Question

At what wavelength does the human body emit the maximum electromagnetic radiation? Use Wien's law from Exercise 14 and assume a skin temperature of${70}^{\circ }F$

Short answer

The wavelength corresponds to the maximum electromagnetic radiation emitted by the human body is $9.848\mu m$.

Step-by-step solution

Given data

Skin temperature$=70°\text{F}$

Formula used

Wein's law can be expressed in terms of temperature (T) and maximum wavelength (${\lambda }_{max}$) such that,

${\lambda }_{max}T=2.898\times {10}^{-3}\text{mK}$

Calculation using Wien's law

Convert the Fahrenheit temperature to Kelvin, such that

$\begin{array}{rcl}T& =& \left((70-32)\times \frac{5}{9}\right)°C+273.15K\\ T& =& 294.26\text{K}\\ & & \end{array}$

Using Wein's law

$$\begin{gathered} {\lambda }_{\max }T=2.898\times {10}^{-3}\text{mK} \\ {\lambda }_{\max }=\frac{2.898\times {10}^{-3}\text{mK}}{294.26K} \\ {\lambda }_{\max }=9.848\times {10}^{-6}\text{m}\left(\frac{1\mu \text{m}}{{10}^{-6}\text{m}}\right) \\ {\lambda }_{\max }=9.848\mu \text{m} \end{gathered}$$

Conclusion 

The wavelength which corresponds to the maximum electromagnetic radiation emitted by the human body is $9.848\mu m$.