Wien's Displacement Law Calculator
How to Use the Wien’s Displacement Law Calculator
This calculator converts temperature into the wavelength where an ideal blackbody emits the most intensely. You enter a temperature in kelvin, and the tool returns the peak wavelength along with its spectral region (infrared, visible, or ultraviolet). If a graph is available in your browser, it also shows the full Planck spectrum with the peak highlighted.
- Enter the temperature in the field above, in kelvin (K). For example, the Sun’s surface is about 5800 K, room temperature is about 300 K.
- Submit the form to compute the peak wavelength λmax.
- Read the result: the calculator reports λmax in meters and in more convenient units (such as micrometers or nanometers) and indicates whether the peak lies in IR, visible, or UV.
- Explore with the plot (when enabled): the blue curve shows the normalized blackbody spectrum and a marker highlights the peak. As temperature increases, the entire curve shifts toward shorter wavelengths.
For users relying on screen readers or with scripts disabled, the numeric output and a text caption summarize the same information as the visual plot: the temperature, the peak wavelength, and the approximate spectral band.
Wien’s Displacement Law in Plain Language
Wien’s displacement law describes how the color of an ideal glowing object changes with temperature. As something gets hotter, the wavelength of light it emits most strongly shifts toward shorter values. That is why warm objects first glow dull red, then orange, then white and bluish as their temperature rises.
Mathematically, the law states that the wavelength of maximum emission λmax is inversely proportional to the absolute temperature T:
Here b is Wien’s displacement constant. In SI units:
λmax = b / T, with b ≈ 2.897 × 10−3 m·K.
If you double the temperature, the peak wavelength is cut in half. If you multiply the temperature by 10, the peak wavelength becomes 10 times smaller.
From Planck’s Law to Wien’s Law
Behind the simple Wien relation lies the more detailed Planck law for blackbody radiation. Planck’s law gives the spectral radiance B(λ, T) at each wavelength λ and temperature T:
Here h is Planck’s constant, c is the speed of light, and k is Boltzmann’s constant. If you differentiate B(λ, T) with respect to λ and set the derivative equal to zero, the resulting equation has a single positive solution for the peak wavelength. That solution simplifies to the Wien form λmax = b / T, where the constant b is derived from fundamental constants and a numerical factor obtained from the derivative.
The calculator typically uses the compact Wien formula for speed, and visualization components often use the full Planck expression to draw the entire curve correctly while still placing the peak at the location predicted by Wien’s law.
Interpreting the Results
When you enter a temperature, the main output is the peak wavelength λmax. To understand what that means physically, it helps to relate it to common spectral regions:
- Infrared (IR): wavelengths longer than about 700 nm (0.7 μm). Warm everyday objects and room-temperature scenes peak in the mid-infrared, far beyond what human eyes can see.
- Visible light: roughly 400–700 nm. Peaks in this range correspond to glowing objects whose color we can see directly (from red through violet).
- Ultraviolet (UV): wavelengths shorter than about 400 nm. Very hot objects, such as some types of stars, have peaks in the UV.
Even if the peak is outside the visible range, the object can still emit some visible light; the law only tells you where the emission is strongest, not where it starts or stops. For example, a 3000 K filament has a peak in the infrared but still emits significant visible red and orange light.
Worked Examples
1. Room-Temperature Object (~300 K)
Imagine an object at approximately room temperature, T = 300 K. Using λmax = b / T:
λmax ≈ (2.897 × 10−3 m·K) / (300 K) ≈ 9.66 × 10−6 m = 9.66 μm.
This peak is in the mid-infrared. That is why thermal cameras designed for building inspections or night vision operate in the infrared instead of visible light.
2. The Sun’s Surface (~5800 K)
For the Sun’s effective surface temperature, T ≈ 5800 K:
λmax ≈ (2.897 × 10−3 m·K) / (5800 K) ≈ 5.0 × 10−7 m = 500 nm.
A wavelength of about 500 nm lies in the green part of the visible spectrum. The Sun emits strongly across the entire visible range, which is why sunlight appears roughly white, but the maximum of the idealized blackbody curve is near green.
3. Hot Kiln (~1200 K)
Consider a kiln heated to T = 1200 K. Applying Wien’s law:
λmax ≈ (2.897 × 10−3 m·K) / (1200 K) ≈ 2.41 × 10−6 m = 2.41 μm.
This peak is in the near-infrared. To the human eye, the kiln appears dull red because the visible tail of the spectrum is still substantial in the red portion even though the true maximum lies in the infrared.
Typical Temperatures and Peak Wavelengths
| Object / Scenario | Temperature T (K) | λmax (approx.) | Spectral Region |
|---|---|---|---|
| Cold sky or deep space | 3 | ∼ 1 mm | Microwave / far IR |
| Room-temperature object | 300 | ∼ 9.7 μm | Mid-IR |
| Hot kiln or furnace | 1200 | ∼ 2.4 μm | Near-IR |
| Sun’s surface | 5800 | ∼ 500 nm | Visible (green) |
| Very hot star | 20000 | ∼ 145 nm | Ultraviolet |
This table can guide you when choosing reasonable temperature ranges for the calculator and interpreting which band the peak will fall into.
Assumptions, Limitations, and Good Practices
To use the results correctly, keep the following points in mind:
- Ideal blackbody assumption: The formula assumes a perfect blackbody emitter with emissivity of 1 at all wavelengths. Real materials deviate from this, so their peak emission can shift and their spectra can have features that Wien’s law does not capture.
- Temperature in kelvin only: The input must be an absolute temperature in kelvin, not degrees Celsius or Fahrenheit. To convert from °C, add 273.15 to get K before entering the value.
- Physically meaningful range: The law is intended for T > 0 K. The calculator should reject negative temperatures and may clamp or warn on extremely large values where numerical rounding becomes significant.
- Peak wavelength vs. perceived color: The peak of the blackbody curve is not the same as the perceived color of an object, which depends on the entire spectrum and on human vision response. The calculator reports λmax, not color temperature in the photographic sense.
- Educational, not a full radiative-transfer model: The tool is designed for teaching, quick estimation, and sanity checks in physics and astronomy. It does not include line absorption/emission, atmospheric effects, or detailed material properties.
- Graph interpretation: If a visualization is shown and it disappears or is replaced by a warning, it usually indicates an invalid or unsupported input (for example, T ≤ 0 K). Correct the temperature and recompute.
For more advanced analyses involving non-blackbody spectra, wavelength-dependent emissivity, or radiative heat transfer in complex geometries, you would need specialized software or more detailed models than Wien’s displacement law alone.
Related Concepts and Further Exploration
Wien’s law is one of several key relations that describe thermal radiation. It complements the Stefan–Boltzmann law, which connects temperature to total radiated power, and Planck’s law, which describes the full spectral distribution. When used together, these tools give a rich picture of how hot objects emit energy across the electromagnetic spectrum.
If your toolkit includes other calculators for Planck’s law, Stefan–Boltzmann flux, or color temperature, linking them conceptually with this Wien’s displacement calculator can help build intuition across temperature, spectrum, and brightness.
Spectrum Conductor Resonance Lab
Drop beneath the calculation to tune prisms and photodiodes in real time. Drag or tap across the canvas—or use your arrow keys—to align the lab’s color filter with incoming stellar pulses before they slip past the detector. Hold balance for a full session to feel how temperature, wavelength, and power interlock.
Short bursts on the arrows nudge the filter gently; sweeping your finger across the canvas grants faster jumps for emergency catches.
Chosen calculator & fit: Wien’s displacement law already ties a single temperature to a vivid hue. Turning that relationship into an active detector lab lets visitors feel how the peak wavelength shifts and why precision filters matter when measuring stars or kilns.
Game concept: Spectrum Conductor Resonance Lab puts you at the console during a 70–90 second photon storm. Procedural wave trains derived from the calculator’s peak wavelength glide down the canvas, and your job is to slide a tunable prism so the detector resonates as each pulse crosses the focus rail. The score and glow respond instantly, building a mini narrative from calm alignment to frantic tuning and, finally, a triumphant cadence of matched colors.
- Mechanics: Drag, tap, or press ←/→/A/D to shift the filter band; press Space for a timed resonance burst that stabilizes jitter but costs charge if mistimed. Keyboard Enter or the overlay button starts play.
- Feedback: Parallax starfields, shimmering waveforms, particle sparks, adaptive coaching tips, and live readouts of target wavelength and detector balance make every adjustment tactile.
- Procedural variety: Target wavelengths wobble around the current temperature, with storm fronts, double pulses, and calm periods ensuring no two runs share the same beat.
Technical approach: A requestAnimationFrame loop with delta timing, object pools, ResizeObserver scaling, and prefers-reduced-motion awareness keeps the canvas fluid at 60 FPS. Calculator submissions feed updated temperatures into the game’s scenario generator, while localStorage preserves best scores and the overlay summarizes educational insights at the end of each session.