SCIENCE · INVESTIGATE AND EXPLAIN

Separate a thermal spectrum’s peak, area and net exchange

A dense thermal emitter can produce a broad spectrum rather than isolated atomic lines. For an ideal blackbody, temperature determines the spectral shape. Wien’s displacement law λpeak T=b locates the maximum of intensity per unit wavelength. Stefan–Boltzmann emission P=AσT⁴ gives the total emitted power over all wavelengths for area A. A peak height or one narrow spectral band is not that total.

Learning outcome: Compare 3000 and 6000 K at one area, then double area at fixed temperature. Predict peak and total-power ratios. Retain the distinction among emitted, absorbed and net power and the finite plotting window.

Starting recap

Use scientific notation, signed energy differences and vector momentum. Distinguish a stipulated quantum or nuclear model from actual apparatus and label optional numerical extensions.

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Try

At 4000 K and area 2 cm², λpeak≈724.443 nm and emitted power≈2903.23 W. Doubling area doubles power with unchanged peak. Raising temperature from 3000 to 4000 K multiplies total emission by (4/3)⁴≈3.16049.

Open the experiment

Predict

Does a peak moving into a shorter-wavelength band prove that all emission has that wavelength?

Your prediction and reason

Experiment

Compare 3000 and 6000 K at one area, then double area at fixed temperature. Predict peak and total-power ratios. Retain the distinction among emitted, absorbed and net power and the finite plotting window.

Compare continuum peak and total thermal power

Worked teaching model · synthetic values · Ideal blackbody emission into a 300 K radiative environment, no convection or conduction model. Wien peak is for spectrum per wavelength. Supplied Planck spectrum is visualization enrichment; absolute spectral exitance per µm is not normalized. No heated object or optical exposure experiment.

Absolute temperature: 4000 K Emitting area: 2 cm²

Reset keeps your prediction, captures and practice history.

Supplied Planck wavelength spectrum — not normalized

1: selected 4000 K — solid line
2: 3000 K reference — long dashes

Accessible coordinates for this panel

These tables sample up to 82 displayed coordinates per series, including both endpoints. Fine curves may contain additional calculated points between table entries; use the ledger for exact analytic minima, frequencies or other named evidence. Values are synthetic, not instrument readings.

selected 4000 K
vacuum wavelength (µm)	spectral exitance (W/m² per µm)
0.1	0.0089486
0.14	48.345
0.18	4151.5
0.22	57592
0.26	309040
0.3	955800
0.34	2095000
0.38	3658600
0.42	5464500
0.46	7303400
0.5	9001000
0.54	10443000
0.58	11574000
0.62	12384000
0.66	12893000
0.7	13138000
0.74	13162000
0.78	13008000
0.82	12718000
0.86	12326000
0.9	11864000
0.94	11354000
0.98	10818000
1.02	10269000
1.06	9720300
1.1	9179600
1.14	8653300
1.18	8145800
1.22	7660000
1.26	7197600
1.3	6759500
1.34	6345900
1.38	5956500
1.42	5590900
1.46	5248000
1.5	4927100
1.54	4626900
1.58	4346400
1.62	4084500
1.66	3839900
1.7	3611700
1.74	3398700
1.78	3199900
1.82	3014300
1.86	2841100
1.9	2679300
1.94	2528200
1.98	2387000
2.02	2255000
2.06	2131500
2.1	2016000
2.14	1907800
2.18	1806500
2.22	1711600
2.26	1622600
2.3	1539000
2.34	1460600
2.38	1387000
2.42	1317800
2.46	1252800
2.5	1191600
2.54	1134000
2.58	1079700
2.62	1028600
2.66	980360
2.7	934870
2.74	891930
2.78	851380
2.82	813060
2.86	776830
2.9	742560
2.94	710120
2.98	679400
3	664650

3000 K reference
vacuum wavelength (µm)	spectral exitance (W/m² per µm)
0.1	5.5545e-8
0.14	0.0092255
0.18	5.3133
0.22	247.46
0.26	3071
0.3	17566
0.34	61613
0.38	155950
0.42	314550
0.46	538740
0.5	817660
0.54	1132500
0.58	1462000
0.62	1786000
0.66	2088500
0.7	2358100
0.74	2587800
0.78	2774700
0.82	2918800
0.86	3022200
0.9	3088200
0.94	3121000
0.98	3125100
1.02	3104800
1.06	3064300
1.1	3007500
1.14	2937700
1.18	2858000
1.22	2771000
1.26	2678800
1.3	2583200
1.34	2485900
1.38	2388000
1.42	2290500
1.46	2194200
1.5	2099800
1.54	2007700
1.58	1918300
1.62	1831900
1.66	1748500
1.7	1668400
1.74	1591500
1.78	1517900
1.82	1447500
1.86	1380300
1.9	1316300
1.94	1255300
1.98	1197200
2.02	1141900
2.06	1089400
2.1	1039500
2.14	992080
2.18	947040
2.22	904260
2.26	863640
2.3	825070
2.34	788450
2.38	753670
2.42	720650
2.46	689290
2.5	659500
2.54	631190
2.58	604300
2.62	578740
2.66	554450
2.7	531340
2.74	509370
2.78	488470
2.82	468570
2.86	449640
2.9	431610
2.94	414440
2.98	398080
3	390190

Each plot rescales independently. Compare values and units rather than apparent screen angles. Different panels can represent different quantities and are not silently overlaid.

Calculated evidence ledger — original synthetic model
Quantity	Value	Unit
Wavelength-spectrum peak	724.44	nm
Emitted power over all wavelengths	2903.2	W
Absorbed 300 K environment power	0.09186	W
Net radiative loss in declared environment	2903.1	W
Power / same-area 3000 K emission	3.1605	ratio

Higher temperature shifts the per-wavelength peak shorter and increases total emitted power; doubling area changes power, not the peak.

Wien’s law and Stefan–Boltzmann scaling are core; the supplied Planck curve is visualization enrichment, not a required calculus derivation. The plotted wavelength window is finite, while ledger power includes the entire spectrum. Peak position depends on whether a spectrum is per wavelength or per frequency; do not transform one peak by c/λ and call it the other maximum. Thermal equilibrium means balanced absorption/emission, not zero emission.

Higher temperature shifts the per-wavelength peak shorter and increases total emitted power; doubling area changes power, not the peak.

Observation to attach to a capture

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

Understand

Begin with evidence

Connect the model

Increasing temperature shifts the wavelength peak shorter and increases total emission strongly. Doubling emitting area doubles total power but does not change the peak wavelength. The displayed curves use an absolute common spectral scale, not separately normalized heights, so changes in magnitude remain visible. Their finite wavelength window does not include the entire all-spectrum power reported in the ledger.

Interpret the ledger

Classical equipartition applied without quantum restrictions predicted an unphysical high-frequency divergence, often called the ultraviolet catastrophe. Planck’s quantized exchange, in units hf, resolves that failure. The supplied numerical Planck curve is optional enrichment; the required comparisons use the two scaling laws and the physical distinction between discrete exchange and a continuous distribution of photon frequencies.

Test the boundary

The workbench separately subtracts absorbed radiation from ideal 300 K surroundings: Pnet=Aσ(T⁴−300⁴). Thermal equilibrium means equal incoming and outgoing power, not cessation of emission. Its per-wavelength maximum cannot be converted into the per-frequency maximum just by c/λ because spectral density transforms with the interval variable. These virtual hot emitters are not instructions to heat objects or view intense sources.

Worked original case

Predict, compare and retain evidence

Does a peak moving into a shorter-wavelength band prove that all emission has that wavelength? Compare 3000 and 6000 K at one area, then double area at fixed temperature. Predict peak and total-power ratios. Retain the distinction among emitted, absorbed and net power and the finite plotting window.

Changed-case practice

A blackbody changes from 3000 to 4500 K at unchanged area. Find peak-wavelength and emitted-power ratios.

Check the reasoning after your attempt

Peak ratio is 2/3; total-emission ratio is 1.5⁴=5.0625. Net power would also require surroundings, so it is not exactly the same ratio.

Check absorption as well as emission

An ideal blackbody absorbs all incident radiation; that is the defining idealization, not the claim that it emits only black or invisible light. At thermal equilibrium, absorbed and emitted powers balance while each can remain nonzero.

Predict and check: an ideal surface absorbs 12 W from its surroundings and emits 12 W, with no other transfer. Its net radiative change is zero. If a changed incoming packet supplies only 9 W while the initial emission is still 12 W, the initial net radiative loss is 3 W and equilibrium has been disturbed. A real partly reflecting surface needs its absorption and emission properties specified rather than silently borrowing the perfect-blackbody premise. (Current clause 15.4.A.2.)

Reflect and source-check

Why does zero net thermal exchange not mean zero emitted photons?

Original prose and supplied numerical evidence; checked 30 October 2026 against the current AP Physics 2 framework, fall 2026 clarifications and OpenStax. Virtual-only: no physical source, exposure or apparatus is requested. Public examples and model captures are not independent mastery or supervised laboratory participation.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.

Apply

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

Why does zero net thermal exchange not mean zero emitted photons?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Infer a photon from an energy-level difference, not a level label Next: Distinguish photon energy, count rate and missing stopping evidence

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