Wiens Law Interactive Calculator

If you need to figure out the wavelength at which a hot object radiates most strongly—or if you want to know what temperature produces a specific peak wavelength—Wien's displacement law does the job with a simple formula. The calculator below covers peak emission wavelength, frequency, photon energy, or blackbody temperature, working from either temperature or wavelength as your starting point. You'll see this used in thermal imaging, industrial pyrometry, and astronomy—anywhere non-contact temperature measurement or spectral analysis is useful. Details on the equations, real calculations, and practical notes are further down the page.

What is Wien's Law?

Wien's law (or Wien's displacement law) says that as an object's temperature increases, the wavelength where it emits most strongly shifts to shorter values. With this law, you can find the peak wavelength for any blackbody if you know the temperature—or you can work backward from a wavelength and get the temperature.

Simple Explanation

When you heat up a piece of metal, it turns dull red, then orange, then white as it gets hotter. That color change happens because the hottest parts emit at shorter and shorter wavelengths—moving from red towards blue—while cooler objects emit mostly in the red or even in infrared, outside visible light. Wien's law gives the number to tell you exactly where the peak falls for any temperature.

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Blackbody Radiation Diagram

Wien's Law Calculator

How to Use This Calculator

1. Pick which calculation you want—to get peak wavelength from a temperature, or the other way around, or to work with frequency, energy, or spectral radiance.
2. Put your known value in the box—usually temperature in Kelvin, or wavelength in nanometers, depending on mode.
3. If you want to see how it works, load an example.
4. Click Calculate to get the result.

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Equations & Variables

The formula below gives you the peak wavelength from a known temperature.

Here is the formula to get the peak frequency from temperature.

This formula gives blackbody spectral radiance at any wavelength and temperature.

This one gives the photon energy at the peak wavelength.

Variable Definitions:

• λ_peak = Peak wavelength of blackbody radiation spectrum (m, nm, μm)
• f_peak = Peak frequency of blackbody radiation spectrum (Hz, THz)
• T = Absolute temperature of blackbody (K)
• b = Wien's displacement constant = 2.897771955 × 10⁻³ m·K
• α = Dimensionless constant for frequency form ≈ 2.821439372
• h = Planck constant = 6.62607015 × 10⁻³⁴ J·s
• c = Speed of light in vacuum = 299,792,458 m/s
• k_B = Boltzmann constant = 1.380649 × 10⁻²³ J/K
• B_λ(λ,T) = Spectral radiance (W/(m²·sr·m))
• E_photon = Energy of photon at peak wavelength (J, eV)

Simple Example

The sun's photosphere is about 5778 K. Plug that into Wien's law for wavelength:

• Temperature: 5778 K
• Peak wavelength: 2.898 × 10⁻³ / 5778 = 501.5 nm (green-yellow)
• Peak frequency: ~340.8 THz
• Spectral region: Green

Theory & Practical Applications

Blackbody Radiation and Wien's Law Fundamentals

Wien's displacement law gives a direct link between an object's temperature and where its emission spectrum hits its maximum. The law comes from differentiating Planck's law, and the result is that as temperature increases, the peak wavelength shifts down toward the ultraviolet. Cooler objects peak in the infrared or even longer. In practice, this basic relationship is the backbone of non-contact temperature measurement, with practical uses from measuring red-hot steel to analyzing distant stars or even the 2.7 K cosmic background. Just keep in mind: the math covers a huge range of situations, but real materials don't always behave exactly like ideal blackbodies.

The difference between the wavelength and frequency forms of Wien's law isn't just academic—it actually matters when you design or pick optical detectors or cameras. The peak wavelength from λ_peak = b/T will not match the wavelength you'd get by converting the frequency peak via c = λf. The reason is the change of variables: the transformation adds a λ² factor. For a 5778 K blackbody, the wavelength peak is at 501.5 nm (green), while the frequency peak sits at 879.9 nm (near-infrared). If you're building or specifying an infrared camera, you need to be aware that detector optimization depends on which spectral form you're using.

Industrial Pyrometry and Temperature Measurement

Wien's law is at the core of non-contact pyrometry. When you need to measure temperature in harsh conditions, like molten metal, glass furnaces, or engine exhausts, it's usually done optically. A ratio pyrometer compares light intensity at two wavelengths to back out temperature, which helps cut down on errors from uncertain emissivity. If you can assume the emissivity doesn't vary strongly with wavelength (a so-called gray body), the error from not knowing the true emissivity is reasonably small—a 10% error in emissivity is only about 7 K of temperature error, even at 1500 K. This is good enough for most process control work.

Some materials, especially alloys or oxidized metals, change emissivity with wavelength. Here, you need either a multi-wavelength pyrometer or to fit the Planck curve directly to your data at three or more points. This lets you solve for both the temperature and the way emissivity changes with wavelength. It's a common approach in very high-temperature applications, like measuring turbine blades running above 1800 K, where surface chemistry can change optical properties across the near-IR and mid-IR as oxide layers form.

Astronomical Spectroscopy and Stellar Classification

In astronomy, Wien's law is a quick way to estimate a star's photospheric temperature—just locate the peak in its spectrum. For the sun, this gives about 5778 K from a peak around 501.5 nm (green-yellow). Hot blue stars, like Rigel, have peaks deep in the ultraviolet, and cool red dwarfs peak out into the infrared. Simple color-photometry systems (like UBV filters) use this temperature-color link to classify stars or galaxies at scale. For anything more accurate, though, you need to fit the whole spectrum and account for effects like dust, which skews the observed peak by scattering and reddening the light. In very dusty environments, astronomers use near-infrared because visible light can't get through—so the measured peak could land far from what's actually being emitted at the surface.

Thermal Imaging and Infrared Camera Design

For typical temperatures around you—say, 20–40 °C—Wien's law puts the peak emission right in the 8-12 μm long-wave infrared range. That's why thermal cameras for building inspections, medical imaging, or surveillance usually use detectors tuned there. Hotter sources—engines, furnaces, jets—shift their peaks towards 3–5 μm (mid-wave IR), so MWIR cameras are used instead. But it's not just about the peak: atmospheric absorption and detector technology also matter. There's strong water vapor absorption outside these "windows", and you need the right detector for your wavelength band (InSb for MWIR, microbolometers for LWIR). Advanced cameras sometimes measure in several bands at once to capture both temperature and material data, but in complex scenes (like flames), you may need to do full spectral fitting because real objects rarely behave as true blackbodies.

Cosmic Microwave Background and Precision Cosmology

The cosmic microwave background (CMB) is about the best real-world blackbody example, with a temperature measured at 2.725 K and a peak emission at roughly 1.06 mm, right where Wien's law predicts. Satellites like COBE, WMAP, and Planck all confirmed this, measuring power at frequencies from 30 to 857 GHz. Getting these data is not easy: the CMB signal is dwarfed by Earth's own thermal emission, so instruments cool sensors and use calibrated references to detect just the tiny difference. The best of these data match Planck and Wien's law so closely that deviations teach us about the early universe—tiny shifts tell cosmologists about the formation and composition of the cosmos, testing physical laws over huge time and length scales.

Worked Example: Designing an Infrared Pyrometer for Aluminum Casting

Problem: An aluminum foundry wants to monitor temperature of molten metal, which usually sits at 973 K (700°C). The factory has an MWIR camera pointed at 3.9 μm, but contrasts are poor. The task: (a) determine the actual peak emission wavelength for that temperature, (b) figure out if the existing camera wavelength is ideal, (c) compare radiance at both peak and camera wavelength assuming an emissivity ε = 0.21 for aluminum, and (d) decide if a different detector or wavelength band makes more sense.

Solution:

(a) Peak wavelength from Wien's law:

λ_peak = b / T = (2.897771955 × 10⁻³ m·K) / (973 K) = 2.979 × 10⁻⁶ m = 2.979 μm

So the theoretical peak is at about 3 μm—short wave infrared.

(b) Optimal detector wavelength:

Even though emission peaks at 3 μm, practical detectors also need to consider atmospheric absorption and what hardware is available. Air absorbs a lot near the 2.7–2.9 μm range (water vapor). So even if 2.98 μm would maximize signal, you may get more real-world signal in, say, the 3.4–3.8 μm window, or around 4.6–5.1 μm, which have less atmospheric loss, especially if the foundry is steamy.

(c) Spectral radiance calculation using Planck's law:

B_λ(λ,T) = ε × (2hc²) / (λ⁵[exp(hc/λk_BT) − 1])

At λ = 3.9 μm (existing camera):

Exponential term: exp(hc/λk_BT) = exp[(6.626×10⁻³⁴)(2.998×10⁸) / ((3.9×10⁻⁶)(1.381×10⁻²³)(973))] = exp(3.804) = 44.87

B_λ(3.9μm, 973K) = 0.21 × [2(6.626×10⁻³⁴)(2.998×10⁸)² / ((3.9×10⁻⁶)⁵(44.87 − 1))]

B_λ(3.9μm, 973K) = 0.21 × [1.191×10⁻¹⁶ / (9.150×10⁻²⁹ × 43.87)] = 0.21 × 2.967×10¹² = 6.23×10¹¹ W/(m²·sr·m)

At λ_peak = 2.979 μm:

Exponential term: exp(hc/λk_BT) = exp[(6.626×10⁻³⁴)(2.998×10⁸) / ((2.979×10⁻⁶)(1.381×10⁻²³)(973))] = exp(4.978) = 145.4

B_λ(2.979μm, 973K) = 0.21 × [1.191×10⁻¹⁶ / ((2.979×10⁻⁶)⁵(144.4))] = 0.21 × 4.813×10¹² = 1.01×10¹² W/(m²·sr·m)

You'll get about 62% more radiance at the Wien peak compared to the 3.9 μm camera wavelength.

(d) Detector recommendation:

The 3.9 μm camera picks up less signal than possible at the peak, but not drastically less. Switching to LWIR (8–12 μm) would make things worse; Planck radiance is about 1800× lower there at this temperature, so you lose a lot of usable signal. A better route is to tweak the MWIR region—go as close as practical to the peak, within a good atmospheric window and available detectors (3.4–3.6 μm, or possibly 4.6–4.8 μm if humidity is a problem). You might also use two-color pyrometry (compare 3.4 μm and 3.8 μm bands) to cancel out some uncertainty from emissivity, especially when absolute calibration is less critical than tracking changes.

Semiconductor Manufacturing and Wafer Temperature Control

In rapid thermal processing (RTP) for chipmaking, silicon wafers get heated from 300 K up to 1450 K in seconds while keeping the temperature spread across the wafer within a couple of kelvin. The emission peak starts near 9.7 μm (room temp) and moves to about 2 μm (maximum process temp). You have a trade-off—either switch between different detector bands as you ramp, or pick a wavelength around 0.9–1.0 μm that gives good enough signal at both ends. The situation is trickier because silicon is transparent above 1.1 μm, so at longer wavelengths you're measuring temperature averaged over a depth, not just the surface. For best accuracy, most RTP systems either measure above the silicon bandgap (at 950 nm) or use a backside pyrometry setup and correct with models for the actual surface temp.

Frequently Asked Questions

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