Blackbody Radiation Calculator
Computes the spectral radiance of a perfect blackbody at a given temperature and wavelength using Planck's law. Use it when studying thermal emission from stars, heated materials, or designing infrared sensors.
About this calculator
Planck's law describes how a perfect blackbody emits electromagnetic radiation across wavelengths at a given temperature. The spectral radiance B is given by: B(λ, T) = (2hc² / λ⁵) / (exp(hc / λk_B T) − 1), where h = 6.626×10⁻³⁴ J·s is Planck's constant, c = 3×10⁸ m/s is the speed of light, k_B = 1.381×10⁻²³ J/K is Boltzmann's constant, λ is wavelength in metres, and T is temperature in Kelvin. At shorter wavelengths or lower temperatures, the exponential denominator grows large and radiance drops sharply. At the peak wavelength (given by Wien's displacement law, λ_max = 2.898×10⁻³ / T), radiance is maximised. This formula is foundational in astrophysics, climate science, and optical engineering.
How to use
Suppose you want the spectral radiance of a blackbody at T = 5778 K (the Sun's surface) at λ = 500 nm (5×10⁻⁷ m). Step 1 — compute the numerator: 2 × 6.626×10⁻³⁴ × (3×10⁸)² = 1.191×10⁻¹⁶. Step 2 — divide by λ⁵ = (5×10⁻⁷)⁵ = 3.125×10⁻³²: gives 3.81×10¹⁵. Step 3 — compute the exponent: (6.626×10⁻³⁴ × 3×10⁸) / (5×10⁻⁷ × 1.381×10⁻²³ × 5778) ≈ 4.97. Step 4 — denominator: exp(4.97) − 1 ≈ 143.3. Step 5 — B ≈ 3.81×10¹⁵ / 143.3 ≈ 2.66×10¹³ W·m⁻²·sr⁻¹·m⁻¹.
Frequently asked questions
What is Planck's law and why does it describe blackbody radiation?
Planck's law was derived by Max Planck in 1900 to resolve the ultraviolet catastrophe predicted by classical physics. It models electromagnetic radiation emitted by an idealised body that absorbs all incoming radiation. The key insight is that energy is quantised — photons carry discrete packets of energy hf — which suppresses radiation at very short wavelengths. This single formula correctly predicts the entire thermal emission spectrum of stars, ovens, and any heated object.
How does temperature affect the peak wavelength of blackbody radiation?
As temperature increases, the peak of the emission spectrum shifts to shorter (bluer) wavelengths, following Wien's displacement law: λ_max = 2.898×10⁻³ / T. A cool star at 3000 K peaks in the infrared (~970 nm), while the Sun at 5778 K peaks in visible green (~500 nm). Very hot objects like blue supergiants (T > 20 000 K) peak in the ultraviolet. This shift is why heated metal glows red, then orange, then white as temperature rises.
What units does spectral radiance use and how should I interpret the result?
Spectral radiance from Planck's law is expressed in watts per square metre per steradian per metre (W·m⁻²·sr⁻¹·m⁻¹). It tells you how much power is emitted per unit surface area, per unit solid angle, and per unit wavelength interval. To obtain total power over a wavelength range, you would integrate B(λ,T) over λ. When working with nanometres instead of metres, be careful to convert both the wavelength input and scale the output accordingly by a factor of 10⁻⁹.