Energy and momentum in EM waves

Electromagnetic waves carry energy and momentum through their electric and magnetic fields. A field solver should therefore produce a ledger, not merely snapshots. If total energy in a closed ideal domain changes, investigate the numerical method. If a wave reaches an open boundary, record the outgoing flux as a transfer rather than calling it unexplained loss.

Think like a programmer

Compute diagnostics from the same field arrays that the update advances. Keep energy density per cell, Poynting-vector samples, accumulated boundary flux, source work, and total energy in a serializable ledger. The renderer can color a field or arrow, but it must not invent energy transport independently of the model arrays.

Model checklist

Inputs: E and B field arrays, ε and μ, cell volume or area convention, source current, boundary policy, and time step.
State: Local energy density, Poynting flux samples, stored-energy total, boundary-transfer accumulator, and source-work accumulator.
Rule: Reduce field samples into energy and flux, then update a ledger with source and boundary terms.
Output: Energy distribution, transport direction, momentum-flow proxy, and conservation residual.
Check: Closed source-free periodic or reflecting domain has near-constant total energy; open-boundary energy decrease matches integrated outgoing flux.

For a linear vacuum field, local electromagnetic energy density is

\[u=\frac{1}{2}\left(\varepsilon_0|\mathbf E|^2+\frac{|\mathbf B|^2}{\mu_0}\right)\]

The Poynting vector gives energy-flow direction and power per area:

\[\mathbf S=\frac{1}{\mu_0}\mathbf E\times\mathbf B\]

For the earlier plane-wave fixture with E along y and B along z, S points along +x. A swapped field direction or sign reverses S, so this is a stronger test than confirming that the two arrays have sine-shaped values.

For a finite grid, keep a discrete ledger:

\[U^{n+1}-U^n\approx W_{\mathrm{sources}}-F_{\mathrm{boundary}}\Delta t+R_{\mathrm{numerical}}\]

U is the summed cell energy, Wsources is energy added by modeled sources, Fboundary is outward flux through the boundary, and Rnumerical is the residual to report—not a term to silently discard. In a source-free periodic domain, the expected boundary flux is zero, so total energy drift measures the numerical method. In an absorbing or open domain, a falling interior total can be correct only when outward flux accounts for it.

const energyDensity = 0.5 * (permittivity * dot(electric, electric) + dot(magnetic, magnetic) / permeability);
const poynting = scaleVector(cross(electric, magnetic), 1 / permeability);
const conservationResidual = nextStoredEnergy - storedEnergy - sourceWork + boundaryOutflow * timeStep;

The exact vector and grid conventions differ by dimension, but the contract stays the same. Use a source-free plane wave to check transport direction. Use a closed boundary to check drift. Use an open boundary to check that the ledger balances. Report all three alongside resolution and time-step data.

Try this experiment

Prediction: In a source-free open domain, energy leaving through a measured boundary should reduce stored interior energy without appearing as numerical dissipation.

Imagine a wave packet approaching the right boundary. Predict the signs of storedEnergy change and boundaryOutflow. Then compare an open boundary with a wrapping boundary and explain which ledger term changes.

Where this model breaks

These formulas use an ideal linear medium and simple field-energy convention. Dispersive, lossy, nonlinear, anisotropic, or strongly material-coupled media need extra energy terms and careful constitutive modeling. Discrete curl stencils, absorbing layers, source injection, and coarse grids can all create residuals; a small total drift does not prove local fields are accurate.

Summary

EM-wave simulations need an energy ledger. Derive energy density and Poynting flux from the evolved fields, account for source work and boundary transfer, and make the remaining conservation residual visible as a numerical diagnostic.

Glossary

Energy flux: energy crossing area per time.
Poynting vector: electromagnetic energy-flux vector.
Momentum: directional conserved quantity carried by fields and matter.
Boundary transfer: ledger entry for energy leaving or entering a domain.
Conservation residual: mismatch remaining after modeled energy terms are accounted for.

Self-check

Which vector gives EM energy-flow direction?
Why is a falling interior total not automatically an error in an open domain?
What does a source-free closed-domain energy drift diagnose?

Sources

D. J. Griffiths, Introduction to Electrodynamics, field energy, momentum, and Poynting theorem.
OpenStax, University Physics Volume 2, Energy and Momentum of Electromagnetic Waves.

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 Energy and Momentum in EM Waves, 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 Energy and Momentum in EM Waves 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 Energy and Momentum in EM Waves into a test

Add local field-energy and boundary-flux diagnostics to electromagnetic wave 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.

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