The total emissivity of zirconia and tungsten can't be directly calculated without a spectral emissivity plot. However, using Planck's law, one can estimate the spectral emissive power of tungsten at a specific wavelength (e.g., 3 µm). Comparing a human's radiative power to an incandescent filament relies on the Stefan-Boltzmann Law, with the filament emitting significantly more power due to its higher temperature (~2,400 K compared to human skin at ~300 K).
The question involves calculating the total emissivity of zirconia and tungsten and the spectral emissive power of a tungsten filament at a specific wavelength, using concepts from thermodynamics and blackbody radiation.
To calculate the total emissivity (e) for each material, you would typically need to integrate the spectral emissivity over all wavelengths and then divide by the integral of a blackbody's emissivity over the same wavelength range. However, the information given in the question is insufficient to perform these calculations directly.
The spectral emissive power of a tungsten filament can be found using Planck's law, which relates the radiation emitted at a certain wavelength to the temperature of the blackbody. The relevant equation for spectral emissive power E(λ, T) at temperature T and wavelength λ is given by:
E(λ, T) = (2πhc^2/λ^5) / (e^(hc/λkT) - 1)
where h is Planck's constant, c is the speed of light, k is Boltzmann's constant, and λ is the wavelength in meters. To find the power per unit wavelength, you would normalize this by multiplying by the emissivity e of tungsten at that wavelength. In this case, we're given that at T = 6000 K and λ = 1 µm, the emissive power is given as 3.73 × 10^13 W/m² per meter of wavelength. If you have a plot or table of emissivity versus wavelength for tungsten, you can find its emissivity at λ = 3 µm and multiply it by the formula above (adjusted for the correct temperature and wavelength) to get the emissive power at that specific point.
Regarding the comparison between an incandescent filament at ~2,400 K and human skin at ~300 K, the filament radiates much more power per unit area. This can be estimated using the Stefan-Boltzmann Law which states that the power radiated per unit area of a blackbody is proportional to the fourth power of the absolute temperature (I ∝ T⁴). Therefore, the filament emits tremendously more power due to its higher temperature.