Question

You were lost while hiking outside wearing only a bathing suit. a) Calculate the power radiated from your body, assuming that your body's surface area is about $2.00{\mathrm{m}}^2$ and your skin temperature is about ${33.0}^{\circ }\mathrm{C}.$ Also, assume that your body has an emissivity of 1.00 . b) Calculate the net radiated power from your body when you were inside a shelter at ${20.0}^{\circ }\mathrm{C}$. c) Calculate the net radiated power from your body when your skin temperature dropped to ${27.0}^{\circ }\mathrm{C}$.

Short answer

Question: Calculate the net radiated power from the body when the skin temperature is lowered to 27.0°C. Answer: The net radiated power from the body when the skin temperature is lowered to 27.0°C is approximately -32.2 W, which means the body is absorbing power at this temperature.

Step-by-step solution

Convert the temperatures to Kelvin

To work with the Stefan-Boltzmann law, we must convert the given temperatures from Celsius to Kelvin. To do this, add 273.15 to the Celsius temperature: Skin Temperature: ${33.0}^{\circ }\mathrm{C}+273.15=306.15\mathrm{K}$ Shelter Temperature: ${20.0}^{\circ }\mathrm{C}+273.15=293.15\mathrm{K}$ Lower Skin Temperature: ${27.0}^{\circ }\mathrm{C}+273.15=300.15\mathrm{K}$

Calculate the power radiated from the body outside

Using the Stefan-Boltzmann law, we can calculate the radiated power $P$: $P=e\cdot A\cdot \sigma \cdot T^4$ Where: $e=1.00$ (emissivity) $A=2.00{\mathrm{m}}^2$ (surface area) $\sigma =5.67\times {10}^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$ (Stefan-Boltzmann constant) $T=306.15\mathrm{K}$ (skin temperature) Using these values: $P=1.00\cdot 2.00\cdot 5.67\times {10}^{-8}\cdot (306.15{)}^4$ $P\approx 1159.4\mathrm{W}$ The power radiated from the body when lost outside is approximately $1159.4\mathrm{W}$.

Calculate the net radiated power from the body inside the shelter

Using the Stefan-Boltzmann law, we can calculate the radiated power from the body when inside the shelter. This time, we must subtract the power absorbed from the shelter: $P_{\text{net}}=P_{\text{body}}-P_{\text{shelter}}$ Calculate the power radiated by the shelter: $P_{\text{shelter}}=e\cdot A\cdot \sigma \cdot T_{\text{shelter}}^4$ With $T_{\text{shelter}}=293.15\mathrm{K}$: $P_{\text{shelter}}=1.00\cdot 2.00\cdot 5.67\times {10}^{-8}\cdot (293.15{)}^4$ $P_{\text{shelter}}\approx 892.3\mathrm{W}$ Now calculate the net power radiated: $P_{\text{net}}=1159.4\mathrm{W}-892.3\mathrm{W}\approx 267.1\mathrm{W}$ The net radiated power from the body when inside the shelter is approximately $267.1\mathrm{W}$.

Calculate the net radiated power when the skin temperature is lowered

When the skin temperature drops to ${27.0}^{\circ }\mathrm{C}$ or $300.15\mathrm{K}$, we need to recalculate the power radiated from the body: $P_{\text{new}}=e\cdot A\cdot \sigma \cdot (300.15{)}^4\approx 860.1\mathrm{W}$ Calculating the new net radiated power: $P_{\text{net, new}}=860.1\mathrm{W}-892.3\mathrm{W}\approx -32.2\mathrm{W}$ The net radiated power from the body when the skin temperature is lowered to ${27.0}^{\circ }\mathrm{C}$ is approximately $-32.2\mathrm{W}$, which means the body is absorbing power at this temperature.

Key concepts

Thermal Radiation

Thermal radiation is a way in which energy travels from one place to another. It involves the emission of electromagnetic waves, especially in the infrared spectrum. Every object with a temperature above absolute zero emits thermal radiation. These waves can carry energy without needing a medium to travel through, meaning they can even move through the vacuum of space.

In everyday life, think about how you can feel the warmth of the sun or a fire even if you aren't in direct contact with it. Your body itself is a source of thermal radiation. Your temperature gives off energy that radiates into the surrounding environment. This emitted energy depends heavily on your temperature and also on the surface characteristics of your body, particularly its emissivity.

Understanding thermal radiation is crucial in various fields. It plays a vital role in heating and cooling processes, engineering, and even helps in our understanding of celestial bodies.

Emissivity

Emissivity is a measure of a material's ability to emit thermal radiation. It is a crucial factor in the Stefan-Boltzmann law, affecting how much energy an object is radiating. The emissivity value ranges from 0 to 1, usually without units, where a perfect black body, which emits maximum possible radiation, has an emissivity of 1.

Different materials have different emissivity values. For example:

• A body with **high emissivity** (close to 1) is very efficient at emitting thermal radiation.
• Materials with **low emissivity** (close to 0) are poor emitters and will radiate much less energy compared to black bodies at the same temperature.

Knowing the emissivity of a body is essential for precise calculations of radiated energy, as shown in thermal imaging and in designing heating systems. In our exercise, the human body was assumed to have an emissivity of 1, aligning with the idea of a perfect emitter for simplicity in calculations.

Temperature Conversion

When you measure temperature, there are various scales available, but Celsius and Kelvin are commonly used in scientific calculations. Kelvin is particularly important in thermodynamics because it starts at absolute zero, where all thermal motion ceases.

To convert from Celsius to Kelvin, which often happens in physics problems involving the Stefan-Boltzmann law, simply add 273.15 to the Celsius value:

• For example, a temperature of ${33.0}^{\circ }\mathrm{C}$ becomes $306.15\mathrm{K}$.
• It's the same for the shelter at ${20.0}^{\circ }\mathrm{C}$, which converts to $293.15\mathrm{K}$.

By converting to Kelvin, you can ensure calculations are consistent and align with the laws of thermodynamics. It eliminates the confusion that might arise from negative temperatures in the Celsius scale.

Net Radiated Power

The concept of net radiated power is about understanding the difference in energy emitted and absorbed by an object. According to Stefan-Boltzmann law, you calculate the power radiated by an object using its emissivity, surface area, and temperature raised to the fourth power, and then multiply by the Stefan-Boltzmann constant, which is $5.67\times {10}^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$.

While calculating net radiated power, consider both the power your body emits and the power absorbed from the surrounding environment, like inside a shelter. Here's a simple way to express it:

• The net power ($P_{\text{net}}$) is found by subtracting the power absorbed from the surroundings from the power emitted by the body.
• For example, if your body emits $1159.4\mathrm{W}$ outside with no absorption and absorbs $892.3\mathrm{W}$ inside a shelter, the net power becomes $1159.4\mathrm{W}-892.3\mathrm{W}=267.1\mathrm{W}$.

Thus, the net radiated power effectively tells you how much energy is leaving an object, considering energy gains from the environment. It's particularly useful in understanding heat exchange dynamics and ensuring comfort in different environments.