A blackbody calculator is a tool used to calculate the spectral radiance of a blackbody at a specific wavelength and temperature. This is important in fields like astronomy, physics, and engineering to understand how objects emit radiation based on their temperature.
Purpose and Functionality
The blackbody calculator uses Planck's law to determine the spectral radiance of a blackbody. Spectral radiance is the amount of radiation energy emitted by the blackbody per unit area, per unit wavelength, per unit solid angle.
Formula
The formula to calculate the spectral radiance B(λ,T)B(\lambda, T)B(λ,T) at a specific wavelength λ\lambdaλ and temperature TTT is:
B(λ,T)=2hc2λ5⋅1ehcλkT−1B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1}B(λ,T)=λ52hc2⋅eλkThc−11
Where:
- B(λ,T)B(\lambda, T)B(λ,T) = Spectral radiance (W/m²/sr/m)
- hhh = Planck's constant (6.626 × 10⁻³⁴ Js)
- ccc = Speed of light (3 × 10⁸ m/s)
- λ\lambdaλ = Wavelength (m)
- kkk = Boltzmann constant (1.381 × 10⁻²³ J/K)
- TTT = Temperature (K)
Inputs
- Wavelength (λ\lambdaλ): The wavelength at which the spectral radiance is to be calculated, in meters (m).
- Temperature (T): The temperature of the blackbody, in Kelvin (K).
Calculation Steps
- Input the Wavelength (λ\lambdaλ): For example, let's use λ=500×10−9\lambda = 500 \times 10^{-9}λ=500×10−9 meters (500 nm, which is in the visible spectrum).
- Input the Temperature (T): For example, let's use T=6000T = 6000T=6000 K (approximate temperature of the Sun).
- Calculate the Spectral Radiance B(λ,T)B(\lambda, T)B(λ,T):
- Compute the exponent term: hcλkT\frac{hc}{\lambda kT}λkThc Using the provided constants: h=6.626×10−34 Jsh = 6.626 \times 10^{-34} \text{ Js}h=6.626×10−34 Js c=3×108 m/sc = 3 \times 10^8 \text{ m/s}c=3×108 m/s λ=500×10−9 m\lambda = 500 \times 10^{-9} \text{ m}λ=500×10−9 m k=1.381×10−23 J/Kk = 1.381 \times 10^{-23} \text{ J/K}k=1.381×10−23 J/K T=6000 KT = 6000 \text{ K}T=6000 KhcλkT=(6.626×10−34)×(3×108)(500×10−9)×(1.381×10−23)×6000≈4.8\frac{hc}{\lambda kT} = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^8)}{(500 \times 10^{-9}) \times (1.381 \times 10^{-23}) \times 6000} \approx 4.8λkThc=(500×10−9)×(1.381×10−23)×6000(6.626×10−34)×(3×108)≈4.8
- Evaluate the exponent: e4.8−1≈119.87e^{4.8} - 1 \approx 119.87e4.8−1≈119.87
- Calculate the spectral radiance: B(λ,T)=2(6.626×10−34)(3×108)2(500×10−9)5⋅1119.87≈3.74×1013 W/m2/sr/mB(\lambda, T) = \frac{2(6.626 \times 10^{-34})(3 \times 10^8)^2}{(500 \times 10^{-9})^5} \cdot \frac{1}{119.87} \approx 3.74 \times 10^{13} \text{ W/m}^2\text{/sr/m}B(λ,T)=(500×10−9)52(6.626×10−34)(3×108)2⋅119.871≈3.74×1013 W/m2/sr/m
Example Calculation
Using the provided constants and input values:
- Calculate the exponent term: hcλkT≈4.8\frac{hc}{\lambda kT} \approx 4.8λkThc≈4.8
- Evaluate the exponent: e4.8−1≈119.87e^{4.8} - 1 \approx 119.87e4.8−1≈119.87
- Calculate the spectral radiance: B(λ,T)≈3.74×1013 W/m2/sr/mB(\lambda, T) \approx 3.74 \times 10^{13} \text{ W/m}^2\text{/sr/m}B(λ,T)≈3.74×1013 W/m2/sr/m
Thus, the spectral radiance at a wavelength of 500 nm and a temperature of 6000 K is approximately 3.74×1013 W/m2/sr/m3.74 \times 10^{13} \text{ W/m}^2\text{/sr/m}3.74×1013 W/m2/sr/m.
Information Table
| Parameter | 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 constant (kkk) | 1.381×10−231.381 \times 10^{-23}1.381×10−23 J/K |
| Example Wavelength (λ\lambdaλ) | 500×10−9500 \times 10^{-9}500×10−9 m |
| Example Temperature (T) | 6000 K |
| Spectral Radiance (B(λ,T)B(\lambda, T)B(λ,T)) | 3.74×10133.74 \times 10^{13}3.74×1013 W/m²/sr/m |
Conclusion
The blackbody calculator is a useful tool for calculating the spectral radiance of a blackbody based on its temperature and the wavelength of interest. It applies Planck's law to provide precise measurements, which are critical in many scientific and engineering applications. By understanding the spectral radiance, we can gain insights into the thermal properties of objects and their emission of radiation.