Before Maxwell, electricity and magnetism were two subjects. His achievement was to write them as four statements about a single field and notice that, together, they are inconsistent unless one term is added — the displacement current. Put it in, and the equations stop merely describing fields pinned to charges: they describe a disturbance that can let go of its source and travel, at a speed built from two constants measured with magnets and capacitors. That speed came out equal to the speed of light. Every scene below is a real 3-D field you can rotate with a drag; scroll through them and the argument builds from a single charge to light itself.
Charge Makes a Field — Gauss's Law
TIER A · DIRECTThe law
\(\displaystyle\oint_S \mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{enc}}}{\varepsilon_0}\), equivalently \(\nabla\!\cdot\!\mathbf{E}=\rho/\varepsilon_0\). Electric field diverges from charge.In plain words
The flux of \(\mathbf{E}\) out of any closed surface counts the charge inside, and nothing else. Grow or move the sphere — wrap the same charge, get the same number.In the 3-D scene
The true field as arrows in space + streamlines. Grow the translucent Gaussian sphere; the panel sums \(\oint\mathbf{E}\cdot d\mathbf{A}\) over it and reports \(Q_{\text{enc}}/\varepsilon_0\). Add a second charge for a dipole.Magnetism Has No Source — Closed Loops & Ampère
TIER A · DIRECTThe law
\(\displaystyle\oint_S \mathbf{B}\cdot d\mathbf{A}=0\) always \((\nabla\!\cdot\!\mathbf{B}=0)\); a steady current circulates \(\mathbf{B}\): \(\displaystyle\oint_C \mathbf{B}\cdot d\boldsymbol{\ell}=\mu_0 I_{\text{enc}}\).In plain words
No "magnetic charge" exists for lines to start on, so every \(\mathbf{B}\) line is a closed loop. Magnetism answers not to a source-point but to a current threading a loop.In the 3-D scene
The field \(B=\mu_0 I/2\pi r\) of a straight wire, drawn as the nested closed rings it really forms — orbit to see they never end. The Amperian ring reports \(\oint\mathbf{B}\cdot d\boldsymbol\ell=\mu_0 I\).Each Field Manufactures the Other
TIER A · DIRECTThe law
Faraday: \(\nabla\!\times\!\mathbf{E}=-\partial\mathbf{B}/\partial t\). Maxwell's new term: \(\nabla\!\times\!\mathbf{B}=\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t\) even with no current — the displacement current.In plain words
A magnetic field that changes in time drives a circulating electric field; a changing electric field drives a circulating magnetic field. Neither needs a charge or a wire — only change.In the 3-D scene
A bar magnet swings through a coil: the changing flux drives an induced current, \(\text{EMF}=-d\Phi_B/dt\), reversing with the magnet's motion (Lenz). Switch to the charging capacitor to watch the displacement current curl \(\mathbf{B}\) where no wire runs.The Field Lets Go — Electromagnetic Waves
TIER A · DIRECTThe law
In empty space the four equations combine to \(\nabla^2\mathbf{E}=\mu_0\varepsilon_0\,\partial^2\mathbf{E}/\partial t^2\): a wave equation, speed \(c=1/\sqrt{\mu_0\varepsilon_0}\).In plain words
A solution is a transverse ripple: \(\mathbf{E}\) and \(\mathbf{B}\) at right angles, in step, marching together. Their sizes are locked, \(E=cB\), moving at \(c\) — set only by \(\mu_0,\varepsilon_0\).In the 3-D scene
The exact plane wave: red \(\mathbf{E}\) vertical, blue \(\mathbf{B}\) horizontal, both \(\propto\sin(kz-\omega t)\), streaming along \(z\) at \(c\). Orbit to see they are truly perpendicular. Slide across the spectrum; \(E_0/B_0\) stays pinned at \(c\).Four lines, one field, and light for free
Read top to bottom, the page is one argument. Gauss's law says the electric field is sourced — it points away from charge, and the flux through any surface counts what is inside. Its magnetic twin says there is no such source for \(\mathbf{B}\): the lines close on themselves, and only currents — and, it turns out, changing electric fields — make them curl. Faraday and Ampère–Maxwell couple the two, and Maxwell's displacement current is the term that turns a pair of static rules into a self-propelling loop.
Strip away every charge and current and the four equations collapse into a wave equation whose speed is fixed by two lab constants:
That number is the speed of light, derived without ever mentioning light — two subjects that looked unrelated turn out to be one, and the union predicts a whole spectrum of radiation nobody had asked for. Everything downstream, from your phone to starlight to the photon, lives inside these four lines.
Boundary of validity
Gauss
The point/line‑charge field is exact; the numeric \(\oint\mathbf{E}\cdot d\mathbf{A}\) is a surface sum over the Gaussian sphere and matches \(Q_{\text{enc}}/\varepsilon_0\). Streamlines are integrated along the true field, not drawn by hand.
Ampère / ∇·B=0
\(B=\mu_0 I/2\pi r\) exact for a straight wire. Circulation \(\oint\mathbf{B}\cdot d\boldsymbol\ell\to\mu_0 I_{\text{enc}}\) and surface flux \(\to 0\), computed around the ring each frame.
Induction & wave
EMF \(=-d\Phi_B/dt\) is the exact derivative of the coil flux; the plane wave is the exact vacuum solution with \(E=cB\), \(\omega/k=c\). \(c,\lambda,\nu,h\nu\) recompute from \(\mu_0,\varepsilon_0,h\).
The 3-D slice
Vectors are sampled on a finite grid and streamlines are finite; both are honest samples of the continuous field, not the whole of it. Retardation and radiation‑zone falloff belong to the radiating‑charge entry.
Constants
\(\varepsilon_0=8.8542\times10^{-12}\), \(\mu_0=1.25664\times10^{-6}\), \(h=6.626\times10^{-34}\) (CODATA). \(c\) and every wavelength/energy are derived on load — no stored lookups.
Honesty tier
Tier A throughout: arrows are the sampled vector field, loops are integrated streamlines, the wave is the analytic solution. No field‑line cartoon the mathematics does not put there.
References
- Maxwell, J. C. (1865). A Dynamical Theory of the Electromagnetic Field. Phil. Trans. R. Soc. 155, 459.
- Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press.
- Faraday, M. (1832). Experimental Researches in Electricity. Phil. Trans. R. Soc. 122, 125.
- Ampère, A.-M. (1826). Théorie des phénomènes électro-dynamiques.
- Gauss, C. F. (1813). the flux/divergence law (Gauss's law in modern form).
- Heaviside, O. (1884–1885). recasting of Maxwell's equations into the modern four. The Electrician.
- Hertz, H. (1888). Über elektrodynamische Wellen im Lufte… Ann. Phys. 34, 610. (waves confirmed)
- Poynting, J. H. (1884). On the Transfer of Energy in the Electromagnetic Field. Phil. Trans. R. Soc. 175, 343.
- Griffiths, D. J. Introduction to Electrodynamics, 4th ed., ch. 7–9.
- Jackson, J. D. Classical Electrodynamics, 3rd ed., ch. 6–7.
- CODATA 2018 recommended values (\(\varepsilon_0,\mu_0,c,h\)).