Heat and Thermodynamics 6 Question 18
23. A human body has a surface area of approximately $1 m^{2}$. The normal body temperature is $10 K$ above the surrounding room temperature $T _0$. Take the room temperature to be $T _0=300 K$. For $T _0=300 K$, the value of $\sigma T _0^{4}=460 Wm^{-2}$ (where $\sigma$ is the Stefan Boltzmann constant). Which of the following options is/are correct?
(2017 Adv.)
(a) If the body temperature rises significantly, then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths
(b) If the surrounding temperature reduces by a small amount $\Delta T _0<T _0$, then to maintain the same body temperature the same (living) human being needs to radiate $\Delta W=4 \sigma T _0^{3} \Delta T _0$ more energy per unit time
(c) The amount of energy radiated by the body in $1 s$ is close to $60 J$
(d) Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation
Show Answer
Answer:
Correct Answer: 23. (b, d)
Solution:
- Assumption $e=1$ [black body radiation]
$$ P=\sigma A\left(T^{4}-T _0^{4}\right) $$
(c) $P _{\text {rad }}=\sigma A T^{4}=\sigma \cdot 1 \cdot\left(T _0+10\right)^{4}$
$$ \begin{gathered} =\sigma \cdot T _0^{4}\left(1+\frac{10}{T _0}\right)^{4} \quad\left[T _0=300 K \text { given }\right] \\ =\sigma \cdot(300)^{4} \cdot\left(1+\frac{40}{300}\right) \approx 460 \times \frac{17}{15} \approx 520 J \\ P _{\text {net }}=520-460 \approx 60 W \end{gathered} $$
$\Rightarrow$ Energy radiated in $1 s=60 J$
$$ \begin{aligned} \text { (b) } & & P & =\sigma A\left(T^{4}-T _0^{4}\right) \\ & & d P & =\sigma A\left(0-4 T _0^{3} \cdot d T\right) \\ \text { and } & & d T & =-\Delta T \\ \Rightarrow & & d P & =4 \sigma A T _0^{3} \Delta T \end{aligned} $$
(d) If surface area decreases, then energy radiation also decreases.
NOTE
While giving answer (b) and (c) it is assumed that energy radiated refers the net radiation. If energy radiated is taken as only emission, then (b) and (c) will not be included in answer.
