Maxwell's Equations

A 3D Visual Journey Through Electromagnetism

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James Clerk Maxwell

Scottish physicist (1831–1879) who unified electricity, magnetism, and optics.

1855 — On Faraday's Lines of Force

1861 — Introduces the displacement current

1865 — Predicts EM waves travel at the speed of light

1873 — A Treatise on Electricity and Magnetism

"The most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics."
— Richard Feynman

The Four Equations

Each connects electric fields E, magnetic fields B, charges ρ, and currents J.

I. Gauss (E)

$$\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}$$

II. Gauss (B)

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

III. Faraday

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

IV. Ampère–Maxwell

$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}$$

Divergence (∇·F)

$$\nabla\cdot\mathbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$

Measures how much a field spreads out from a point
Positive → source (like a sprinkler)
Negative → sink (like a drain)
Zero → incompressible flow, no net outflow

Watch the 3D particle fountain: particles stream radially outward from a source — that's positive divergence in action.

Curl (∇×F)

$$\nabla\times\mathbf{F}=\begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\\partial_x&\partial_y&\partial_z\\F_x&F_y&F_z\end{vmatrix}$$

Measures circulation around a point
Nonzero curl → rotation / vortex present
Zero curl → irrotational (conservative) field

The 3D vortex shows particles spiraling around an axis — that's nonzero curl.

I. Gauss's Law for Electricity

Differential

$$\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}$$

Integral

$$\oint_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{enc}}}{\varepsilon_0}$$

E — Electric field (N/C)
ρ — Volume charge density (C/m³)
ε₀ — Permittivity of free space ≈ 8.85×10⁻¹² F/m
Field lines originate on + charges, terminate on − charges

Gauss's Law — Physical Meaning

Enclose a charge in any closed surface. The total flux depends only on the enclosed charge.

Key Insights

Surface shape doesn't matter — only enclosed charge
No charge inside → net flux = 0
Recovers Coulomb's law for a sphere:

$$E(4\pi r^2)=\frac{q}{\varepsilon_0}\;\Rightarrow\;E=\frac{q}{4\pi\varepsilon_0 r^2}$$

The 3D view shows flux arrows piercing a transparent Gaussian sphere.

II. Gauss's Law for Magnetism

Differential

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

Integral

$$\oint_S\mathbf{B}\cdot d\mathbf{A}=0$$

No magnetic monopoles — no isolated N or S poles
B-field lines always form closed loops
Every line leaving a surface must re-enter it
Cut a magnet → two smaller magnets, not isolated poles

The Monopole Mystery

	Electric	Magnetic
Monopoles?	Yes (charges)	Never observed
Divergence	$$\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$$	$$\nabla\cdot\mathbf{B}=0$$
Lines	Begin/end on charges	Always closed loops

If monopoles existed:

$$\nabla\cdot\mathbf{B}=\mu_0\rho_m$$

Dirac (1931): one monopole → explains charge quantization
GUTs predict them, but none detected yet

III. Faraday's Law of Induction

Differential

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

Integral

$$\oint_C\mathbf{E}\cdot d\mathbf{l}=-\frac{d\Phi_B}{dt}$$

A changing B field creates a circulating E field
The minus sign is Lenz's law: induced E opposes the change
Foundation of generators, transformers, induction

Faraday in Action — The Generator

Three ways to change magnetic flux:

Change B magnitude — vary field strength
Change area — expand or contract the loop
Change angle — rotate the loop in the field

Applications

Generators — rotating coil in B field → AC voltage
Transformers — changing I in primary → voltage in secondary
Wireless charging — oscillating flux couples coils

The 3D coil rotates inside the B field — watch the EMF graph respond in real time.

IV. Ampère–Maxwell Law

Differential

$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}$$

Integral

$$\oint_C\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\text{enc}}+\mu_0\varepsilon_0\frac{d\Phi_E}{dt}$$

μ₀J — currents produce circulating B (original Ampère)
μ₀ε₀∂E/∂t — changing E also produces B (Maxwell's addition)

Displacement Current

A charging capacitor: current flows in the wire but stops at the plates.

The Problem

Surface through wire: $$\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I$$ ✓
Surface between plates: $$\oint\mathbf{B}\cdot d\mathbf{l}=0$$ ✗

Maxwell's Fix

$$J_d=\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}$$

The changing E between plates acts as a "virtual current" — continuity restored.

Deriving Electromagnetic Waves

In free space (ρ=0, J=0), take the curl of Faraday's law:

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

Substitute Ampère–Maxwell (J=0):

$$\nabla^2\mathbf{E}=\mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$

This is the wave equation! With speed:

$$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}\approx 3\times10^8\;\text{m/s}$$

Electromagnetic Wave

E and B oscillate perpendicular to each other and to the direction of propagation.

$$\mathbf{E}=E_0\sin(kz-\omega t)\,\hat{x}$$ $$\mathbf{B}=B_0\sin(kz-\omega t)\,\hat{y}$$

Transverse wave — E⊥B⊥propagation
$$E_0/B_0=c$$ — fixed ratio
Self-sustaining: changing E → B → E → ...
No medium needed — propagates through vacuum

The 3D wave shows E (blue, vertical) and B (pink, horizontal) in phase, propagating along z.

The Speed of Light

Maxwell computed c from two lab-measurable constants:

$$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=299{,}792{,}458\;\text{m/s}$$

Constant	Value	Measured from
μ₀	4π×10⁻⁷ T·m/A	Force between wires
ε₀	8.854×10⁻¹² F/m	Capacitor experiments
c	2.998×10⁸ m/s	Computed from μ₀ε₀

Light is an electromagnetic wave. Maxwell unified optics with electromagnetism.

The Electromagnetic Spectrum

All EM waves obey Maxwell's equations — they differ only in wavelength and frequency.

$$c=\lambda f$$

Radio — communication, broadcasting

Microwaves — cooking, radar, WiFi

Infrared — thermal imaging

Visible — human vision (400–700nm)

Ultraviolet — sterilization

X-rays — medical imaging

Gamma — nuclear processes

Applications & Modern Impact

Nearly every modern technology traces back to Maxwell.

📡

Telecom

Radio, cell, satellite

⚡

Power Grid

Generators, transformers

💻

Computing

Circuits, signal propagation

🏥

Medical

MRI, X-ray, CT

🔬

Photonics

Lasers, fiber optics

🛰️

Space & GPS

Deep-space comms

The Symmetry of Maxwell's Equations

Notice the elegant pairing between electric and magnetic phenomena:

$$\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$$
Sources: charges

$$\nabla\cdot\mathbf{B}=0$$
No magnetic sources

$$\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$$
Changing B → E

$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\partial\mathbf{E}/\partial t$$
Current + changing E → B

The curl equations are nearly mirror images — the asymmetry (ρ vs 0, J vs 0) reflects that magnetic monopoles don't exist. If they did, perfect symmetry would be restored.

Summary

Equation	Form	Meaning
Gauss (E)	$$\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$$	Charges source E
Gauss (B)	$$\nabla\cdot\mathbf{B}=0$$	No monopoles
Faraday	$$\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$$	Changing B → E
Ampère–Maxwell	$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\partial\mathbf{E}/\partial t$$	Current + changing E → B

$$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

Light is an electromagnetic wave.

Four equations. Two fields. One unified theory.