Blackbody Radiation
Separate a thermal spectrum’s peak from its complete area and total emitted power.
AP Physics 2 Topic 15.4 · College Board CED printed p. 142, 143 / PDF p. 149, 150 · See this topic in the course outline
A warm object radiates a broad spectrum. The most prominent wavelength and total emitted power answer different questions.
Before you start
- Thermal energy
- Kelvin temperature
- Graph area and ratios
Thermal energy becomes a continuous radiation spectrum
Matter spontaneously transfers some internal thermal energy into electromagnetic radiation. A blackbody is an ideal absorber of all incident radiation. At thermal equilibrium it also emits radiation while its temperature remains constant because energy entering and leaving balance. “Black” refers to ideal absorption, not an insistence that a hot body must look dark.
A blackbody's continuous emission spectrum depends only on its temperature, not on a collection of isolated atomic transition lines. Real bodies approximate the ideal to varying degrees; their surface emissivity can make the emitted distribution differ. A filament's broad thermal spectrum and a dilute gas's separated line spectrum should not be modeled as the same thing.
Read the axis before comparing curves
Plot spectral intensity per unit wavelength on the vertical axis and wavelength on the horizontal axis. The curve's peak gives the wavelength with greatest emitted power per area per wavelength interval; it is not the only emitted wavelength. Adding the contributions from all wavelength intervals—graphically, the area under the entire curve—gives total emitted intensity. Multiplying that total by emitting surface area gives total power.
Classical continuous-energy predictions fail to reproduce the measured distribution, especially at short wavelengths. Planck's law uses quantized energy exchanges and gives a finite thermal spectrum. We use its qualitative shape without deriving it: as temperature rises, the curve increases and the peak shifts toward shorter λ. At a fixed T, doubling the surface area doubles total power but does not move the spectral peak or change the intensity per unit emitting area.
| Temperature (K) | Peak λ (μm) | Total emitted intensity relative to 3000 K |
|---|---|---|
| 3000 | 0.966 | 1 |
| 4500 | 0.644 | 5.0625 |
| 6000 | 0.483 | 16 |
A sketch using these values places the hotter peak farther left and gives its entire curve greater area. Do not equate the numerical peak height with total emitted power or read a wavelength-plot peak as if it were a frequency-plot peak.
| λ (μm) | 3000 K spectral intensity | 6000 K spectral intensity |
|---|---|---|
| 0.25 | 0.0000 | 0.2615 |
| 0.483 | 0.0069 | 1.0000 |
| 0.967 | 0.0312 | 0.4043 |
| 2 | 0.0117 | 0.0504 |
| 4 | 0.0016 | 0.0044 |
| Temperature | Peak λ (μm) | Peak height relative to 3000 K | Total intensity relative to 3000 K |
|---|---|---|---|
| 3000 | 0.967 | 1 | 1 |
| 6000 | 0.483 | 32 | 16 |
The vertical scale divides both spectral intensities by the hotter peak height. Shading marks the area under each curve in the displayed interval; the complete area includes the long-wavelength tails beyond 4 μm. The exact total-area ratio 16 follows σT⁴. The 32 peak-height ratio illustrates why one peak height cannot stand in for that area. The construction uses the Planck shape; an AP response uses its qualitative curve and Wien/Stefan–Boltzmann laws, without deriving that shape.
Use absolute temperature and distinguish total from net power
Wien's displacement law is . A hotter blackbody peaks at shorter wavelength. Use kelvins: temperature ratios in Celsius do not apply.
The total emitted power of an ideal blackbody is , where σ = 5.67 × 10⁻⁸ W/(m²·K⁴) and A is its emitting surface area. Doubling T at fixed A multiplies P by 16; doubling A at fixed T multiplies P by 2. This is emitted power, not automatically the net energy loss. In an environment a body also absorbs radiation; a simple blackbody net-loss model would subtract the incoming term. The AP comparison must state which power is being used.
Peak and power scale differently
An ideal blackbody has area 0.020 m² and temperature 3000 K. Find peak wavelength and total emitted power; compare with 6000 K at the same area.
λmax = b/T = (2.90 × 10⁻³)/3000 = 9.67 × 10⁻⁷ m = 967 nm.
Why this step? Wien’s law gives the peak of the wavelength distribution, not a single emitted color.
P = AσT⁴ = (0.020)(5.67 × 10⁻⁸)(3000)⁴ = 9.19 × 10⁴ W.
Why this step? Emitted power sums over the entire spectrum and the emitting area.
At 6000 K the peak wavelength halves to 483 nm and P increases by 2⁴ = 16 to 1.47 × 10⁶ W.
Why this step? Temperature ratios control the two laws differently.
Answer and check
967 nm and 9.19 × 10⁴ W at 3000 K; 483 nm and 1.47 × 10⁶ W at 6000 K.
| λ (μm) | 3000 K spectral intensity | 6000 K spectral intensity |
|---|---|---|
| 0.25 | 0.0000 | 0.2615 |
| 0.483 | 0.0069 | 1.0000 |
| 0.967 | 0.0312 | 0.4043 |
| 2 | 0.0117 | 0.0504 |
| 4 | 0.0016 | 0.0044 |
| Temperature | Peak λ (μm) | Peak height relative to 3000 K | Total intensity relative to 3000 K |
|---|---|---|---|
| 3000 | 0.967 | 1 | 1 |
| 6000 | 0.483 | 32 | 16 |
The vertical scale divides both spectral intensities by the hotter peak height. Shading marks the area under each curve in the displayed interval; the complete area includes the long-wavelength tails beyond 4 μm. The exact total-area ratio 16 follows σT⁴. The 32 peak-height ratio illustrates why one peak height cannot stand in for that area. The construction uses the Planck shape; an AP response uses its qualitative curve and Wien/Stefan–Boltzmann laws, without deriving that shape.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
Two identical-temperature blackbodies have areas A and 3A. Compare their peak wavelength, intensity per wavelength, and total emitted power.
Worked answer and reason
Same peak and same spectral intensity per unit emitting area; the larger body emits three times total power.
Surface area multiplies power but does not alter the temperature-dependent spectrum per area.
Question 2
A graph’s vertical axis is W/(m²·μm). Does its maximum height equal total intensity in W/m²?
Worked answer and reason
No. Sum across wavelength intervals or take the graph’s area to obtain total intensity; the peak height is per unit wavelength.
The units reveal why peak and integrated quantity differ.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- A blackbody emits only at its peak wavelength.
- It emits a continuous distribution; Wien’s law locates the greatest spectral contribution, not the only wavelength.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 7 · graphical
- Problem 8 · mixed
- Problem 18 · experimental
Scientific calculator workspace
Keep intermediate precision and report units. This calculator supports arithmetic and elementary functions.
Scientific calculator
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For the teacher
- Require both peak-position and integrated-area comments on a spectral graph. Do not infer true temperature from an uncalibrated phone photograph’s color.
Source objectives and scope
- 15.4.A — Describe the electromagnetic radiation emitted by an object due to its temperature.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- AP Physics 2 Course and Exam Description, official Unit 15 — Official topic/LO/EK scope and boundaries; local pinned pages 143–158, equations and relevant clauses visually checked on 143–144,146,149–150,152–158.
- AP Physics 2 Course and Exam Description Clarifications, effective Fall 2026 — 15.7.B.1 current definition includes decay to a lower energy level of the same nucleus.
- OpenStax University Physics 3, Blackbody Radiation — Verification of Wien peak and integrated spectral intensity versus Stefan–Boltzmann total power.
- OpenStax University Physics 3, Photoelectric Effect — Verification of threshold, maximum kinetic energy, intensity distinctions, and retarding-potential experiment.
- OpenStax University Physics 3, The Compton Effect — Verification of positive electron-at-rest shift and photon energy/momentum accounting.
- OpenStax University Physics 3, Nuclear Binding Energy — Verification of all-nucleon mass defect and atomic-mass convention.
- OpenStax University Physics 3, Radioactive Decay — Verification of exponential survival, half-life, and nuclear transformation balances.
- OpenStax University Physics, Volume 3: 6.1 Blackbody Radiation — Primary textbook verification of the wavelength-based blackbody spectral shape, the distinction between peak height and complete spectrum area, Wien’s displacement law, and Stefan–Boltzmann emitted intensity and power.
Glossary and deeper reading · 12 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Wavelength
Wavelength is the spatial distance between neighboring points at the same phase of a periodic wave. It connects propagation speed and frequency through speed equal to frequency times wavelength. For example, neighboring crests of a sinusoidal water wave are separated by one wavelength.
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Power
Power is the rate at which energy is transferred or converted. It is measured in watts, with one watt equal to one joule per second. For example, two motors can lift the same load through the same height but have different average powers if they take different times.
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Kelvin temperature scale
The kelvin scale measures thermodynamic temperature from absolute zero. A temperature interval of one kelvin has the same size as one degree Celsius, but their zero points differ. For example, temperature ratios in the ideal gas law must use kelvin values rather than Celsius values.
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Temperature
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact. For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample. For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.
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Electromagnetic wave
An electromagnetic wave contains oscillating electric and magnetic fields that transport energy. In the plane-wave vacuum model, both fields are perpendicular to each other and to the propagation direction. For example, visible light and radio waves can both travel through empty space.
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Thermal radiation
Thermal radiation is electromagnetic energy emitted by matter because of its temperature. It can transfer energy through a vacuum, and net radiative transfer between ordinary bodies is from hotter to cooler. For example, sunlight warming a surface does not need air to carry the energy across space.
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Emission spectrum
An emission spectrum shows the frequencies or wavelengths of radiation emitted by a source. An excited dilute gas produces characteristic bright lines associated with allowed energy-level transitions. For example, comparing several observed lines with known element spectra can help identify which atoms are present.
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Emissivity
Emissivity describes a surface’s thermal-radiation emission compared with an ideal blackbody at the same temperature. It is dimensionless and can depend on wavelength, surface condition, and temperature. For example, two surfaces of equal area at the same temperature can radiate different powers because their emissivities differ.
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Wave intensity
Wave intensity is the time-averaged energy-transfer rate per area perpendicular to propagation. It is measured in watts per square meter and differs from total power because area matters. For example, spreading the same sound power over a larger area reduces the intensity there.
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Quantization
Quantization means a quantity can take only certain allowed values in the stated quantum system. Bound-state boundary conditions can restrict energy to discrete levels rather than a continuous range. For example, an electron bound in a hydrogen atom cannot have an arbitrary energy between its allowed levels.
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Frequency
Frequency is the number of complete repeating cycles per unit time. It is measured in hertz, meaning cycles per second, and is the reciprocal of the period. For example, a vibrating string with a higher frequency produces more cycles each second.
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Wien’s displacement law
Wien’s displacement law relates a blackbody spectrum’s peak wavelength to its kelvin temperature. The peak wavelength is inversely proportional to temperature when the spectrum is expressed per unit wavelength. For example, a hotter blackbody peaks at a shorter wavelength, though it still emits across a broad range.