Question

The human body has a surface area of approximately $1.8 \mathrm{m}^{2}$, a surface temperature of approximately $30^{\circ} \mathrm{C}$, and a typical emissivity at infrared wavelengths of e = 0.97. If we make the approximation that all photons are emitted at the wavelength of peak intensity, how many photons per second does the body emit?

Solution

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To get the number of photons emitted by a human body per second we need two variables, energy of a single photon and energy emitted in one second. First we get the latter by employing Stefan-Boltzman's law, $Q=e\sigma A T^4 \Delta t$ so we can write $Q=0.97\times 5.67 \times 10^{-8} \times 1.8 \times (303.15)^4$ $Q=836 \textrm{J}$ Now, we can get the $\lambda_{peak}$ $\lambda_{peak}=\frac{2.9 \times 10^{-3}}{303.15}$ $\lambda_{peak}=9570 \textrm{nm}$ From here the energy of single photon is $E_{ph}=\frac{hc}{\lambda_{peak}}$ $E_{ph}=\frac{6.63\times 10^{-34}\times 3\times 10^8}{9570 \times 10^{-9}}$ $E_{ph}=2\times 10^{-20} \textrm{J}.$ Finally, the number of photons is $N=\frac{Q}{E_{ph}}$ $N=417\times 10^{20}$