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Chapter 36: Problem 23

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

Expert verified
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

01

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
02

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
03

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.

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