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 \(70^{\circ} \mathrm{F}\).

Short answer

The wavelength at which the human body (with a skin temperature of \(70^{\circ} \mathrm{F}\)) emits the maximum electromagnetic radiation can be found using Wien's Law after converting the temperature from Fahrenheit to Kelvin. The computation involves substituting the known values into the formula.

Step-by-step solution

Convert Fahrenheit to Kelvin

To change temperature from Fahrenheit to Kelvin, the formula to be used is \[T(K) = \frac{5}{9}(T(F) - 32) + 273.15\] So the temperature in Kelvin is: \[T(K) = \frac{5}{9}(70 - 32) + 273.15\]

Apply Wien's Law

Once the temperature in Kelvin is obtained, Wien's Law can be applied. The formula is \[λ_{max} = \frac{b}{T}\], where b is the Wien's constant equal to \(2.898 \times 10^{-3} m.K\). Substitute the values for T(K) obtained in step 1 and b into the equation to find \(λ_{max}\).

Compute \(λ_{max}\)

Finally, after substituting the values into the Wien's Law formula, compute the result which will give the maximum wavelength \(λ_{max}\) at which the human body emits the maximum radiation.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that spreads out as it travels through space. It can be visualized as waves composed of both electric and magnetic fields oscillating perpendicular to each other, moving at the speed of light. This radiation includes a vast spectrum of wavelengths and frequencies, from very long radio waves to very short gamma rays. The human body itself emits infrared radiation because of the heat it produces. Understanding electromagnetic radiation is crucial in many fields, including astronomy, telecommunications, and medicine.

For instance, when discussing the heat from a human body or the light emitted by a star, we are referring to types of electromagnetic radiation. This also relates directly to Wien’s Law, which connects the temperature of an object to the peak wavelength of the radiation it emits.

Blackbody Radiation

Blackbody radiation refers to a hypothetical object that is a perfect emitter and absorber of all forms of electromagnetic radiation. In reality, no perfect blackbodies exist, but many objects act approximately like blackbodies. For example, stars, including our sun, can be treated as blackbodies to simplify the study of the radiation they emit.

Blackbody radiation is characterized by a spectrum that depends only on the object's temperature. An important aspect of this kind of radiation is its predictability, as described by Wien's Law. This allows us to estimate the temperature of an object based on the radiation it emits, which is invaluable in understanding the physical properties of stars and other astronomical objects.

Temperature Conversion

Temperature conversion is crucial when working with different temperature scales. The most commonly used scales are Celsius, Fahrenheit, and Kelvin. Each scale has its own applications; for example, Celsius is widely used in most of the world for daily weather reports, while Fahrenheit is primarily used in the United States. The Kelvin scale, however, is the standard unit of measure for temperature in the sciences.

To convert Fahrenheit to Kelvin, which is necessary for applying Wien’s Law, you can use the conversion formula \(T(K) = \frac{5}{9}(T(F) - 32) + 273.15\). It is important to make this conversion because Wien's Law operates with temperatures measured in Kelvin only. Failing to correctly convert temperatures can lead to inaccurate calculations of emitted radiation wavelengths.

Kelvin Scale

The Kelvin scale is an absolute temperature scale that measures temperature from absolute zero, theoretically the coldest possible temperature, where molecular movement stops. One Kelvin represents the same temperature change as one degree Celsius, but they have different starting points: 0K is -273.15°C. Unlike Celsius and Fahrenheit, the Kelvin scale does not utilize the term 'degree' and is written simply as 'Kelvin' (K).

In scientific contexts, especially in physics and astronomy, the Kelvin scale is used because many formulas, including Wien's Law, are based on absolute temperature measurements. This scale eliminates the negative temperature values that can complicate calculations, making it a standardized unit for scientific research.