Your body, at a temperature of about \(37^{\circ} \mathrm{C}\left(98.6^{\circ} \mathrm{F}\right)\), emits radiation in the infrared region of the spectrum. a. What is the peak wavelength, in micrometers, of your emitted radiation? b. Assuming an exposed body surface area of \(0.25 \mathrm{m}^{2}\), how many watts of power do you radiate?

Wien's Displacement Law helps explain why your body emits radiation primarily in the infrared spectrum. This law states that the wavelength at which the emission of a blackbody spectrum peaks is inversely proportional to its temperature in Kelvin. In equation form, it is given by: \[ \text{Peak Wavelength} (\text{micrometers}) = \frac{2898 \text{(micrometer K)}}{T(K)} \]
The constant 2898 comes from empirical data. By converting body temperature from Celsius to Kelvin and applying this equation, you can find the peak emission wavelength. For a human body at 37°C, converting to Kelvin gives 310.15 K. Substituting into Wien's Displacement Law gives a peak wavelength of about 9.34 micrometers. This falls within the infrared range, which is why we often think of body heat as 'infrared radiation.'

The Stefan-Boltzmann Law is essential for calculating the total power radiated by a blackbody. This law states that the power radiated per unit area of a blackbody is proportional to the fourth power of its absolute temperature. The formula is: \[ P = e \times A \times \sigma \times T^4 \]
Here, \text{e} is the emissivity which measures the efficiency of radiation (for human skin, it's 0.97), A is the surface area (here 0.25 m²), *** is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \text{W/m}^2\text{K}^4\), and T is the absolute temperature in Kelvin. By substituting the values into this formula, you can calculate the total power radiated by a human body. For a body at 310.15 K with a surface area of 0.25 m², this calculates to approximately 129.12 Watts.

Infrared radiation is a type of electromagnetic radiation with wavelengths longer than visible light but shorter than microwaves. Human bodies emit infrared radiation as part of their natural heat dissipation.
Infrared light has wavelengths between 700 nm to 1 mm, and it is invisible to the naked eye but can be felt as heat. This form of radiation is crucial in various technologies like thermal imaging cameras and medical diagnostics. The peak wavelength of human radiation, calculated using Wien's Displacement Law, falls into the infrared range. This is why thermal imaging devices can 'see' the heat radiated by human bodies.

Temperature conversion is essential in physics and many other sciences. In this exercise, the body temperature was initially given in Celsius but needed to be converted to Kelvin. The conversion formula is straightforward: \[ T(K) = T(°C) + 273.15 \]
For example, a body temperature of 37°C converts to 310.15 K. Using Kelvin is crucial because many physical laws, including Wien's Displacement Law and the Stefan-Boltzmann Law, require absolute temperature. This conversion ensures that the calculations for peak wavelength and radiated power are accurate.
Remember, always convert to Kelvin when dealing with thermal radiation problems to avoid errors.