The Electromagnetic Wave Equation
How do light, radio waves, and X-rays travel through empty space? In this lesson, we derive the wave equation from Maxwell's equations and build intuition by visualizing every concept with live Python code you can edit and run.
What you will learn: Starting from Maxwell's four equations, we show that electric and magnetic fields satisfy a wave equation whose solutions travel at the speed of light. We then explore superposition, interference, standing waves, and wave packets - all with interactive code labs.
Prerequisites: Basic calculus (partial derivatives), familiarity with vectors (curl, divergence). No coding experience required - just edit the numbers and see what happens.
Lesson Outline
1.Maxwell's Equations - The Starting Point
2.Deriving the Wave Equation
3.The Plane Wave Solution
4.Visualizing Electric & Magnetic Fields
5.Superposition & Interference
6.Standing Waves
7.Wave Packets & Group Velocity
8.Key Equations Summary
1. Maxwell's Equations - The Starting Point
Everything in classical electromagnetism follows from four equations published by James Clerk Maxwell in 1865. In free space (no charges, no currents), they simplify to a beautifully symmetric set:
Maxwell's Equations in Free Space
Gauss's law - no free charges create or terminate field lines in vacuum
Gauss's law for magnetism - magnetic monopoles do not exist
Faraday's law - a changing magnetic field induces an electric field
Ampere-Maxwell law - a changing electric field creates a magnetic field
Key Insight
Look at equations III and IV together: a changing B creates E, and a changing E creates B. This mutual induction is the engine that sustains electromagnetic waves - the fields regenerate each other as they propagate through space.
Physical Constants
- ε₀ = 8.854 × 10⁻¹² F/m - permittivity of free space
- μ₀ = 4π × 10⁻⁷ H/m - permeability of free space
- c = 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s - speed of light
Notation
- ∇ · - divergence (how much a field spreads out)
- ∇ × - curl (how much a field circulates)
- ∂/∂t - partial derivative with respect to time
The Misconception
Sound needs air, water waves need water - so light must need the “luminiferous aether,” a medium filling all space.
The Reality
The Michelson-Morley experiment (1887) proved no aether exists. Maxwell's equations show E and B regenerate each other - the fields are the medium.
2. Deriving the Wave Equation
We now show that Maxwell's equations in free space imply that both E and B satisfy a wave equation. This is one of the most beautiful results in all of physics.
Step-by-Step Derivation
Step 1: Take the curl of Faraday's law
Apply ∇× to both sides of equation III:
Step 2: Substitute Ampère-Maxwell law
Replace ∇ × B using equation IV:
Step 3: Use the vector identity
The vector identity ∇ × (∇ × E) = ∇(∇ · E) - ∇²E, and since ∇ · E = 0 (Gauss's law):
Step 4: The wave equation
Rearranging gives:
This is the electromagnetic wave equation. The same derivation with the roles of E and B swapped gives an identical equation for B.
The Speed of Light Falls Out
Compare with the standard wave equation ∇²f = (1/v²) ∂²f/∂t². The wave speed is v = 1/√(μ₀ε₀). Plug in the measured values of μ₀ and ε₀, and you get v ≈ 3 × 10⁸ m/s - the speed of light. This was Maxwell's remarkable prediction: light is an electromagnetic wave.
The Misconception
Students treat the electromagnetic wave equation as a special formula for visible light.
The Reality
The same equation describes radio waves, microwaves, infrared, visible, UV, X-rays, and gamma rays. The only difference is the frequency. The entire electromagnetic spectrum is one phenomenon.
3. The Plane Wave Solution
The simplest solution to the wave equation is the plane wave - a sinusoidal oscillation that extends infinitely in the transverse directions and propagates in one direction:
E₀ - amplitude (maximum field strength)
k = 2π/λ - wavenumber (spatial frequency)
ω = 2πf - angular frequency
λ - wavelength
f - frequency (cycles per second)
φ - initial phase
The dispersion relation connects them: ω = ck, so c = fλ
To verify that this is indeed a solution, substitute into the wave equation. The spatial second derivative gives -k²E, and the temporal second derivative gives -ω²E. The equation ∇²E = μ₀ε₀ ∂²E/∂t² becomes -k² = -μ₀ε₀ω², which is satisfied when ω/k = 1/√(μ₀ε₀) = c.
Live Lab: Plane Electromagnetic Wave
Visualize E(x,t) = E0 sin(kx - ωt) as a function of position at a fixed time. Try changing the wavelength and amplitude.
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Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
4. Visualizing Electric & Magnetic Fields
In an electromagnetic wave, the electric field E and the magnetic field B are always perpendicular to each other and to the direction of propagation. This is called a transverse wave.
Transversality
Both E and B oscillate perpendicular to the propagation direction k. If the wave travels along x, then E might point along y and B along z.
In Phase
In vacuum, E and B reach their maxima and zero crossings at the same points in space and time. They are perfectly in phase.
Amplitude Ratio
The magnitudes are related by E₀ = cB₀. Since c is huge, E is much larger in SI units (V/m vs Tesla).
The Misconception
Because E and B are perpendicular in space, students conclude they must also be shifted 90° in time (like in an LC circuit).
The Reality
In a plane wave in vacuum, E and B are in phase - they reach zero and peak at the same locations. The spatial perpendicularity is about direction, not timing.
Live Lab: E and B Fields of a Plane Wave
The electric and magnetic fields oscillate perpendicular to each other and to the propagation direction. Edit the code to change the phase or see it at different times.
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Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
5. Superposition & Interference
The wave equation is linear - if E₁ and E₂ are individual solutions, then E₁ + E₂ is also a solution. This is the principle of superposition, and it explains all interference phenomena.
Constructive Interference
When two waves are in phase (crests aligned with crests), their amplitudes add. The resulting wave has larger amplitude.
Destructive Interference
When two waves are out of phase (crests aligned with troughs), they cancel. If equal amplitude, complete cancellation.
Beats
When two waves with slightly different frequencies interfere, the result oscillates at the average frequency with an amplitude that varies slowly at the beat frequency f_beat = |f₁ - f₂|. This is exactly what you hear when tuning a guitar string against a reference.
Live Lab: Superposition of Two Waves
Two waves of slightly different frequencies produce beats. Change f2 to see how the beat pattern changes.
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Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
6. Standing Waves
When two identical waves travel in opposite directions, their superposition creates a standing wave - a pattern that oscillates in time but does not propagate.
Mathematical Form
Two counter-propagating plane waves:
Their sum, using the identity sin(a) + sin(b) = 2 sin((a+b)/2) cos((a-b)/2):
Nodes
Points where sin(kx) = 0, so the field is always zero. Located at x = nλ/2.
Antinodes
Points where |sin(kx)| = 1, so the field reaches maximum amplitude. Located at x = (2n+1)λ/4.
Standing waves are fundamental to resonators, laser cavities, and musical instruments. A laser cavity of length L supports only wavelengths satisfying L = nλ/2 - the resonance condition.
Live Lab: Standing Wave
A standing wave forms when two counter-propagating waves interfere. Notice the nodes (zero amplitude) and antinodes (maximum amplitude).
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Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
7. Wave Packets & Group Velocity
A single plane wave extends infinitely - it has a well-defined frequency but no localization. Real signals (laser pulses, photons, radio bursts) are wave packets: superpositions of many frequencies that form a localized pulse.
Phase Velocity vs Group Velocity
Phase Velocity (v_p)
The speed at which individual wave crests move:
In vacuum, v_p = c for all frequencies.
Group Velocity (v_g)
The speed at which the envelope (and energy) moves:
In a dispersive medium, v_g ≠ v_p. Information travels at v_g.
Heisenberg's Uncertainty Principle (Classical Version)
A wave packet cannot be simultaneously narrow in space and narrow in frequency. Mathematically: Δx · Δk ≥ 1/2. A shorter pulse requires a broader range of frequency components. This is a mathematical consequence of Fourier analysis - it applies to water waves and radio signals too, not just quantum mechanics.
Live Lab: Wave Packet & Group Velocity
A wave packet is a superposition of many frequencies forming a localized pulse. Try changing the number of components or the spread Δk.
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Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
Live Lab: Phase Velocity vs Group Velocity
In a dispersive medium, the phase velocity and group velocity differ. Watch how the envelope (group) moves at a different speed than the carrier (phase).
Preparing preview
Running once automatically on first load
Learn by Experimenting
This is a safe playground for learning! Try changing:
- • Colors: Modify color values to see different palettes
- • Numbers: Adjust sizes, positions, or data ranges
- • Labels: Update titles, axis names, or legends
Edit the code, run it, then open the full data visualization tool to continue with your own dataset.
The Half-Truth
In anomalous dispersion regions, v_g can mathematically exceed c or even become negative. Students conclude information can travel faster than light.
The Full Picture
When v_g > c, the group velocity ceases to represent the signal velocity. The actual signal front always travels at ≤ c (the Sommerfeld-Brillouin result). No information violates causality.
8. Key Equations Summary
| Concept | Equation |
|---|---|
| Wave equation | ∇²E = μ₀ε₀ ∂²E/∂t² |
| Speed of light | c = 1/√(μ₀ε₀) |
| Plane wave | E = E₀ sin(kx - ωt) |
| Dispersion relation | ω = ck |
| Standing wave | E = 2A sin(kx) cos(ωt) |
| Phase velocity | v_p = ω/k |
| Group velocity | v_g = dω/dk |
| E-B relationship | E₀ = cB₀ |
Visualize It Yourself
Upload your own data or describe any physics simulation in plain English. Plotivy generates the Python code and the plot instantly - from wave optics to quantum mechanics.
Polarization of Light
Linear, circular, and elliptical polarization. Jones vectors, Malus' law, and waveplates - all with interactive simulations.
Coming soon