Question

At what wavelength does the human body emit the maximum electromagnetic radiation? Use Wien's law from Exercise 79 and assume a skin temperature of \(70^{\circ} \mathrm{F}\)

Short answer

The wavelength at which the human body emits maximum electromagnetic radiation can be found using this formula, but the exact numerical value depends on the result from step 1.

Step-by-step solution

Convert temperature from Fahrenheit to Kelvin

The formula to convert temperature from Fahrenheit to Kelvin is \( K = \frac{5}{9} ( F - 32) + 273.15 \). Substituting \( F = 70 \) into the formula, we get Kelvin value as \( K = \frac{5}{9} (70 - 32) + 273.15 \)

Use Wien's law to find wavelength

Wien's law is given by \( \lambda_{\mathrm{max}} = \frac{b}{T} \), where \( b = 2.8977729 \times 10^{-3} \mathrm{m}\cdot \mathrm{K} \) is Wien's constant. Plug in the values of \( b \) and temperature in Kelvin obtained from step 1 into this equation to find the wavelength at which the human body emits maximum electromagnetic radiation.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a type of energy that travels through space as waves. This energy spans a broad spectrum, known as the electromagnetic spectrum, which includes various types of radiation such as radio waves, microwaves, infrared radiation, visible light, ultraviolet light, X-rays, and gamma rays. The key characteristic of electromagnetic radiation is its wavelength or frequency. Shorter wavelengths correspond to higher frequencies and higher energy, whereas longer wavelengths indicate lower frequencies and lower energy.

The human body, like all objects with a non-zero temperature, emits electromagnetic radiation. This is primarily in the infrared region of the spectrum, which is why infrared cameras can detect people even in complete darkness. The wavelength at which a body emits the most energy is influenced by its temperature, following Wien's Law. Wien's Law helps to determine this peak wavelength, as can be seen in the context of the exercise involving the human body.

Temperature Conversion

Temperature conversion is essential in physics, as different calculations require specific temperature units. For the exercise, we need to convert the body temperature from Fahrenheit to Kelvin. Kelvin is the standard unit for temperature in scientific calculations, as it is an absolute scale based on molecular movement.

The conversion formula from Fahrenheit to Kelvin is \[ K = \frac{5}{9} ( F - 32) + 273.15 \]This converts Fahrenheit, which is more commonly used in everyday life in certain parts of the world, to Kelvin, a scale without negative numbers, which starts at absolute zero. It's important to remember that temperatures in Kelvin are always higher by 273.15 as compared to Celsius.

By ensuring your temperature is converted correctly, you set up for more accurate results when performing scientific calculations such as finding peak electromagnetic radiation wavelengths.

Wavelength Calculation

To find the peak wavelength at which the human body emits radiation, you use Wien's Law. This law gives a clear mathematical model to connect temperature with peak emission wavelength. Wien's Law is expressed as \[ \lambda_{\mathrm{max}} = \frac{b}{T} \]where \( \lambda_{\mathrm{max}} \) is the peak wavelength and \( T \) is the temperature in Kelvin.

In this formula, \( b \) is Wien's constant, approximately \( 2.8977729 \times 10^{-3} \ \mathrm{m} \cdot \mathrm{K} \). This constant makes it possible to consider any body's temperature to find out the peak wavelength of electromagnetic radiation it emits. Typically, cooler bodies, like humans, emit longer wavelengths, primarily in the infrared range.

Once you have converted the temperature of the human body from Fahrenheit to Kelvin using the simple formula provided, substitute it into Wien’s Law. This final step allows you to calculate the precise wavelength emanating most intensely from the body. The result will usually be within the infrared spectrum, emphasizing the importance of temperature in defining electromagnetic radiation behavior.