EM wave solutions
In source-free ideal space, coupled electric and magnetic fields support waves that propagate at vacuum light speed. A plane wave is the simplest regression fixture: its frequency, wavelength, field directions, phase relationship, and energy transport all have known answers before a grid solver is introduced.
Think like a programmer
Model checklist
- Inputs
- Initial E and B field arrays, Δx, Δt, material constants ε and μ, boundary policy, and a known wavevector.
- State
- Staggered electric and magnetic field samples and accumulated boundary-energy transfer.
- Rule
- Apply coupled curl updates under a stated stability condition.
- Output
- Later fields, propagation speed, phase, energy density, and flux.
- Check
- E, B, and propagation are mutually perpendicular; numerical speed approaches 1/√(εμ); closed-domain energy stays within stated tolerance.
In a vacuum plane wave propagating along +x, one possible orientation is
The phase relation gives
\[c=\frac{\omega}{k}=\frac{1}{\sqrt{\varepsilon_0\mu_0}},\qquad \mathbf E\cdot\mathbf B=0\]For the displayed orientation, E is y-directed, B is z-directed, and E × B points in +x. Those direction checks are as important as matching a sine curve: swapping components or a curl sign can make a moving pattern while reversing energy transport.
const phase = waveNumber * positionX - angularFrequency * time;
const electricY = electricAmplitude * Math.sin(phase);
const magneticZ = electricY / vacuumLightSpeed;
const transverseDot = electricY * 0 + magneticZ * 0;
The last line is intentionally trivial for this coordinate-specific fixture: it encodes the expected zero dot product. In a general 3D solver, use vector dot and cross products, then compare the numerical propagation speed by tracking a phase feature or packet peak over time. Repeat with smaller Δx and Δt; a stable but wrong wave speed is still wrong.
Try this experiment
Prediction: Halving grid spacing while also choosing a stable smaller time step should reduce phase error for the same physical travel distance.
Start with a sine wave whose wavelength spans several cells. Predict what happens when it is represented by too few cells, then compare a coarse and refined run at the same time. Name the diagnostic that distinguishes a phase-speed error from a boundary-energy leak.Where this model breaks
Summary
Use an analytic transverse plane wave as an EM-grid regression fixture. Verify direction, phase, speed, energy, and refinement behavior together; none of those checks alone establishes that a field solver is reliable.
Glossary
- Transverse: perpendicular to propagation direction.
- Phase: position within a repeating cycle.
- Wave packet: localized combination of wave components.
- Curl: spatial rotation-like derivative that couples electric and magnetic updates.
- Numerical dispersion: grid-induced dependence of simulated wave speed on wavelength.
Self-check
- Which directions do E and B have in the stated +x plane-wave fixture?
- Why can a travelling visual still hide a curl-sign error?
- What comparison reveals numerical phase-speed error?
Sources
- OpenStax, University Physics Volume 2, Electromagnetic Waves.
- D. J. Griffiths, Introduction to Electrodynamics, source-free waves and energy flow.
Model contract
Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.
\[\text{observable output} = f(\text{inputs},\,\text{state})\]- Inputs
- Quantities you set or measure, with units and useful bounds.
- State
- Values the program must retain to reproduce the next result.
- Rule
- The relationship or update that turns inputs and state into a result.
- Check
- A known limit, unit check, invariant, or measured result that can expose a bad model.
Implement the idea as a model
For EM Wave Solutions, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Maxwell’s Equations and EM Waves, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.
Guided experiment
Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.
Where this model breaks
This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.
Summary
Treat EM Wave Solutions as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.
Glossary
- Input: a measured value or chosen parameter supplied to a model.
- State: the smallest set of values needed to continue or reproduce a model.
- Validation: comparing an output with a known result, limit, invariant, or measurement.
Self-check
- Which values are inputs, and which values must remain state?
- What observable result would tell you the model is behaving as expected?
- Which assumption would you test first before applying the model to a real system?
Model review: turn EM Wave Solutions into a test
Use analytic transverse waves as speed, phase, and energy regression cases for field solvers.
- Name the inputs and units that the maxwell’s equations and em waves model needs.
- Separate the state you must keep from values you can calculate when needed.
- Write one rule that maps the current state and inputs to an observable result.
- Choose a limiting case, unit check, invariant, or known result before trusting an output.
- State one assumption you would change before using this simplified model for a real decision.