You were lost while hiking outside wearing only a bathing...

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 $27.0^{\circ} \mathrm{C}$.

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 $27.0^{\circ} \mathrm{C}$.

Key Concepts

Stefan-Boltzmann Law

This law relates the power radiated per unit area of a body to the fourth power of its absolute temperature. It is fundamental in understanding thermal radiation and is used to compute the power emitted by an object based on its temperature, surface area, and emissivity.

Emissivity

Emissivity is a measure of how effectively a surface emits thermal radiation compared to a perfect blackbody. It ranges from 0 to 1, with 1 indicating a perfect emitter. This concept is crucial when calculating radiated power, as it adjusts the idealized results of the Stefan-Boltzmann law to account for real-world material properties.

Blackbody Radiation

Blackbody radiation refers to the theoretical maximum radiation an object can emit at a given temperature if it absorbs all incident radiation. The concept serves as a benchmark in thermal physics, and real objects are often compared to blackbodies through their emissivity.

Net Radiative Heat Transfer

Net radiative heat transfer is the difference between the power radiated by an object and the power it absorbs from its surroundings. This concept is particularly important in environments where the ambient temperature is non-negligible, as it determines the actual rate of heat loss or gain.

Temperature Conversion (Celsius to Kelvin)

Since the Stefan-Boltzmann law requires absolute temperature, converting temperatures from Celsius to Kelvin is essential. This conversion is performed by adding 273.15 to the Celsius temperature, ensuring accurate calculations of radiated power.

Transcript

00:01 So here the power radiated by one's body would be equal to stefan boltzmann's constant, multiplied by the immissivity, multiplied by the surface area of the human body, multiplied by the temperature of the human body, raised to the fourth power.

00:25 So this is stefan boltzman's equation, and we can say that the power absorbed by the body, say power absorbed, and we can say power absorbed.

00:35 This would be just to have the subscripts be a little bit shorter.

00:42 This would be equalling to sigma, epsilon, a, t, knot.

00:50 4th here, t not would be essentially the indoor shelter temperature.

00:58 And so we can say that then the net power would be equalling to the difference between the two, sigma epsilon a multiplied by the temperature of the body raised to the fourth minus the shelter temperature raised to the fourth and essentially at this point we can solve for part a this would be power radiated this would be equaling then 5 .67 times 10 to the negative 8th watts per kelvin raised to the fourth meters squared, multiplied by the emissivity of one assumed, times two square meters, being the surface area of the human body, multiplied by 306 kelvin, being the average temperature of the human body, raised to the fourth power, and the power radiated is then equaling to approximately 994 watts.

02:13 We can then say for part b, the net power would be 5 .67 times 10 to the negative 8th watts per kelvin to the fourth meter squared multiplied by 1, multiplied by 2 .00 square meters, multiplied by 2 .00 square meters, multiplied by by 306 kelvin raised to the fourth power minus 293 kelvin or 20 degrees celsius raised to the fourth power.

02:52 And we find that then the net radiated power from the body being equal to 159 watts...

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