SCIENCE · INVESTIGATE AND EXPLAIN

Keep carrier averaging separate from pulse integration

A plane wave carries energy through its Poynting vector. For a sinusoidal carrier, cycle averaging gives intensity epsilon0c Eamplitude²/2. Now let that carrier amplitude have a slowly varying envelope Epeak sin(πt/T) over a pulse of duration T. The carrier frequency is stipulated 1 GHz while T is in microseconds, so the envelope is slow relative to the carrier.

Learning outcome: Compare doubled amplitude with doubled duration and derive each scaling before revealing values. Retain both averaging steps, area and momentum idealization. Explain what additional material and geometry evidence would be needed to infer deposited heat or detector response.

Starting recap

Review flux integrals, linear differential equations, initial conditions, complex numbers and vector derivatives. Keep source constraints, RMS conventions and calibration roles explicit.

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Try

For Epeak=2 V/m and T=2 microseconds, Ipeak=2epsilon0c and fluence is 2epsilon0c×10⁻⁶ J/m². With the internally consistent teaching constants this is about 5.305165×10⁻⁹ J/m². The selected area intercepts about 5.305165×10⁻¹¹ J; perfect reflection transfers twice the absorbing momentum for that same intercepted energy.

Open the experiment

Predict

At fixed duration, doubling peak electric carrier amplitude changes pulse energy by:

Your prediction and reason

Experiment

Compare doubled amplitude with doubled duration and derive each scaling before revealing values. Retain both averaging steps, area and momentum idealization. Explain what additional material and geometry evidence would be needed to infer deposited heat or detector response.

Integrate pulse energy and transferred momentum

Worked teaching model · synthetic values · Locally plane vacuum wave, carrier frequency1 GHz and slowly varying sine envelope over duration T. Cycle-averaged intensity I(t)=epsilon0 c Epeak² sin²(pi t/T)/2 for0≤t≤T. Uniform intercepted area.01 m²; compare ideal complete absorption and perfect normal reflection. No exposure, detector response or material-safety claim.

Peak electric carrier amplitude: 2 V/m Pulse duration: 2 microseconds

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Cycle-averaged envelope, not resolved carrier oscillations

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: sin-squared envelope (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Cycle-averaged envelope, not resolved carrier oscillations

sin-squared envelope

time (microseconds)	intensity (W/m²)
0	0
0.025	8.17703e-6
0.05	3.26577e-5
0.075	7.32911e-5
0.1	0.0001298266
0.125	0.0002019158
0.15	0.0002891142
0.175	0.0003908841
0.2	0.0005065982
0.225	0.0006355429
0.25	0.0007769234
0.275	0.0009298679
0.3	0.001093434
0.325	0.001266612
0.35	0.001448335
0.375	0.001637483
0.4	0.001832889
0.425	0.002033349
0.45	0.002237627
0.475	0.002444463
0.5	0.002652582
0.525	0.002860702
0.55	0.003067538
0.575	0.003271815
0.6	0.003472275
0.625	0.003667682
0.65	0.00385683
0.675	0.004038553
0.7	0.004211731
0.725	0.004375297
0.75	0.004528241
0.775	0.004669622
0.8	0.004798567
0.825	0.004914281
0.85	0.005016051
0.875	0.005103249
0.9	0.005175338
0.925	0.005231874
0.95	0.005272507
0.975	0.005296988
1	0.005305165
1.025	0.005296988
1.05	0.005272507
1.075	0.005231874
1.1	0.005175338
1.125	0.005103249
1.15	0.005016051
1.175	0.004914281
1.2	0.004798567
1.225	0.004669622
1.25	0.004528241
1.275	0.004375297
1.3	0.004211731
1.325	0.004038553
1.35	0.00385683
1.375	0.003667682
1.4	0.003472275
1.425	0.003271815
1.45	0.003067538
1.475	0.002860702
1.5	0.002652582
1.525	0.002444463
1.55	0.002237627
1.575	0.002033349
1.6	0.001832889
1.625	0.001637483
1.65	0.001448335
1.675	0.001266612
1.7	0.001093434
1.725	0.0009298679
1.75	0.0007769234
1.775	0.0006355429
1.8	0.0005065982
1.825	0.0003908841
1.85	0.0002891142
1.875	0.0002019158
1.9	0.0001298266
1.925	7.32911e-5
1.95	3.26577e-5
1.975	8.17703e-6
2	0

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Peak cycle-averaged intensity	0.005305165	W/m²
Pulse duration	2.00000e-6	s
Integrated energy per area	5.30516e-9	J/m²
Energy intercepted by selected area	5.30516e-11	J
Absorbing-surface momentum transfer	1.76839e-19	kg m/s
Perfect normal-reflector momentum transfer	3.53678e-19	kg m/s
Average intensity over pulse duration	0.002652582	W/m²

Integrating the envelope gives a second factor of one half beyond the carrier average.

A plane wave has instantaneous Poynting magnitude epsilon0 c E². Averaging the rapid sinusoidal carrier gives epsilon0 c Eamplitude²/2. The envelope amplitude is Epeak sin(pi t/T); its squared integral over the pulse is T/2. Thus fluence=epsilon0 c Epeak²T/4. Complete absorption transfers U/c momentum and perfect normal reflection 2U/c under the stated idealization. Neither relation determines a material response or exposure limit.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Integrating the envelope gives a second factor of one half beyond the carrier average.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State geometry, sources and initial data

Derive the field or circuit relation

The cycle-averaged envelope intensity is Ipeak sin²(πt/T), where Ipeak=epsilon0c Epeak²/2. Its integral over the duration is Ipeak T/2 because the squared envelope contributes another average factor. Multiplying by the uniformly intercepted area 0.01 m² gives intercepted pulse energy. Confusing carrier averaging with envelope averaging would double the result.

Audit energy, uncertainty and domain

Under ideal complete absorption the transferred momentum is U/c; perfect normal reflection gives 2U/c. These are specified momentum-transfer idealizations, not material efficiency, heating, exposure or biological predictions. The display shows the averaged envelope rather than individual carrier cycles, and its endpoints are exactly zero by the stated pulse shape.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At fixed amplitude and intercepted area, duration changes from 1 to 3 microseconds. Find the pulse-energy ratio.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic evidence checked 31 October 2026 against the primary explanation. Constants are rounded teaching values and are mutually consistent where Maxwell relations require it. No live circuit, high-voltage, energized-inductor, magnet, beam or optical exposure activity is requested. Retain first work, criterion-linked critique, revision and newly assigned evidence; public numerical feedback is not practical or learner certification.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which source, initial-state, calibration or field-law premise makes the result conditional, and what unused evidence would challenge it?

Your explanation and limitations

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