Chapter 9

Maxwell's Equations

The complete set of equations governing all classical electromagnetic phenomena.

9.1 The Complete Maxwell Equations

Maxwell's genius was to add the displacement current$\epsilon_0 \partial\mathbf{E}/\partial t$ to Ampere's law, completing the system. The four equations in SI units (differential form):

I. Gauss's Law (E)

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

Electric field sourced by charge density

II. No Magnetic Monopoles

$$\nabla \cdot \mathbf{B} = 0$$

Magnetic field lines never begin or end

III. Faraday's Law

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

Changing B induces curling E

IV. Ampere-Maxwell Law

$$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$

Currents and changing E source curling B

Together with the Lorentz force law $\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$, Maxwell's equations completely describe all classical electromagnetic phenomena — from the attraction of magnets to the propagation of light across the universe.

9.2 The Displacement Current

Ampere's law $\nabla \times \mathbf{B} = \mu_0\mathbf{J}$ is inconsistent with charge conservation when applied to a capacitor being charged. The divergence of$\nabla\times\mathbf{B}$ must be zero (curl is divergence-free), but$\nabla\cdot\mathbf{J} = -\partial\rho/\partial t \neq 0$ in general.

Maxwell added the displacement current to restore consistency:

$$\mathbf{J}_d = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$$$\nabla \cdot (\mathbf{J} + \mathbf{J}_d) = \nabla\cdot\mathbf{J} + \epsilon_0\nabla\cdot\frac{\partial\mathbf{E}}{\partial t} = -\frac{\partial\rho}{\partial t} + \frac{\partial\rho}{\partial t} = 0 \checkmark$$

9.3 The Wave Equation for Light

In vacuum ($\rho = 0$, $\mathbf{J} = 0$), taking the curl of Faraday's law and using Ampere-Maxwell:

$$\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t}(\nabla \times \mathbf{B}) = -\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$$$\nabla(\underbrace{\nabla\cdot\mathbf{E}}_{=0}) - \nabla^2\mathbf{E} = -\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$

$$\boxed{\nabla^2\mathbf{E} = \mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} = \frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}}$$

This is the wave equation! The speed of propagation is$c = 1/\sqrt{\mu_0\epsilon_0} = 2.998\times10^8\,\text{m/s}$ — the speed of light. Maxwell identified light as an electromagnetic wave.

9.3.1 Potentials and Gauge

We write $\mathbf{B} = \nabla\times\mathbf{A}$ and$\mathbf{E} = -\nabla V - \partial\mathbf{A}/\partial t$. In the Lorenz gauge ($\nabla\cdot\mathbf{A} + \mu_0\epsilon_0\dot{V} = 0$), the potentials satisfy decoupled wave equations:

$$\Box^2 V \equiv \left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)V = -\frac{\rho}{\epsilon_0}$$$$\Box^2 \mathbf{A} = -\mu_0\mathbf{J}$$

where $\Box^2$ is the d'Alembertian operator. These are inhomogeneous wave equations — the sources $\rho$ and $\mathbf{J}$ drive the potential waves.

Simulation: FDTD EM Wave Propagation

1D Finite-Difference Time-Domain (FDTD) simulation of Maxwell's equations, demonstrating Gaussian pulse propagation at speed $c$ and the correct dispersion relation $\omega = ck$.

Maxwell FDTD: EM Wave Propagation

1D FDTD simulation of Maxwell's equations: Gaussian pulse propagating at c, dispersion relation ω=ck.

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

# ──────────────────────────────────────────────
# Wave equation from Maxwell's equations:
# Demonstrate that E and B satisfy the wave eq.
# Show 1D EM wave propagation (FDTD scheme)
# ──────────────────────────────────────────────
c = 3e8          # speed of light [m/s]
mu0 = 4*np.pi*1e-7
eps0 = 1/(mu0*c**2)

# 1D FDTD (Yee scheme) in vacuum
# E_y, B_z components; wave propagates in x
Nx = 400
dx = 1e-3    # 1 mm cell size
dt = dx / (2*c)   # Courant condition: dt < dx/c

Ey = np.zeros(Nx)
Bz = np.zeros(Nx)

n_steps = 1200
# Source: Gaussian pulse at center
src = Nx // 4
sigma_t = 40   # width in time steps
snapshots = {}

for n in range(n_steps):
    # Update B from curl E (Faraday)
    Bz[:-1] -= dt/dx * (Ey[1:] - Ey[:-1])

# Update E from curl B (Ampere-Maxwell)
    Ey[1:] += c**2 * dt/dx * (Bz[1:] - Bz[:-1])

# Gaussian source injection
    Ey[src] += np.exp(-0.5*((n - 3*sigma_t)/sigma_t)**2)

# ABC boundaries (absorbing)
    Ey[0]  = Ey[1]
    Ey[-1] = Ey[-2]

if n in [200, 500, 1100]:
        snapshots[n] = Ey.copy()

x = np.arange(Nx) * dx * 100   # cm

fig, axes = plt.subplots(2, 2, figsize=(13, 9))
fig.patch.set_facecolor('#0a0a1a')
colors = ['#60a5fa', '#f59e0b', '#34d399']
for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

for idx, (step, Ey_snap) in enumerate(snapshots.items()):
    axes[0, idx % 2 if idx < 2 else 0].plot(x, Ey_snap, color=colors[idx], lw=2, label=f't = {step*dt*1e9:.1f} ns')
    axes[0, idx % 2 if idx < 2 else 0].set_xlabel('x [cm]', color='#94a3b8')
    axes[0, idx % 2 if idx < 2 else 0].set_ylabel('$E_y$ [arb]', color='#94a3b8')
    axes[0, idx % 2 if idx < 2 else 0].tick_params(colors='#94a3b8')
    axes[0, idx % 2 if idx < 2 else 0].grid(alpha=0.2, color='#334155')
axes[0,0].set_title('EM Pulse Propagation (FDTD)', color='white', fontsize=11)
axes[0,0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)

# Last snapshot on axes[0,1]
axes[0,1].plot(x, list(snapshots.values())[2], color='#34d399', lw=2, label=f't={list(snapshots.keys())[2]*dt*1e9:.1f} ns')
axes[0,1].set_xlabel('x [cm]', color='#94a3b8'); axes[0,1].set_ylabel('$E_y$ [arb]', color='#94a3b8')
axes[0,1].set_title('Late-time Wave Profile', color='white', fontsize=11)
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,1].grid(alpha=0.2, color='#334155')

# Dispersion relation: omega vs k
k_arr = np.linspace(0, 2*np.pi/dx/4, 300)
omega_arr = c * k_arr
omega_num = 2/dt * np.arcsin(c*dt/(2*dx) * 2*np.abs(np.sin(k_arr*dx/2)))
axes[1,0].plot(k_arr/1000, omega_arr/1e12, color='#60a5fa', lw=2, label='Analytic: ω=ck')
axes[1,0].plot(k_arr/1000, omega_num/1e12, 'r--', lw=2, label='FDTD numerical')
axes[1,0].set_xlabel('k [1/mm]', color='#94a3b8'); axes[1,0].set_ylabel('ω [THz]', color='#94a3b8')
axes[1,0].set_title('Dispersion Relation ω vs k', color='white', fontsize=11)
axes[1,0].tick_params(colors='#94a3b8')
axes[1,0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[1,0].grid(alpha=0.2, color='#334155')

# Speed of light check
lambda_vals = np.linspace(400e-9, 700e-9, 100)
f_vals = c / lambda_vals
axes[1,1].plot(lambda_vals*1e9, f_vals/1e12, color='#a78bfa', lw=2)
axes[1,1].set_xlabel('Wavelength λ [nm]', color='#94a3b8'); axes[1,1].set_ylabel('Frequency f [THz]', color='#94a3b8')
axes[1,1].set_title('Visible Light: f = c/λ', color='white', fontsize=11)
axes[1,1].tick_params(colors='#94a3b8')
axes[1,1].fill_between(lambda_vals*1e9, 0, f_vals/1e12, alpha=0.15, color='#a78bfa')
axes[1,1].grid(alpha=0.2, color='#334155')

plt.tight_layout()
plt.savefig('maxwell_fdtd.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
print(f"Speed of light: c = {c:.3e} m/s = 1/sqrt(mu0*eps0)")
print(f"Verification: 1/sqrt(mu0*eps0) = {1/np.sqrt(mu0*eps0):.3e} m/s")

Click Run to execute the Python code

First run will download Python environment (~15MB)

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