Applications: Electromagnetic Waves

Learning Objectives

By the end of this section, you will be able to:

1. Recognize the same wave equation $u_{tt} = c^{2}\,u_{xx}$ hiding inside Maxwell's four equations.
2. Derive the wave equation for the electric and magnetic fields in vacuum starting from Faraday's and Ampère–Maxwell's laws.
3. Predict the speed of light from two laboratory constants $\mu_0$ and $\varepsilon_0$.
4. Write down a plane-wave solution and check that $E\perp B\perp \hat{k}$ and $|E|/|B|=c$.
5. Distinguish linear, circular and elliptical polarization and explain each as a phase relation between two orthogonal components.
6. Compute the energy density and Poynting vector of a plane wave.
7. Connect the universal wave equation to radio, fibre-optic communication, medical imaging, and wireless power.

The Big Picture: From Strings to Light

“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.” — James Clerk Maxwell, 1864

For six chapters we have been studying one equation written in many disguises:

On a guitar string, $u$ was the vertical displacement and $c$ was the wave speed $\sqrt{T/\rho}$. On a drumhead $u$ was the height of the membrane. In a sound wave it was air pressure. In each case the same shape — a travelling sinusoid — emerged because the second time-derivative of $u$ balanced a restoring force proportional to the second space-derivative.

In 1864 James Clerk Maxwell discovered that light itself obeys exactly this equation. The displacement $u$ is replaced by an electric field $\mathbf{E}$, the “tension” by the magnetic constant $\mu_0$, the “density” by the electric constant $\varepsilon_0$, and the wave speed by

Historical Context: Maxwell's Synthesis

Three giants laid the groundwork. Carl Friedrich Gauss formulated the divergence laws relating field flux to enclosed sources. Michael Faraday showed by experiment that a time-varying magnetic field induces an electric field. And André-Marie Ampère related currents to circulating magnetic fields. Each of these laws was beautiful, but they sat beside each other like separate puzzle pieces.

In 1861 Maxwell noticed something subtle: Ampère's law as it stood violated charge conservation when applied to a charging capacitor. To fix it he added a term proportional to the time derivative of the electric field — the famous displacement current. That single correction made the equations internally consistent and, completely unexpectedly, produced a wave equation. The speed of that wave matched Fizeau and Foucault's 1850s measurement of the speed of light. 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.”

Twenty-three years later, in 1887, Heinrich Hertz generated and detected radio waves in his laboratory — the first artificial electromagnetic waves — using a spark gap to launch them and a wire loop to receive them. Within a decade Marconi was sending radio across oceans. The wave equation had become a technology.

The Question That Launched a Revolution

Here is the question we want to answer in this section, framed as concretely as possible:

We already know the wave equation describes vibrations on strings and drums. Does it also describe the propagation of electric and magnetic disturbances through empty space?

The answer is yes, and the proof is almost suspicious in its elegance: the wave equation simply falls out when you take the curl of Faraday's law and substitute the Ampère–Maxwell law into it. There is no force balance, no continuum-mechanics derivation. The mathematics of fields alone is enough.

Maxwell's Equations in Empty Space

Strip away charges and currents (we will study a region of vacuum far from any source). Maxwell's four equations become:

Deriving the Wave Equation for E and B

We can squeeze the wave equation out of Maxwell's equations in about five lines. Take the curl of Faraday's law:

Use the vector identity $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^{2}\mathbf{E}$ on the left, and substitute Ampère–Maxwell on the right:

In vacuum $\nabla\cdot\mathbf{E}=0$, so the first term on the left vanishes. Rearrange:

This is the wave equation, component by component. Each cartesian component of $\mathbf{E}$ obeys $u_{tt} = c^{2}\,\nabla^{2}u$ with $c^{2} = 1/(\mu_0\varepsilon_0)$. The same manipulation starting from the curl of Ampère–Maxwell gives the identical equation for $\mathbf{B}$.

The Speed Emerges: c = 1/√(μ₀ε₀)

Let's actually compute the number. The two constants were already known in the lab decades before Maxwell:

Compute:

That number — to four digits — is the speed of light measured by Fizeau in 1849 with a rotating toothed wheel: 3.13 × 10⁸ m/s. Use the slider below to see how the speed changes when you replace vacuum by a material with different relative permittivity and permeability.

Plane Waves: The Simplest Solution

D'Alembert taught us that any function of $x - ct$ solves the 1D wave equation. The simplest interesting choice is a sinusoid:

Substitute into $E_{tt} = c^{2} E_{xx}$:

The wave equation enforces a relationship between frequency and wave number — the dispersion relation. In vacuum every plane wave moves at the same speed $c$, regardless of wavelength. (In a medium with frequency-dependent $\varepsilon$ — like glass — different colors travel at different speeds. That is what splits white light into a rainbow.)

Three relations always hold for a vacuum plane wave:

• Phase speed: $v_p = \omega/k = c$.
• Field amplitudes: $|\mathbf{E}|/|\mathbf{B}| = c$.
• Geometry: $\mathbf{E}$, $\mathbf{B}$, and the propagation direction $\hat{\mathbf{k}}$ form a right-handed triad — $\hat{\mathbf{k}} = \hat{\mathbf{E}}\times\hat{\mathbf{B}}$.

Visualizing the Coupled E–B Wave

Slide the wavelength and amplitude controls below. Notice that the crests of $\mathbf{E}$ (orange) line up perfectly with the crests of $\mathbf{B}$ (cyan) — they are in phase, not 90° apart as you might guess from inductor/capacitor intuition. They oscillate at right angles to each other and both perpendicular to the direction of travel: a textbook transverse wave.

Polarization: The Tip of E Draws a Curve

Because $\mathbf{E}$ is perpendicular to the propagation direction, it has two independent components in the transverse plane — say $E_y$ and $E_z$. The full electric field of a plane wave is:

The two amplitudes $A_y, A_z$ and the relative phase $\delta$ are three free numbers. They decide the polarization state — the curve the tip of $\mathbf{E}$ traces when you look along the propagation direction.

Use the explorer to feel the relationship. Pick a preset, then drag the amplitudes and phase to interpolate between states.

Energy and the Poynting Vector

An EM wave carries energy. The instantaneous energy density (in J/m³) is the sum of an electric and a magnetic contribution:

For a plane wave with $|\mathbf{B}| = |\mathbf{E}|/c$, the two terms are equal — the wave splits energy 50/50 between electric and magnetic forms.

Energy flows in the direction of propagation at speed $c$. The flow per unit area is the Poynting vector:

The right-hand rule on $\mathbf{E}\times\mathbf{B}$ always points along $\hat{\mathbf{k}}$ — the direction the wave is going. Time-averaging over one cycle for a sinusoidal plane wave gives the convenient form $\langle S\rangle = \tfrac{1}{2}\varepsilon_0 c\,E_0^{2}$. The sun delivers about $1361~\text{W/m}^{2}$ of $\langle S\rangle$ at the top of Earth's atmosphere — the “solar constant”.

The Electromagnetic Spectrum

The wave equation has no preferred frequency — only the dispersion relation $\omega = ck$. So solutions exist at every wavelength, from kilometres-long radio waves to femtometre-short gamma rays. They are all the same kind of object; the labels (“radio”, “visible”, “X-ray”) only reflect how we generate, detect, and use them.

Worked Example: Plane Wave at 1 GHz

Work this through with pen and paper before you peek at the solution. It pulls together every formula in the section.

Python Implementation

Now let's confirm the entire chain numerically. We will construct a plane wave by hand, plot E and B side by side, and use finite differences to check that $\partial^{2}E/\partial t^{2}$ really equals $c^{2}\,\partial^{2}E/\partial x^{2}$ at every grid point.

1import numpy as np
2import matplotlib.pyplot as plt
3
4# Fundamental constants
5mu0  = 4 * np.pi * 1e-7        # vacuum permeability  [H/m]
6eps0 = 8.854187817e-12         # vacuum permittivity  [F/m]
7c    = 1.0 / np.sqrt(mu0 * eps0)   # predicted speed of light [m/s]
8
9print(f"c (from Maxwell) = {c:.6e} m/s")
10print(f"c (measured)     = 2.997924e+08 m/s")
11
12# A plane wave traveling in +x with E along y
13#   E_y(x, t) = E0 * cos(k x - omega t)
14#   B_z(x, t) = (E0 / c) * cos(k x - omega t)
15
16E0     = 1.0                    # amplitude of E [V/m]
17lam    = 0.6                    # wavelength [m]   -> radio range
18k      = 2 * np.pi / lam        # wave number  [rad/m]
19omega  = c * k                  # dispersion relation: omega/k = c
20freq   = omega / (2 * np.pi)    # ordinary frequency [Hz]
21
22# Discrete grid
23x  = np.linspace(0.0, 2.0 * lam, 400)
24dx = x[1] - x[0]
25dt = 0.5 * dx / c               # well inside the CFL limit (dt < dx/c)
26t_grid = np.arange(0.0, 5e-9, dt)   # 5 nanoseconds total
27
28# Snapshot the wave at a single time
29t0  = 1e-9                      # 1 ns
30Ey  = E0 * np.cos(k * x - omega * t0)
31Bz  = (E0 / c) * np.cos(k * x - omega * t0)
32
33# ── Verify the wave equation numerically: u_tt = c^2 u_xx ──────────
34def second_derivative_x(u, dx):
35    """Central difference for d^2 u / dx^2 on a 1D grid."""
36    d2u = np.zeros_like(u)
37    d2u[1:-1] = (u[2:] - 2 * u[1:-1] + u[:-2]) / dx**2
38    return d2u
39
40# Take a tiny time step to estimate u_tt by finite differences
41Ey_minus = E0 * np.cos(k * x - omega * (t0 - dt))
42Ey_plus  = E0 * np.cos(k * x - omega * (t0 + dt))
43
44u_tt = (Ey_plus - 2 * Ey + Ey_minus) / dt**2
45u_xx = second_derivative_x(Ey, dx)
46
47residual = u_tt - c**2 * u_xx
48print(f"max |u_tt - c^2 u_xx| = {np.max(np.abs(residual[2:-2])):.3e}")
49
50# Side-by-side plots
51fig, axes = plt.subplots(2, 1, figsize=(10, 6), sharex=True)
52
53axes[0].plot(x, Ey, color="#f97316", linewidth=2, label="E_y(x, t)")
54axes[0].plot(x, Bz * c, color="#06b6d4", linewidth=2,
55             linestyle="--", label="c · B_z(x, t)")
56axes[0].axhline(0, color="#94a3b8", linewidth=0.5)
57axes[0].set_ylabel("Field amplitude")
58axes[0].set_title(f"Plane wave at t = {t0*1e9:.1f} ns, λ = {lam} m")
59axes[0].legend(loc="upper right")
60axes[0].grid(True, alpha=0.3)
61
62axes[1].plot(x, c**2 * u_xx, color="#22c55e", linewidth=2,
63             label="c² ∂²E/∂x²")
64axes[1].plot(x, u_tt, color="#a855f7", linewidth=2, linestyle="--",
65             label="∂²E/∂t²")
66axes[1].set_xlabel("position x [m]")
67axes[1].set_ylabel("second derivative")
68axes[1].set_title("Wave equation check (curves should coincide)")
69axes[1].legend(loc="upper right")
70axes[1].grid(True, alpha=0.3)
71
72plt.tight_layout()
73plt.show()
74
75# Print frequency and time period
76print(f"\nλ = {lam} m,  f = {freq/1e6:.3f} MHz,  T = {1/freq*1e9:.3f} ns")
77print(f"E0 / B0 = c = {E0 / (E0/c):.6e} m/s  (ratio always = c)")

PyTorch Implementation

Finite differences are nice, but they leak truncation error. Let's use automatic differentiation to compute the second derivatives exactly. PyTorch's autograd treats every operation as a node in a graph, applies the chain rule symbolically, and gives us results down to machine epsilon.

1import torch
2import math
3
4# Constants
5mu0  = 4 * math.pi * 1e-7
6eps0 = 8.854187817e-12
7c    = 1.0 / math.sqrt(mu0 * eps0)
8
9# Plane-wave parameters
10lam   = 0.6                         # wavelength [m]
11k     = 2 * math.pi / lam           # wave number
12omega = c * k                       # dispersion relation
13
14def E(x, t):
15    """Plane-wave E component: E_y(x, t) = cos(k x - omega t)."""
16    return torch.cos(k * x - omega * t)
17
18# Pick a point in spacetime where we will check the wave equation
19x = torch.tensor(0.2, requires_grad=True)
20t = torch.tensor(1e-9, requires_grad=True)
21
22u = E(x, t)
23
24# First derivatives (held in the graph for further differentiation)
25du_dx, du_dt = torch.autograd.grad(
26    outputs=u, inputs=(x, t),
27    create_graph=True,
28)
29
30# Second derivatives
31d2u_dx2 = torch.autograd.grad(du_dx, x, create_graph=True)[0]
32d2u_dt2 = torch.autograd.grad(du_dt, t, create_graph=True)[0]
33
34# Wave-equation residual:  u_tt - c^2 u_xx  should be 0
35residual = d2u_dt2 - c**2 * d2u_dx2
36
37print(f"u            = {u.item():.6e}")
38print(f"u_xx         = {d2u_dx2.item():.6e}")
39print(f"u_tt         = {d2u_dt2.item():.6e}")
40print(f"c^2 u_xx     = {(c**2 * d2u_dx2).item():.6e}")
41print(f"residual     = {residual.item():.3e}    (≈ 0 means the wave eq. holds)")
42
43# ── Symbolic identity: u_tt + c^2 u_xx = 0 for cos(kx-omega t) iff omega = c k
44ratio = (-d2u_dt2 / d2u_dx2).sqrt()
45print(f"sqrt(-u_tt / u_xx) = {ratio.item():.6e}  =  c = {c:.6e}")

Applications

Common Pitfalls

Test Your Understanding

Five quick questions to check the intuition. Tap an answer to see the explanation.

Summary

We took the universal wave equation $u_{tt} = c^{2}\,u_{xx}$ — the same one that governs strings and drums — and showed that Maxwell's four field equations, applied in vacuum, force both $\mathbf{E}$ and $\mathbf{B}$ to obey it with wave speed

Three lessons to carry forward:

1. Universality. The wave equation is not about strings or drums or light — it is the mathematical signature of any system where a restoring response is proportional to spatial curvature. Recognizing it lets you import every tool you have built (D'Alembert solution, separation of variables, normal modes, Fourier series, finite differences, autograd) directly into a new physical domain.
2. Light is not special. It is just the EM solution at frequencies that happen to interact with the rods and cones in our eyes. Radio waves, microwaves, X-rays, and gamma rays are the same kind of object obeying the same wave equation.
3. From two lab constants, the speed of light. The number $c = 1/\sqrt{\mu_0\varepsilon_0}$ is the most spectacular “unintended consequence” in the history of physics: two pieces of static-electricity bookkeeping combine to predict the speed of light to four-digit accuracy.