = 0 \quad \text{(no monopoles)}$$

$$\oint \vec{E}\cdot d\vec{l} = -\frac{d\Phi_B}{dt} \quad \text{(Faraday)}$$

$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0\epsilon_0\frac{d\Phi_E}{dt} \quad \text{(Ampere-Maxwell)}$$

Maxwell's equations can also be written in terms of the scalar and vector potentials$\Phi$ and $\vec{A}$, where$\vec{B} = \nabla \times \vec{A}$ and$\vec{E} = -\nabla\Phi - \partial\vec{A}/\partial t$. The potentials are not unique: the gauge freedom$\vec{A} \to \vec{A} + \nabla\chi$,$\Phi \to \Phi - \partial\chi/\partial t$ leaves the physical fields unchanged.

In the Lorenz gauge ($\nabla\cdot\vec{A} + \mu_0\epsilon_0\partial\Phi/\partial t = 0$), both potentials satisfy the wave equation with sources, making the radiation problem manifestly covariant under Lorentz transformations. This gauge structure is the prototype for all modern gauge theories in particle physics.

4.7 Historical Context

Michael Faraday (1831): Discovered electromagnetic induction and introduced the concept of "lines of force" — the precursor to the modern field concept. Faraday's physical intuition, despite lacking mathematical training, provided the foundation for Maxwell's theory.

James Clerk Maxwell (1865): Published "A Dynamical Theory of the Electromagnetic Field," unifying electricity, magnetism, and optics. Maxwell wrote: "We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

Heinrich Hertz (1887): Experimentally generated and detected electromagnetic waves using spark-gap oscillators and resonant circuits, confirming Maxwell's predictions. He showed that EM waves could be reflected, refracted, and polarized, just like light.

John Henry Poynting (1884): Derived the expression for electromagnetic energy flow, now called the Poynting vector. His theorem provided the mathematical framework for understanding how energy moves through electromagnetic fields.

Oliver Heaviside (1884): Reformulated Maxwell's original 20 equations (in quaternion notation) into the compact four-equation form we use today, using vector calculus notation. He also independently discovered the Poynting vector.

4.7b The Displacement Current: Maxwell's Key Insight

Maxwell's addition of the displacement current term was motivated by a fundamental inconsistency in the original Ampere's law. Consider charging a capacitor: the current flows through the wire but stops at the capacitor plates. Applying Ampere's law to a loop around the wire gives $\oint \vec{B}\cdot d\vec{l} = \mu_0 I$, but a surface passing between the plates encloses no current at all.

Maxwell resolved this by noting that the changing electric field between the plates acts as an effective current:

Displacement Current

$$\vec{J}_d = \epsilon_0\frac{\partial\vec{E}}{\partial t}$$

Between the capacitor plates, $\vec{J}_d$ equals the conduction current in the wire. Adding $\vec{J}_d$ to Ampere's law makes$\nabla\cdot(\nabla\times\vec{B})$ identically zero, restoring consistency with charge conservation:$\nabla\cdot\vec{J} + \partial\rho/\partial t = 0$.

The displacement current is essential for electromagnetic waves. Without it, there would be no coupling between time-varying $\vec{E}$ and $\vec{B}$fields, and no wave propagation. It is remarkable that this purely theoretical addition, motivated by mathematical consistency, led to the prediction of electromagnetic waves and the identification of light as an electromagnetic phenomenon.

4.8 Python Simulation

This simulation visualizes the electromagnetic plane wave, the Poynting vector, the electromagnetic spectrum, and the energy density.

Maxwell's Equations: EM Waves, Poynting Vector, and EM Spectrum

Python

script.py133 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Maxwell's Equations and Electromagnetic Waves
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle("Maxwell's Equations: EM Waves", fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: EM plane wave (E and B fields) ---
ax1 = axes[0, 0]
z = np.linspace(0, 4 * np.pi, 500)
k = 1.0
omega = 1.0
t_snap = 0

E = np.cos(k * z - omega * t_snap)
B = np.cos(k * z - omega * t_snap)

ax1.plot(z / np.pi, E, color='#22d3ee', linewidth=2.5, label='E field (x-component)')
ax1.plot(z / np.pi, B, color='#f97316', linewidth=2.5, label='B field (y-component)', linestyle='--')
ax1.fill_between(z / np.pi, 0, E, color='#22d3ee', alpha=0.1)
ax1.fill_between(z / np.pi, 0, B, color='#f97316', alpha=0.1)
ax1.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax1.set_xlabel('z / pi')
ax1.set_ylabel('Field amplitude (normalized)')
ax1.set_title('EM Plane Wave: E and B in Phase')
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Energy density and Poynting vector ---
ax2 = axes[0, 1]
eps0 = 8.854e-12
mu0 = 4 * np.pi * 1e-7
c = 1.0 / np.sqrt(mu0 * eps0)
E0 = 1.0  # normalized

u_E = 0.5 * np.cos(k * z)**2
u_B = 0.5 * np.cos(k * z)**2
u_total = u_E + u_B
S = np.cos(k * z)**2  # Poynting vector magnitude (normalized)

ax2.plot(z / np.pi, u_E, color='#22d3ee', linewidth=2, label='u_E (electric)', alpha=0.8)
ax2.plot(z / np.pi, u_B, color='#f97316', linewidth=2, label='u_B (magnetic)', alpha=0.8)
ax2.plot(z / np.pi, u_total, color='white', linewidth=2, label='u_total', linestyle='--')
ax2.plot(z / np.pi, S, color='#4ade80', linewidth=2, label='|S| (Poynting)', alpha=0.8)
ax2.axhline(y=0.5, color='gray', linewidth=0.5, linestyle=':', alpha=0.5)
ax2.set_xlabel('z / pi')
ax2.set_ylabel('Normalized density')
ax2.set_title('Energy Density and Poynting Vector')
ax2.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: EM spectrum ---
ax3 = axes[1, 0]
bands = [
    ('Radio', 1e3, 1e9, '#ef4444'),
    ('Microwave', 1e9, 3e11, '#f97316'),
    ('Infrared', 3e11, 4e14, '#eab308'),
    ('Visible', 4e14, 7.5e14, '#4ade80'),
    ('UV', 7.5e14, 3e16, '#22d3ee'),
    ('X-ray', 3e16, 3e19, '#a855f7'),
    ('Gamma', 3e19, 3e22, '#ec4899'),
]

y_pos = 0
for name, f_low, f_high, color in bands:
    ax3.barh(y_pos, np.log10(f_high) - np.log10(f_low), left=np.log10(f_low),
             color=color, alpha=0.7, height=0.6, edgecolor='white', linewidth=0.5)
    ax3.text((np.log10(f_low) + np.log10(f_high)) / 2, y_pos, name,
             ha='center', va='center', color='white', fontsize=8, fontweight='bold')
    y_pos += 1

ax3.set_xlabel('log10(frequency / Hz)')
ax3.set_yticks([])
ax3.set_title('Electromagnetic Spectrum')
ax3.set_xlim(2, 23)

# --- Panel 4: Wave propagation snapshot at different times ---
ax4 = axes[1, 1]
z_prop = np.linspace(0, 8 * np.pi, 500)

for t_val, alpha_val in [(0, 0.3), (np.pi/3, 0.5), (2*np.pi/3, 0.7), (np.pi, 1.0)]:
    E_prop = np.cos(k * z_prop - omega * t_val)
    ax4.plot(z_prop / np.pi, E_prop, linewidth=1.5, alpha=alpha_val, color='#22d3ee',
             label='t = ' + str(round(t_val / np.pi, 2)) + ' pi' if t_val > 0 else 't = 0')

ax4.axhline(y=0, color='white', linewidth=0.3, alpha=0.3)
ax4.set_xlabel('z / pi')
ax4.set_ylabel('E(z, t)')
ax4.set_title('Wave Propagation: Snapshots in Time')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
# Arrow showing propagation direction
ax4.annotate('', xy=(7, 0.8), xytext=(5, 0.8),
            arrowprops=dict(arrowstyle='->', color='#f97316', lw=2))
ax4.text(6, 0.9, 'propagation', color='#f97316', fontsize=9, ha='center')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("MAXWELL'S EQUATIONS: NUMERICAL RESULTS")
print("=" * 60)
print("\n--- Fundamental Constants ---")
print("  epsilon_0 = " + str(eps0) + " F/m")
print("  mu_0 = " + str(round(mu0, 10)) + " H/m")
print("  c = 1/sqrt(mu_0 * eps_0) = " + str(round(c, 1)) + " m/s")
print("  c (exact) = 299792458 m/s")
print("\n--- EM Spectrum ---")
for name, f_low, f_high, _ in bands:
    lam_low = c / f_high
    lam_high = c / f_low
    print("  " + name + ": " + str(round(f_low, 1)) + " - " + str(round(f_high, 1)) + " Hz")
print("\n--- Plane Wave Properties ---")
print("  E and B perpendicular to k and to each other")
print("  |E| = c|B|")
print("  Time-averaged intensity: I = (1/2) eps_0 c E_0^2")
print("  Radiation pressure (absorption): P = I/c")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

Wave equation: $\nabla^2\vec{E} = \mu_0\epsilon_0\partial_t^2\vec{E}$
Speed of light: $c = 1/\sqrt{\mu_0\epsilon_0}$
Poynting: $\vec{S} = \vec{E}\times\vec{B}/\mu_0$
Intensity: $I = \frac{1}{2}\epsilon_0 c E_0^2$
Radiation pressure: $P = I/c$

Key Insights

Displacement current completes Maxwell's equations
EM waves are transverse: E, B, k mutually perpendicular
Equal electric and magnetic energy densities
Light is an electromagnetic wave
Poynting's theorem = energy conservation for fields

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