Question

The temperature of your skin is approximately \(35.0^{\circ} \mathrm{C}\). a) Assuming that it is a blackbody, what is the peak wavelength of the radiation it emits? b) Assuming a total surface area of \(2.00 \mathrm{~m}^{2}\), what is the total power emitted by your skin? c) Based on your answer in (b), why don't you glow as brightly as a light bulb?

Short answer

Answer: The peak wavelength of radiation emitted by the skin is approximately 1.07 * 10^-5 m. We don't glow as brightly as a lightbulb even though we emit more power (about 933.17 W) because our bodies emit infrared radiation, which is not visible to the naked eye. Lightbulbs, on the other hand, emit most of their energy in the form of visible light.

Step-by-step solution

Calculate the peak wavelength emitted by the skin

According to Wien's Law, the peak wavelength of the emitted radiation is inversely proportional to the temperature of the object. The formula to calculate the peak wavelength (λ) is: λ = b / T where λ is the peak wavelength, T is the temperature (in Kelvin), and b is Wien's constant (3.3 * 10^-3 m*K). First, let's convert the given temperature from Celsius to Kelvin: T = 35.0 + 273.15 = 308.15 K Now let's use the formula to find the peak wavelength: λ = (3.3 * 10^-3) / 308.15 ≈ 1.07 * 10^-5 m

Calculate the power emitted by the skin

To estimate the power emitted by the skin, we will use the Stefan-Boltzmann Law, which states that the power emitted per unit area is directly proportional to the temperature raised to the fourth power. The formula we will use is: P = σ * A * T^4 where P is the total power, σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W * m^-2 * K^-4), A is the surface area, and T is the temperature (in Kelvin). Given the surface area A = 2.00 m^2 and the body temperature of T = 308.15 K, we can now calculate the power emitted by the skin: P = (5.67 * 10^-8 W * m^-2 * K^-4) * (2.00 m^2) * (308.15 K)^4 ≈ 933.17 W

Compare the emitted power to that of a light bulb

The calculated power emitted by the skin is about 933.17 W. In comparison, a typical incandescent light bulb emits around 60-100 W. However, lightbulbs emit most of their energy in the form of visible light, while the human body emits infrared radiation, which is not visible to the naked eye. Therefore, although the power emitted by the skin due to blackbody radiation is higher than that of a light bulb, we do not glow as brightly as a light bulb does because the emitted radiation is not in the visible spectrum.

Key concepts

Wien's Law

Wien's Law helps us understand the color of the light emitted by a blackbody radiator based on its temperature. A blackbody is an idealized physical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. When it comes to the human body, our skin emits radiation as a blackbody.

According to Wien's Law, the peak wavelength \[ \lambda = \frac{b}{T}\] where \( \lambda \) is the peak wavelength, \( T \) is the absolute temperature in Kelvin, and \( b \) is a constant approximately equal to \( 2.9 \times 10^{-3} \ m \cdot K \).

By converting the temperature of our skin, from Celsius to Kelvin, we can calculate the dominant wavelength of the emitted radiation. For example, at a temperature of 35°C (or 308.15K), the peak wavelength is approximately 10.7 micrometers, which falls in the infrared spectrum.Understanding this principle allows us to see why different objects at varied temperatures emit different colors of light. It's why a hot iron glows red, while cooler objects, like human skin, are not visible to the naked eye, as they emit primarily infrared radiation.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law tells us how much energy is emitted from a blackbody as a function of its temperature. This law is essential to understanding the radiative power of objects based on their temperature.

The formula is:\[ P = \sigma \cdot A \cdot T^4\] where \( P \) is the total power emitted, \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \ W \cdot m^{-2} \cdot K^{-4}) \), \( A \) represents the area, and \( T \) is the temperature in Kelvin.

For a typical human body with a surface area of about 2 square meters and a temperature around 308.15K, the emitted power is substantial—approximately 933 watts. However, this energy is mostly not visible to us as it emits in the infrared spectrum.

These calculations show why even though the skin emits a lot of energy, it doesn't appear as bright as a light bulb. A light bulb's power is focused in the visible spectrum, unlike the body's infrared emissions.

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation with wavelengths longer than visible light. It is categorized as being between microwaves and visible light on the electromagnetic spectrum.

Human bodies emit infrared radiation due to their temperature. Because this radiation is not visible to the naked eye, it is often associated with heat rather than light.

Some everyday uses of infrared radiation include:

• Remote controls for televisions, using infrared signals.
• Heat lamps, which keep food warm by emitting infrared radiation.
• Thermal imaging cameras, which detect infrared radiation to "see" heat signatures, letting us visualize temperature differences in complete darkness.

Infrared technology also plays a crucial role in astronomy and meteorology, allowing scientists to observe objects and phenomena invisible to the naked eye, by detecting their heat signature.

Understanding infrared radiation enhances our appreciation of how objects like the human body interact with the electromagnetic spectrum, staying largely unnoticed due to the radiation being outside the visible light range.