If you use a filter that transmits only between 1.95-2.05 um, what is the total power falling ona 1 cm^2 detector 1m away from a 1cm^2 1000K black body?
I know the formula for I and I think I need to integrate something, but I have no idea what formula to use for this problem.
To calculate the total power falling on a detector, you can use the Stefan-Boltzmann law, which describes the power radiated by a black body at a certain temperature. It states that the power radiated per unit area (I) is proportional to the fourth power of the temperature (T) and is given by the formula:
I = σ * T^4
Where:
I is the power radiated per unit area (W/m^2),
σ is the Stefan-Boltzmann constant (≈ 5.67 * 10^-8 W/m^2K^4),
T is the temperature of the black body (in Kelvin).
Now, to determine the power falling on the detector, you need to integrate the intensity of radiation over the specified wavelength range using the Planck's law. Planck's law describes the spectral energy density, which represents the amount of energy radiated at different wavelengths by a black body at a given temperature.
The formula for Planck's law is:
B(λ, T) = C / (λ^5 * [exp(hc / (λkT)) - 1])
Where:
B(λ, T) is the spectral energy density at a given wavelength (W/m^2 * m),
C is a constant (≈ 3.74 * 10^4 W/m^2),
λ is the wavelength (m),
h is Planck's constant (≈ 6.63 * 10^-34 J*s),
c is the speed of light (≈ 3.00 * 10^8 m/s),
k is Boltzmann's constant (≈ 1.38 * 10^-23 J/K),
T is the temperature of the black body (in Kelvin).
To integrate the spectral energy density over the specific wavelength range, you need to determine the limits of integration that correspond to the desired wavelength range of the filter (1.95-2.05 μm).
Once you have evaluated the integral of B(λ, T) over the specified wavelength range, you can multiply it by the surface area of the detector to find the total power falling on the detector.
Please note that the calculations involved may require advanced mathematical techniques and numerical methods, such as numerical integration or software tools like Python or MATLAB, depending on the exact values and precision required.