# L1 Principle — Maxwell's Equations in Vacuum

**ID:** `PWM-L1-em-maxwell-vacuum`
**Status:** ⊙ Testnet (genesis)
**Submitter:** _(wallet address)_

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## 1. Domain

Electromagnetism — classical field theory in vacuum (no sources, no media).

## 2. Statement

Maxwell's equations describe the coupled evolution of electric and magnetic fields in vacuum. Together with the Lorentz force law, they predict every classical electromagnetic phenomenon, including the propagation of light as a transverse wave at speed `c`.

In vacuum, with no charges or currents, the four field equations reduce to a self-consistent wave equation for both **E** and **B**.

## 3. Governing Equations

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

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\qquad
\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
$$

Taking the curl of the third equation and substituting from the fourth yields the wave equation:

$$
\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}
\qquad \text{with} \qquad
c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}
$$

## 4. Variables and Units

| Symbol | Quantity | SI Unit |
|--------|----------|---------|
| **E** | Electric field | V/m |
| **B** | Magnetic field | T |
| ε₀ | Vacuum permittivity | F/m (≈ 8.854 × 10⁻¹² F/m) |
| μ₀ | Vacuum permeability | H/m (= 4π × 10⁻⁷ H/m) |
| c | Speed of light in vacuum | m/s (= 299,792,458 m/s) |

## 5. Conservation Laws and Symmetries

- **Energy conservation:** Poynting's theorem `∂u/∂t + ∇·S = 0` with `u = ½(ε₀E² + B²/μ₀)`.
- **Momentum conservation:** field momentum density `g = ε₀(E × B)`.
- **Lorentz invariance** — the equations are covariant under Lorentz boosts.
- **Gauge invariance** under `Aᵘ → Aᵘ + ∂ᵘΛ`.
- **Symmetries:** time-translation, spatial-translation, rotation, parity (P), time-reversal (T) — but C and CP only when combined with full Maxwell+matter system.

## 6. Validity Regime

Holds for:

- Classical electromagnetic fields at all macroscopic length scales
- Energies below the QED scale (photon energies ≪ 511 keV)
- Vacuum or low-density media (refractive index ≈ 1)

**Breaks down** in: quantum-coherent regimes (need QED), strong-field regimes ≥ 10¹⁸ V/m (Schwinger limit), and curved spacetime (need general-covariant Maxwell).

## 7. References

- Jackson, J. D. *Classical Electrodynamics* (3rd ed., Wiley, 1999), Ch. 6.
- Griffiths, D. J. *Introduction to Electrodynamics* (4th ed., Pearson, 2017), Ch. 7.
- Maxwell, J. C. "A Dynamical Theory of the Electromagnetic Field." *Phil. Trans. R. Soc. London* 155 (1865): 459–512.

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## File Mapping

This bundle consists of two files:

| File | Role | How to regenerate |
|------|------|-------------------|
| `principle.md` | Source of truth — edit this | Human or LLM |
| `principle.json` | Structured metadata for on-chain registry | LLM regenerates from §1 Domain, §2 Statement, §3 Governing Equations, §4 Variables and Units, §5 Conservation Laws, §6 Validity Regime, §7 References |

**Prompt for your LLM after editing `principle.md`:**

> Read `principle.md`. Regenerate `principle.json` so every field matches the Markdown.
> Schema:
> `{ id, name, domain, statement, equations_latex[], variables[{symbol,name,unit}], conservation_laws[], symmetries[], validity_regime, references[{title,authors,year,venue,doi}] }`
> Output only the JSON object — no prose, no fences.