Blackbody Radiation Calculator

A Blackbody Radiation Calculator is a useful tool that helps in calculating the spectral radiance emitted by a blackbody at a given wavelength and temperature. This calculation is based on Planck's law, which describes how much radiation is emitted by a blackbody in thermal equilibrium at a specific temperature.

Purpose and Functionality

The primary purpose of a Blackbody Radiation Calculator is to provide an easy way to calculate the spectral radiance for different wavelengths and temperatures. This is particularly useful in fields like astrophysics, climate science, and materials science, where understanding the radiation properties of objects is crucial.

The Formula

The formula used in the Blackbody Radiation Calculator is derived from Planck's law and is given by:

B(λ,T)=2hc2λ51ehcλkBT−1B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}B(λ,T)=λ52hc2​eλkB​Thc​−11​

Where:

• B(λ,T)B(\lambda, T)B(λ,T) is the spectral radiance.
• λ\lambdaλ is the wavelength.
• TTT is the absolute temperature of the blackbody.
• hhh is Planck's constant (6.626×10−346.626 \times 10^{-34}6.626×10−34 Js).
• ccc is the speed of light in a vacuum (3×1083 \times 10^83×108 m/s).
• kBk_BkB​ is Boltzmann's constant (1.381×10−231.381 \times 10^{-23}1.381×10−23 J/K).

Calculation Steps

1. Convert constants (if not using SI units directly):
   • Planck's constant (hhh): 6.626×10−346.626 \times 10^{-34}6.626×10−34 Js
   • Speed of light (ccc): 3×1083 \times 10^83×108 m/s
   • Boltzmann's constant (kBk_BkB​): 1.381×10−231.381 \times 10^{-23}1.381×10−23 J/K
2. Calculate the spectral radiance:
   • Plug the values of λ\lambdaλ, TTT, hhh, ccc, and kBk_BkB​ into the formula.
   • Compute the exponent part ehcλkBTe^{\frac{hc}{\lambda k_B T}}eλkB​Thc​.
   • Compute the denominator ehcλkBT−1e^{\frac{hc}{\lambda k_B T}} - 1eλkB​Thc​−1.
   • Compute the numerator 2hc2λ5\frac{2hc^2}{\lambda^5}λ52hc2​.
   • Divide the numerator by the denominator to get B(λ,T)B(\lambda, T)B(λ,T).

Example Calculation

Let's calculate the spectral radiance at a wavelength of 500 nm (which is 500×10−9500 \times 10^{-9}500×10−9 m) and a temperature of 3000 K.

1. Convert the wavelength to meters: 500 nm=500×10−9 m500 \text{ nm} = 500 \times 10^{-9} \text{ m}500 nm=500×10−9 m
2. Use the formula: B(500×10−9,3000)=2×6.626×10−34×(3×108)2(500×10−9)51e6.626×10−34×3×108500×10−9×1.381×10−23×3000−1B(500 \times 10^{-9}, 3000) = \frac{2 \times 6.626 \times 10^{-34} \times (3 \times 10^8)^2}{(500 \times 10^{-9})^5} \frac{1}{e^{\frac{6.626 \times 10^{-34} \times 3 \times 10^8}{500 \times 10^{-9} \times 1.381 \times 10^{-23} \times 3000}} - 1}B(500×10−9,3000)=(500×10−9)52×6.626×10−34×(3×108)2​e500×10−9×1.381×10−23×30006.626×10−34×3×108​−11​
3. Calculate the exponent: 6.626×10−34×3×108500×10−9×1.381×10−23×3000=9.58×10−1\frac{6.626 \times 10^{-34} \times 3 \times 10^8}{500 \times 10^{-9} \times 1.381 \times 10^{-23} \times 3000} = 9.58 \times 10^{-1}500×10−9×1.381×10−23×30006.626×10−34×3×108​=9.58×10−1
4. Calculate the denominator: e0.958−1≈1.606e^{0.958} - 1 \approx 1.606e0.958−1≈1.606
5. Calculate the numerator: 2×6.626×10−34×(3×108)2(500×10−9)5≈2.98×10−7\frac{2 \times 6.626 \times 10^{-34} \times (3 \times 10^8)^2}{(500 \times 10^{-9})^5} \approx 2.98 \times 10^{-7}(500×10−9)52×6.626×10−34×(3×108)2​≈2.98×10−7
6. Calculate the spectral radiance: B(500×10−9,3000)=2.98×10−71.606≈1.86×10−7 W⋅m−2⋅sr−1⋅m−1B(500 \times 10^{-9}, 3000) = \frac{2.98 \times 10^{-7}}{1.606} \approx 1.86 \times 10^{-7} \text{ W} \cdot \text{m}^{-2} \cdot \text{sr}^{-1} \cdot \text{m}^{-1}B(500×10−9,3000)=1.6062.98×10−7​≈1.86×10−7 W⋅m−2⋅sr−1⋅m−1

This gives the spectral radiance at a wavelength of 500 nm for a blackbody at 3000 K.

Information Table

Constant	Value
Planck's constant (hhh)	6.626×10−346.626 \times 10^{-34}6.626×10−34 Js
Speed of light (ccc)	3×1083 \times 10^83×108 m/s
Boltzmann's constant (kBk_BkB​)	1.381×10−231.381 \times 10^{-23}1.381×10−23 J/K

Conclusion

The Blackbody Radiation Calculator is a powerful tool for determining the spectral radiance of a blackbody at any given wavelength and temperature. By understanding and using Planck's law, this calculator can aid in various scientific and engineering applications, making it invaluable for research and practical uses alike.