0.0000000567
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0.0000000567
W/(m²·K⁴)
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The Stefan-Boltzmann Constant Calculator computes the thermal radiation power emitted by a body at temperature $$T$$ using the Stefan-Boltzmann law:
$$P = \varepsilon \sigma A T^4$$
where $$\sigma = 5.670\,374\,419 \times 10^{-8}\;\text{W m}^{-2}\text{K}^{-4}$$ is the Stefan-Boltzmann constant, $$A$$ is the surface area, $$\varepsilon$$ is the emissivity (1 for a perfect blackbody), and $$T$$ is the absolute temperature in kelvin. The calculator also applies Wien's displacement law to find the peak emission wavelength:
$$\lambda_{\max} = \frac{b}{T}, \quad b = 2.897\,772 \times 10^{-3}\;\text{m K}$$
This tool is essential for astrophysics (stellar luminosity), thermal engineering (radiative heat loss), climate science (planetary energy balance), and infrared technology.
The Stefan-Boltzmann law integrates the Planck blackbody spectrum over all wavelengths to give the total power radiated per unit area:
$$j^* = \sigma T^4 = \frac{2\pi^5 k_B^4}{15 h^3 c^2} T^4$$
The Stefan-Boltzmann constant $$\sigma$$ is derived from other fundamental constants: $$k_B$$ (Boltzmann), $$h$$ (Planck), and $$c$$ (speed of light). Its value is:
$$\sigma = \frac{2\pi^5 k_B^4}{15 h^3 c^2} = 5.670\,374\,419 \times 10^{-8}\;\text{W m}^{-2}\text{K}^{-4}$$
For a real surface with emissivity $$\varepsilon$$ (0 to 1), the total radiated power is:
$$P = \varepsilon \sigma A T^4$$
The net radiative power accounts for absorption from the surroundings at ambient temperature $$T_{\text{amb}}$$:
$$P_{\text{net}} = \varepsilon \sigma A \left(T^4 - T_{\text{amb}}^4\right)$$
The $$T^4$$ dependence makes radiation extremely sensitive to temperature. Doubling the temperature increases radiation by a factor of 16.
Wien's displacement law gives the wavelength at which the blackbody spectrum peaks:
$$\lambda_{\max} = \frac{b}{T} = \frac{2.898 \times 10^{-3}}{T}\;\text{m}$$
The Sun ($$T \approx 5778\;\text{K}$$) peaks at $$\lambda_{\max} \approx 502\;\text{nm}$$ (green light). A human body ($$T \approx 310\;\text{K}$$) peaks at $$\lambda_{\max} \approx 9.35\;\mu\text{m}$$ (mid-infrared). Incandescent bulbs ($$T \approx 2700\;\text{K}$$) peak at about 1073 nm (near-infrared), which is why most of their energy is wasted as heat.
Applications include computing stellar luminosities ($$L = 4\pi R^2 \sigma T_{\text{eff}}^4$$), designing thermal insulation, modeling planetary climates, and calibrating infrared cameras.
The Total Radiated Power gives the energy per second emitted by the entire surface — for the Sun, this is about $$3.85 \times 10^{26}$$ W (the solar luminosity). The Power per Unit Area (radiative flux) tells you the intensity of emission per square metre. The Net Radiated Power subtracts the radiation absorbed from the environment — this is the actual heat loss rate. The Wien Peak Wavelength tells you what type of radiation dominates: visible, infrared, ultraviolet, etc.
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The Sun (T = 5778 K, R = 6.96 × 10⁸ m, A = 6.09 × 10¹⁸ m²) radiates ~3.85 × 10²⁶ W. Its peak emission is at 502 nm (green), though it appears white due to broad-spectrum emission. The cosmic microwave background (2.725 K) contributes negligible absorption.
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A human body (37°C, ~1.7 m² skin area, ε ≈ 0.98) radiates about 825 W total, but net loss to a 20°C room is only ~95 W because the surroundings radiate back. Peak emission is at 9.34 μm in the mid-infrared — this is what thermal cameras detect.
The Stefan-Boltzmann constant $$\sigma = 5.670\,374\,419 \times 10^{-8}\;\text{W m}^{-2}\text{K}^{-4}$$ appears in the Stefan-Boltzmann law $$P = \sigma A T^4$$, which gives the total power radiated by a blackbody. It is derived from Planck's constant, the Boltzmann constant, and the speed of light: $$\sigma = 2\pi^5 k_B^4 / (15 h^3 c^2)$$.
Emissivity $$\varepsilon$$ (0 to 1) measures how efficiently a surface radiates compared to a perfect blackbody. A blackbody has $$\varepsilon = 1$$. Polished metals have low emissivity ($$\varepsilon \approx 0.03$$–$$0.1$$), while human skin, water, and most non-metals have high emissivity ($$\varepsilon > 0.9$$). Emissivity can depend on wavelength, angle, and temperature.
The $$T^4$$ dependence comes from integrating the Planck function $$B(\nu, T)$$ over all frequencies. Both the peak intensity (proportional to $$T^3$$) and the peak frequency (proportional to $$T$$) increase with temperature, giving a combined $$T^4$$ scaling for total power. This makes radiation the dominant heat transfer mechanism at high temperatures.
Wien's law $$\lambda_{\max} = b/T$$ with $$b = 2.898 \times 10^{-3}\;\text{m K}$$ gives the wavelength at which a blackbody emits most intensely. Hotter objects peak at shorter wavelengths: the Sun (5778 K) peaks in visible green, while room-temperature objects peak in the mid-infrared (~10 μm). This is why hot objects glow red, then white, then blue.
A star's luminosity is $$L = 4\pi R^2 \sigma T_{\text{eff}}^4$$, where $$R$$ is the stellar radius and $$T_{\text{eff}}$$ is the effective temperature. By measuring luminosity (from apparent brightness and distance) and temperature (from spectral analysis), astronomers can determine stellar radii. This places stars on the Hertzsprung-Russell diagram and reveals their evolutionary stage.
Earth absorbs solar radiation ($$\sim 1361\;\text{W/m}^2$$ at the top of the atmosphere) and re-emits thermal radiation at ~255 K effective temperature. The balance $$\sigma T_e^4 = (1-\alpha)S/4$$ (where $$\alpha$$ is albedo, $$S$$ is solar constant) determines Earth's equilibrium temperature. Greenhouse gases trap outgoing infrared radiation, raising the surface temperature above the 255 K radiative equilibrium to ~288 K.
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