4.2 Planck's Curves and Wien's Displacement Law
Key Takeaways
- Planck’s law describes spectral radiance as a function of wavelength (or frequency) and absolute temperature; curves peak and then fall on both the short- and long-wavelength sides
- Wien’s displacement law states λ_max · T ≈ 2898 µm·K, so the wavelength of peak spectral radiance shifts to shorter wavelengths as temperature rises
- A surface near 300 K peaks near 10 µm, which sits in the long-wave infrared (LWIR) atmospheric window used by most industrial electrical and building cameras
- MWIR (roughly 3–5 µm) is preferred for many hotter targets and some gas/optical applications; it is not the default band for ambient electrical and building surveys
- Camera spectral response, not human vision, determines which part of the Planck curve the imager uses to infer temperature
Stefan–Boltzmann gives total power integrated over all wavelengths. Cameras and atmospheric windows care about which wavelengths carry that energy. Planck’s law and Wien’s displacement law explain the spectral shape of thermal radiation, why peak emission moves with temperature, and why Level II electrical and building work overwhelmingly uses long-wave infrared (LWIR) imagers.
From Total Power to Spectral Distribution
A blackbody at temperature T emits a continuous spectrum. Spectral radiance (or spectral exitance, depending on formulation) is high at some wavelengths and low at others. If you plot intensity versus wavelength for a fixed temperature, you get a Planck curve: rising from near zero at very short wavelengths, reaching a maximum, then declining through the infrared toward longer wavelengths.
Key qualitative properties of Planck curves:
| Feature | Behavior as temperature increases |
|---|---|
| Peak height | Increases strongly (more radiance at the peak) |
| Peak location (λ_max) | Moves to shorter wavelength |
| Short-wavelength tail | Rises dramatically (visible glow for very hot objects) |
| Long-wavelength tail | Increases, but less dramatically than the short side |
| Area under the curve (all λ) | Follows Stefan–Boltzmann (∝ T⁴) |
Planck’s law is the microscopic foundation; Stefan–Boltzmann is the integral of Planck’s spectrum over all wavelengths; Wien’s law locates the peak of that spectrum.
Wien’s Displacement Law
Wien’s displacement law for the wavelength of maximum spectral radiance (in the common thermography form) is:
λ_max · T ≈ 2898 µm·K
(Some references use 2897.8 µm·K or state the constant in m·K; for exam work, ≈ 2900 µm·K is a sufficient memory value if options are coarse, but prefer 2898 µm·K.)
Therefore:
λ_max ≈ 2898 / T
with λ_max in micrometers when T is in kelvin.
Worked peak-wavelength examples
| Object / condition | Approx. T | λ_max ≈ 2898/T | Spectral region of peak |
|---|---|---|---|
| Human skin / room ambient surfaces | 300 K | ≈ 9.7 µm | LWIR |
| Warm electrical connection ~70 °C | 343 K | ≈ 8.4 µm | LWIR |
| Boiling water surface ~100 °C | 373 K | ≈ 7.8 µm | Near LWIR / mid-long IR edge |
| Soldering / hot metal ~400 °C | 673 K | ≈ 4.3 µm | MWIR |
| Dull red hot ~700 °C | 973 K | ≈ 3.0 µm | MWIR / near short-wave IR |
| Incandescent filament (thousands of K) | ~2500–3000 K | ~1 µm | Near-IR / visible edge |
Exam takeaway: Everyday electrical gear, building envelopes, roofs, and mechanical equipment near ambient to a few hundred degrees Celsius have peaks in or near the 8–14 µm region. Very hot process equipment, molten materials, and many combustion-related targets shift peak energy into MWIR or shorter bands.
Peak shift intuition without memorizing every number
- Cooler → peak moves right (longer wavelength) on a wavelength plot
- Hotter → peak moves left (shorter wavelength)
- The sun (~5800 K) peaks in the visible; that is the same law that puts room-temperature objects deep in the infrared
You do not need to see glow for an object to radiate intensely in IR. Room-temperature walls radiate strongly near 10 µm even though they look “cold” to the eye.
Reading Planck Curves for Thermographers
When training materials show a family of Planck curves for several temperatures:
- Higher-T curves lie above lower-T curves at essentially all infrared wavelengths of practical interest (more energy at every band once T is high enough).
- The fraction of energy inside a fixed camera band (for example 8–14 µm) changes with temperature even though the camera band is fixed.
- For cold targets, a larger share of energy sits at long wavelengths; for hot targets, more energy shifts into MWIR and SWIR.
- Camera calibration assumes a Planck-shaped blackbody spectrum filtered by the detector/optics response.
Band-limited measurement vs total exitance
A LWIR camera does not “see” the entire Stefan–Boltzmann integral. It integrates radiance over its spectral response. That is still enough to invert temperature for blackbodies because radiance in the band is a unique, monotonic function of T over the camera’s calibrated range. When emissivity is not 1, or when reflected radiation is significant, the inversion needs the Level II parameter set (ε, RAT, atmosphere, distance, window τ).
Atmospheric Windows and Industrial Camera Bands
Earth’s atmosphere is not transparent at all IR wavelengths. Water vapor and CO₂ create absorption bands. Thermography exploits atmospheric windows where transmission is relatively high:
| Band (common industrial labels) | Approx. wavelength range | Typical uses |
|---|---|---|
| SWIR | ~0.9–1.7 µm | Reflected IR, some high-T process, specialty |
| MWIR | ~3–5 µm | Hotter industrial targets, gas imaging (specific filters), some military/aerospace |
| LWIR | ~8–14 µm (often 7.5–13 or 7.5–14) | Electrical, building, roofing, general predictive maintenance |
LWIR 8–14 µm for industrial cameras
Most handheld uncooled microbolometer cameras sold for plant electrical surveys, building diagnostics, and mechanical PdM are LWIR. Reasons that appear on Level II exams and in practice:
- Wien peak for ambient objects sits near 10 µm — maximum photon/energy density for the targets you actually survey.
- Atmospheric transmission in the 8–14 µm window supports useful outdoor and indoor path lengths for typical inspection distances.
- Uncooled detector technology (microbolometers) is mature, relatively affordable, and rugged for field use in LWIR.
- Solar reflection management differs by band; LWIR is widely standardized for building and electrical procedures, with known caveats for outdoor reflection and low-ε metals.
MWIR vs LWIR — when each wins
| Consideration | LWIR advantage | MWIR advantage |
|---|---|---|
| Targets near ambient (buildings, switchgear at modest ΔT) | Strong — peak energy in-band | Weaker signal from cool targets |
| Very hot process / metals | Can work with correct range; may saturate | Often better spectral placement for high T |
| Detector type / cost | Uncooled common | Often cooled photon detectors (higher performance, cost, logistics) |
| Some gas detection | Limited without special filters | Filtered MWIR common for certain gases |
| Electrical / building standards & training examples | Dominant in NETA-style plant IR and building IR | Less common as the default survey camera |
Why most electrical and building work is LWIR: those applications involve surfaces that are typically within tens of degrees of ambient up to moderate elevations under load. Their Planck peaks and practical atmospheric paths align with 8–14 µm uncooled cameras. Specifying a MWIR camera for a routine building air-leakage survey or a 480 V panel route would be unusual unless a specialized target (certain gases, very hot equipment, R&D) drives the choice.
Solar Load, Daylight, and Spectral Confusion
Sunlight is a high-temperature Planck spectrum peaking in the visible/near-IR. Outdoor surveys can pick up reflected solar energy, especially on reflective metals and glass. Band choice and inspection timing (for example post-sunset roof moisture surveys) are partly spectral and partly heat-transfer timing problems. Level II candidates should separate:
- Emitted thermal radiation governed by the target’s own T (Planck/Wien)
- Reflected radiation from sun, sky, lights, and nearby hot objects
- Transmitted radiation through thin films, plastics, or IR windows (next section)
A bright solar reflection is not “the temperature of the sun on the panel”; it is a radiometric contamination that produces an erroneous high apparent temperature if you treat the reading as pure emission.
Practical Exam Scenarios
Scenario A — Building envelope at night. Interior ~21 °C, exterior cold. Surfaces near 270–300 K. Peak near 10 µm. LWIR camera, adequate interior–exterior ΔT, and controlled wind give meaningful patterns. Wien explains why the camera band matches the problem.
Scenario B — Bolted bus connection under load. Surface maybe 40–90 °C. Still LWIR-peaked. Quantitative severity uses corrected temperature and ΔT criteria, not a change of spectral band.
Scenario C — Kiln shell or furnace. Hundreds of °C. Peak moves toward MWIR. Either a high-temperature LWIR range or an MWIR system may be specified; filters and saturation limits matter. Planck/Wien justify rethinking the instrument, not merely the color palette.
Scenario D — “Why can’t I use a security NIR camera for quantitative thermography?” Near-IR security imagers often rely on reflected illumination and are not radiometrically calibrated to Planck thermal emission in LWIR. Spectral response and calibration chain differ from thermographic imagers.
Formulas to Carry Forward
| Law | Formula / statement | Role |
|---|---|---|
| Planck | Spectral radiance = f(λ, T) | Shape of emission spectrum |
| Wien | λ_max · T ≈ 2898 µm·K | Peak location vs temperature |
| Stefan–Boltzmann | E_b = σT⁴ | Total power over all λ |
| Graybody (simple) | Scale blackbody by ε | Real surfaces (overview) |
Summary for Recall
Planck curves show how thermal radiation is distributed with wavelength; hotter objects emit more at every infrared band of interest and shift their peak to shorter wavelengths. Wien’s law quantifies that shift: near 300 K the peak is about 10 µm, squarely in the LWIR 8–14 µm window used by standard industrial cameras. MWIR suits many hotter or specialty applications, but electrical and building thermography is predominantly LWIR because target temperatures, atmospheric windows, and uncooled camera technology align there. Level II competence means connecting spectral theory to camera selection, inspection conditions, and the limits of what a band-limited radiometric reading can represent.
According to Wien’s displacement law (λ_max · T ≈ 2898 µm·K), approximately where does the spectral peak of a 300 K blackbody fall?
As the temperature of a blackbody increases, what happens to the wavelength of maximum spectral radiance?
Why is most electrical switchgear and building-envelope thermography performed with LWIR (approximately 8–14 µm) cameras?