Problem 16 - Physics

Question

A person is standing outdoors in the shade where the temperature is \(28{ }^{\circ} \mathrm{C}\). (a) What is the radiant energy absorbed per second by his head when it is covered with hair? The surface area of the hair (assumed to be flat) is \(160 \mathrm{~cm}^{2}\) and its emissivity is \(0.85 .\) (b) What would be the radiant energy absorbed per second by the same person if he were bald and the emissivity of his head were \(0.65 ?\)

Step-by-Step Solution

Answer

The radiant energy absorbed is 7.60 W with hair and 5.81 W if bald.

Step 1: Convert Temperature to Kelvin

The first step is to convert the given temperature from Celsius to Kelvin, since the Stefan-Boltzmann law requires temperature in Kelvin.\[T = 28^{\circ} \text{C} + 273.15 = 301.15 \text{K}.\]

Step 2: Convert Area to Square Meters

Next, convert the area from square centimeters to square meters because the Stefan-Boltzmann law also requires the area in square meters.\[A = 160 \text{ cm}^2 = 160 \times 10^{-4} \text{ m}^2 = 0.0160 \text{ m}^2.\]

Step 3: Calculate Radiant Energy for Hair

Use the Stefan-Boltzmann law to calculate the radiant energy absorbed per second when the head is covered with hair.The Stefan-Boltzmann law equation is:\[E = \sigma \epsilon A T^4\]where:- \(E\) is the radiant energy absorbed per second,- \(\sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4) \),- \(\epsilon\) is the emissivity,- \(A\) is the area in \(\text{m}^2\),- \(T\) is the temperature in Kelvin.Substitute the values for hair:\[E_{\text{hair}} = 5.67 \times 10^{-8} \times 0.85 \times 0.0160 \times (301.15)^4 = 7.60 \text{ W}.\]

Step 4: Calculate Radiant Energy for Bald Head

Now, calculate the radiant energy absorbed when the head is bald by substituting the emissivity for a bald head into the Stefan-Boltzmann equation.Substitute the bald head values:\[E_{\text{bald}} = 5.67 \times 10^{-8} \times 0.65 \times 0.0160 \times (301.15)^4 = 5.81 \text{ W}.\]

Key Concepts

Emissivity

Emissivity is a crucial concept in understanding how objects absorb and emit radiant energy. It is a dimensionless quantity ranging from 0 to 1. A value of 1 means the object is a perfect blackbody, fully absorbing all incident radiation.
In the exercise, the emissivity values are given as 0.85 for hair and 0.65 for a bald head. This means hair absorbs more radiant energy compared to a bald head.

• The emissivity of an object affects how much energy it either radiates or absorbs.
• Higher emissivity means higher absorption of radiant energy.
• Materials and textures can affect emissivity values, like the difference between hair and skin in the exercise.

Understanding emissivity helps us know why certain surfaces might feel warmer or cooler to the touch, depending on how much of the infrared radiation they absorb or emit.

Radiant Energy Absorption

Radiant energy absorption involves the intake of energy from electromagnetic waves. The Stefan-Boltzmann Law helps calculate this absorbed energy using the formula \(E = \sigma \epsilon A T^4\).
Here, the constant \(\sigma\) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{ W/m}^2 \text{K}^4)\), and it relates to emission from a blackbody.

• Radiant energy absorbed varies with emissivity, area, and temperature.
• In our example, the hair-covered head absorbs more than the bald head due to higher emissivity.
• The larger the area of exposure, the greater the absorption potential.

By manipulating these variables, we can predict how much energy different surfaces absorb under identical conditions, critical for fields like thermal engineering and meteorology.

Temperature Conversion

Temperature conversion is vital when applying the Stefan-Boltzmann law, which requires temperature in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
For instance, the exercise involved converting 28°C to Kelvin:\[T = 28^{\circ} \text{C} + 273.15 = 301.15 \text{K}.\]

• Kelvin is an absolute scale, starting at absolute zero, unlike Celsius.
• Conversion ensures consistency when using formulas derived from thermodynamic principles.
• Always convert temperature before applying it to any thermal formula.

Understanding temperature conversion is foundational in scientific calculations, helping ensure accuracy in heat and energy computations.

Area Conversion

Area conversion is necessary when calculating energy using the Stefan-Boltzmann Law. The law requires areas in square meters to maintain consistency with its formula.
In conversion, for example, from square centimeters to square meters, you multiply by \(10^{-4}\):\[A = 160 \text{ cm}^2 = 160 \times 10^{-4} \text{ m}^2 = 0.0160 \text{ m}^2.\]

• Ensure all measurements use consistent units before any calculation.
• Square centimeters used for smaller areas must be converted to square meters.
• A precise area conversion is crucial for accurate results when using energy formulas.

Familiarity with area conversion aids in adjusting unit measurements, thereby avoiding errors in complex scientific calculations.