Wien Displacement Law Simulator — Peak Wavelength of Black-Body Radiation

The Wien displacement law follows from differentiating the Planck law $B_{\lambda}(\lambda,T) = (2hc^2/\lambda^5) / (\exp(hc/\lambda k_{B}T) - 1)$ with respect to $\lambda$ and setting the derivative to zero.

$$\lambda_{\max}\,T = b,\qquad b = \frac{hc}{k_{B}\,x_{W}} = 2.898\times10^{-3}\ \mathrm{m\cdot K}$$

Here $x_{W} = 4.965114\ldots$ is the numerical solution of $x e^{x} - 5(e^{x}-1) = 0$, $h$ is the Planck constant, $c$ is the speed of light and $k_{B}$ is the Boltzmann constant. Integrating the same Planck spectrum over all wavelengths returns the Stefan-Boltzmann law: $q = \int_{0}^{\infty} \pi B_{\lambda}\,d\lambda = \sigma T^{4}$ with $\sigma = 2\pi^{5}k_{B}^{4}/(15h^{3}c^{2}) = 5.670\times10^{-8}\ \mathrm{W/(m^{2}\cdot K^{4})}$. A real surface is treated as a gray body with $\varepsilon \le 1$ and gives $q = \varepsilon\sigma T^{4}$.

For a radiator of surface area $A$, the total radiated power is $P = q A$, and assuming an isotropic point source the irradiance on a sphere of radius $R$ is $E = P/(4\pi R^{2})$. The point-source approximation requires $R$ to be much larger than the radiator size; in the near field a view factor must be used and the net exchange between two surfaces becomes $Q_{\mathrm{net}} = \varepsilon\sigma A F_{12}(T_{1}^{4}-T_{2}^{4})$.

Internally the simulator draws the spectrum in the dimensionless self-similar form $u(x) = (1/x^{5})/(\exp(x_{W}/x)-1)$ with $x = \lambda/\lambda_{\max}$, so changing $T$ rescales the curve horizontally without changing its shape - the famous Wien-Planck similarity property of black-body radiation.

Solar irradiance and photovoltaic design: the Sun (T = 5800 K) peaks at lambda_max = 500 nm and concentrates about 43% of its radiated energy in the visible band 380-780 nm. Plug the defaults into the tool and you get q = 64.2 MW/m^2; stretch R to the Earth orbit (1.496e11 m) and the irradiance drops to 1.36 kW/m^2 - the solar constant. This sets the Shockley-Queisser limit (about 33%) for silicon cells with a 1.12 eV band gap and a long-wavelength cut-off near 1100 nm, and directly informs the spectral splitting in tandem and perovskite photovoltaics.

Incandescent bulbs and LED lighting: a tungsten filament at 2800 K peaks at 1035 nm in the near infrared, so only the long-wavelength tail of its spectrum reaches the visible band, giving a luminous efficacy of just 10-15 lm/W. Pushing T to 3300 K shortens lambda_max to 878 nm and almost doubles q (factor 1.94), but Arrhenius-driven evaporation collapses the lifetime. White LEDs avoid the constraint entirely by directly exciting narrow visible bands, reaching efficacies above 100 lm/W.

Infrared thermography and ear thermometers: the human body (T = 310 K) peaks at lambda_max = 9.4 um, well into the mid infrared, so VOx microbolometers and InSb/HgCdTe quantum detectors can image people at room temperature. Set T = 310 K in the tool and lambda_max falls off the 0-3000 nm chart to the right, showing why detector and lens designs must be tuned to the 8-14 um atmospheric window. Applications run from power-plant pipe diagnostics through building-envelope inspection to non-contact fever screening.

Stellar classification and colour temperature: astronomers use the colour of starlight to infer surface temperature. Red giants such as Betelgeuse (T = 3500 K, lambda_max = 828 nm), the Sun (G2V, 5800 K, 500 nm), white dwarfs such as Sirius B (25000 K, 116 nm) and blue giants such as Rigel (11000 K, 263 nm) span ultraviolet to infrared peaks. Sweeping T from 100 to 10000 K shows visually that only a narrow window (about 3700-7600 K) places the peak inside the visible band, highlighting how special the Sun is. The Hertzsprung-Russell diagram is essentially a plot of colour temperature against luminosity sigma*T^4 * 4*pi*R_star^2.

The most common mistake is to assume that the wavelength peak and the frequency peak are simply linked by c = lambda * nu. The wavelength distribution $B_{\lambda}$ and the frequency distribution $B_{\nu}$ are linked by a Jacobian $|d\lambda| = |c/\nu^{2}| d\nu$, and therefore peak at different wavelengths even at the same temperature. The wavelength form gives $\lambda_{\max} T = 2898\ \mu\mathrm{m\cdot K}$ (with $x_{W} = 4.965$), while the frequency form gives $\nu_{\max}/T = 58.79\ \mathrm{GHz/K}$ (with $x_{W}' = 2.821$); the corresponding wavelength $c/\nu_{\max} = 5.099\ \mathrm{mm\cdot K}/T$ is 1.76 times longer than the wavelength peak. Always check whether a paper is using the wavelength or frequency form. This simulator uses the wavelength form throughout.

Another pitfall is to claim that real surfaces are not black bodies, so the Wien law does not apply. In the gray-body approximation (eps weakly dependent on wavelength), the peak position is still b/T - emissivity only lowers the height of the whole spectrum. Only for selective emitters with strongly wavelength-dependent eps - cavity emitters, photonic crystals, two-dimensional materials - does the effective peak deviate from the black-body prediction. The simulator treats eps as wavelength-independent and is therefore correct for gray surfaces; selective emitters require integrating $B_{\lambda}(\lambda,T)\,\varepsilon(\lambda)$ explicitly.

Finally, the point-source formula E = P/(4*pi*R^2) does not work at all distances. It is a far-field approximation valid only when the radiator size sqrt(A) is much smaller than R; in the near field the view factor F is much larger than 1/(4*pi*R^2). For a sphere of diameter D evaluated at its surface (R = D/2) the true F is 0.5, several times larger than the point-source prediction 1/(pi*D^2). CAE thermal analyses use the net exchange $Q_{\mathrm{net}} = \varepsilon\sigma A F_{12}(T_{1}^{4}-T_{2}^{4})$ with view factors obtained from geometric integration or closed-form expressions for cuboids, coaxial cylinders and parallel plates. The simulator is appropriate for far-field estimates such as Sun-Earth and for q and P at the radiating surface itself.