Section 02
The Idea, Step by Step
Flip a light switch, heat soup in a microwave, catch a text on your phone — behind all of it is one thing: electric and magnetic fields working together. Maxwell's four equations are just four sentences about how those two fields behave, and together they turn out to be light.
The cast: two fields
An electric field $\vec{E}$ is the push a charge feels — it points away from a "+" and toward a "−". A magnetic field $\vec{B}$ is the push a moving charge or a compass needle feels. Maxwell's achievement was to write down every rule these two fields obey, with no exceptions left over.
The four rules in plain words
1. Gauss for E — electric field lines start on positive charge and end on negative charge, so charge is the "source." 2. Gauss for B — magnetic field lines never start or stop; they always close into loops, because there is no such thing as a lone magnetic charge (a "monopole"). 3. Faraday — a changing magnetic field creates an electric field, which is exactly how every generator and transformer works. 4. Ampère–Maxwell — an electric current, and a changing electric field, each create a magnetic field.
One number that changed physics
Rules 3 and 4 feed each other: a changing $\vec{B}$ makes an $\vec{E}$, whose change makes a $\vec{B}$, and on it goes — a wave that carries itself through empty space. Its speed drops straight out of the two constants of electricity and magnetism:
Speed of the wave
$$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=\frac{1}{\sqrt{(4\pi\times10^{-7})(8.85\times10^{-12})}}\approx3\times10^{8}\;\text{m/s}$$
That is the measured speed of light, so Maxwell concluded light is an electromagnetic wave. Inside matter the wave slows to $v=c/\sqrt{\varepsilon_r\mu_r}$, which is why glass and water bend light.
How the sliders map
The charge density $\rho$ sets the strength of the Gauss-E field lines; the current density $J$ and the E and B amplitudes drive the Ampère–Maxwell and wave views; the frequency sets how fast the fields oscillate; and the medium $\varepsilon_r$ shows the wave slowing below $c$.
Try this in the sim above
Open Gauss E and drag $\rho$ negative — the radiating field lines flip to point inward toward the charge. Switch to Faraday and watch the induced-current arrow reverse direction each time the magnetic field changes sign. Then raise the medium $\varepsilon_r$ toward 80 (water) and watch the $c=1/\sqrt{\mu_0\varepsilon_0}$ readout fall — the very same light, just moving slower through matter.
Section 03
Equations & Derivation
Gauss's Law (Electric)
$$\oint\vec{E}\cdot d\vec{A}=\frac{Q_{\rm enc}}{\varepsilon_0},\quad\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$
Gauss's Law (Magnetic)
$$\oint\vec{B}\cdot d\vec{A}=0,\quad\nabla\cdot\vec{B}=0\;\text{(no magnetic monopoles)}$$
Faraday's Law
$$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt},\quad\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$$
Ampere–Maxwell Law
$$\oint\vec{B}\cdot d\vec{l}=\mu_0\!\left(I_{\rm enc}+\varepsilon_0\frac{d\Phi_E}{dt}\right),\quad\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$$
Speed of Light from Maxwell
$$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=2.998\times10^8\;\text{m/s},\quad\text{EM wave: }v=\frac{c}{\sqrt{\varepsilon_r\mu_r}}$$
The Four Laws
| Symbol | Quantity | Unit/Value |
|---|
| $\nabla\cdot\vec{E}=\rho/\varepsilon_0$ | Gauss E: electric fields diverge from charges | |
| $\nabla\cdot\vec{B}=0$ | Gauss B: no magnetic monopoles | |
| $\nabla\times\vec{E}=-\partial\vec{B}/\partial t$ | Faraday: changing B creates E (induction) | |
| $\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\varepsilon_0\partial\vec{E}/\partial t$ | Ampere-Maxwell: currents and changing E create B | |
1
Maxwell's displacement current. In 1865, Maxwell added $\varepsilon_0\partial\vec{E}/\partial t$ to Ampere's law to ensure charge conservation. This term means changing $\vec{E}$ acts like a current and creates $\vec{B}$ — even in vacuum. This completed the equations and immediately predicted electromagnetic waves at speed $c=1/\sqrt{\mu_0\varepsilon_0}$.
2
EM waves from Maxwell. Taking curl of Faraday's law and substituting Ampere-Maxwell gives $\nabla^2\vec{E}=\mu_0\varepsilon_0\partial^2\vec{E}/\partial t^2$ — the wave equation with speed $c$. Maxwell recognised $c=3\times10^8$ m/s matched the measured speed of light and concluded light is an EM wave.
3
Unification of electricity, magnetism, and light. Maxwell's equations unified three seemingly separate phenomena: electric forces, magnetic forces, and light. They also implied that electric and magnetic fields are different views of the same thing depending on the observer's velocity — a key seed of Einstein's special relativity.
4
Wave impedance. In vacuum: $Z_0=E/H=\sqrt{\mu_0/\varepsilon_0}=377\,\Omega$. This is why antennas and transmission lines are designed with impedances near 377 Ω for efficient radiation. In a medium: $Z=Z_0/\sqrt{\varepsilon_r}$.
Ref: Griffiths — Introduction to Electrodynamics (4th Ed.), Chs. 7–9; Halliday, Resnick & Walker 10th Ed., §32-1; Purcell — Electricity and Magnetism (3rd Ed.).
Section 05
Common Misconceptions
❌ Maxwell's equations are only valid in vacuum.
✅ Maxwell's equations hold in all media with appropriate modifications: $\varepsilon_0\to\varepsilon=\varepsilon_r\varepsilon_0$ and $\mu_0\to\mu=\mu_r\mu_0$. Wave speed in medium: $v=c/\sqrt{\varepsilon_r\mu_r}$. Boundary conditions at interfaces derive from Maxwell's equations and give Snell's law, Fresnel equations, and waveguide modes.
📖 HRW 10th Ed., §33-3; Griffiths — Introduction to Electrodynamics, Ch. 4.
❌ The displacement current is a real current.
✅ Displacement current $\varepsilon_0\partial\vec{E}/\partial t$ is not a flow of charge — no charges move between capacitor plates. It is a mathematical term Maxwell added to ensure charge conservation. However, it has all the physical consequences of a real current: it creates magnetic fields, allows EM waves to propagate, and satisfies continuity equations.
📖 Griffiths — Introduction to Electrodynamics, §7.3.3.
❌ Gauss's Law for magnetism ($\nabla\cdot\vec{B}=0$) is just a mathematical convenience.
✅ $\nabla\cdot\vec{B}=0$ is a deep physical statement: there are no magnetic monopoles. Every experimental search for monopoles has returned null. If a monopole existed, it would appear as a source in $\nabla\cdot\vec{B}=\mu_0\rho_m$ and would modify Faraday's law. Grand unified theories predict monopoles should exist — their absence is a major unsolved problem.
📖 HRW 10th Ed., §29-4; Griffiths — Introduction to Electrodynamics, §5.3.
Misconception research: Arons — A Guide to Introductory Physics Teaching.