Question

Human bodies also glow by the same physics as the sun or a light bulb filament, only it is too far out in the infrared for the human eye to see. For familiar objects (and human skin) all in the neighborhood of \(300 \mathrm{~K}\), what is the approximate wavelength of peak blackbody radiation, in microns?

Short answer

Answer: The approximate wavelength of peak blackbody radiation for familiar objects and human skin at a temperature of 300 K is 9.67 μm.

Step-by-step solution

Write down the known values

We know the temperature \(T = 300 \mathrm{K}\) and Wien's constant \(b = 2.9 \times 10^{-3} \mathrm{m \cdot K}\).

Write down Wien's Law

Wien's Law states that \(\lambda_{max} = \frac{b}{T}\), where \(\lambda_{max}\) is the peak wavelength, \(b\) is Wien's constant, and \(T\) is the temperature.

Substitute the known values to find \(\lambda_{max}\)

We can now substitute the values for \(T\) and \(b\) into Wien's Law: \(\lambda_{max} = \frac{2.9 \times 10^{-3} \mathrm{m \cdot K}}{300 \mathrm{K}}\)

Calculate \(\lambda_{max}\) and convert to microns

Dividing the values, we get: \(\lambda_{max} = 9.67 \times 10^{-6} \mathrm{m}\) Since 1 micron is equal to \(10^{-6} \mathrm{m}\), we can convert this result to microns: \(\lambda_{max} = 9.67 \mathrm{\mu m}\)

Final answer

The approximate wavelength of peak blackbody radiation for familiar objects and human skin at a temperature of \(300 \mathrm{K}\) is \(9.67 \mathrm{\mu m}\).

Key concepts

Wien's Law

Understanding the mechanism behind blackbody radiation is essential for comprehending various phenomena, such as why humans emit infrared radiation. This leads us directly to one of the pivotal laws in this context—Wien's Law.

Wien's Law is a principle in physics that defines the relationship between the temperature of an object and the wavelength at which it emits radiation most strongly. The law is expressed mathematically by the formula \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) represents the peak wavelength of radiation, \( T \) is the absolute temperature of the blackbody in kelvins (K), and \( b \) is a constant of proportionality known as Wien's displacement constant, approximately equal to \( 2.9 \times 10^{-3} \mathrm{m \cdot K} \).

This has fascinating implications: as the temperature of a blackbody increases, the peak wavelength of emitted radiation shifts to shorter wavelengths. For a cool body like a human being, the peak wavelength is in the infrared range, which is invisible to our eyes but can be detected by thermal imaging devices.

Infrared Radiation

When we delve into the types of radiation, one particularly interesting range is infrared radiation. This is the type of electromagnetic radiation, with wavelengths longer than visible light but shorter than microwave radiation. Strikingly, it's exactly the kind of radiation that warm bodies, such as humans, predominantly emit.

Infrared radiation can be classified into three categories based on its wavelength: near-infrared, mid-infrared, and far-infrared. While near-infrared is closest to visible light and can sometimes be seen as red glow using cameras, mid and far-infrared are associated with thermal emissions from objects - the heat that we feel but cannot see.

This radiation plays a crucial role in various applications, including thermal imaging, where cameras capture the infrared radiation emitted by objects to determine their temperature, and in remote controls, where near-infrared light is used to send signals.

Peak Wavelength Calculation

Now, moving onto practical applications, let’s discuss how to calculate the peak wavelength of emitted radiation using Wien's Law, as highlighted in our exercise. The peak wavelength is the wavelength at which the intensity of the radiation is at its highest.

To calculate \( \lambda_{max} \), one must first ascertain the temperature of the blackbody in kelvins. Then, Wien's displacement constant, which is always the same value, is divided by this temperature. The result of this computation gives the peak wavelength in meters.

Converting to Microns

Since many scientific fields prefer using microns (\(\mu m\)) for wavelength measurements, it's common to convert the peak wavelength from meters. The conversion factor is straightforward as \(1 \mu m = 10^{-6} m\). Therefore, multiplying the peak wavelength by \(10^6\) will convert it into microns, providing a more intuitive sense of the scale at which these phenomena occur.

The calculation of the peak wavelength is not just a theoretical exercise; it has practical applications in science and technology, such as in designing thermal cameras and understanding the heat signatures of different objects.