A human body has a surface area of approximately 1   m 2 . The normal body temperature is 10   K…
A human body has a surface area of approximately . The normal body temperature is above the surrounding room temperature . Take the room temperature to be . For , the value of (where is the Stefan-Boltzmann constant). Which of the following options is/are correct?
The amount of energy radiated by the body in 1 second is close to 60 joules
If the surrounding temperature reduces by a small amount $\Delta T_0 \ll T_0$, then to maintain the same body temperature the same (living) human being needs to radiate $\Delta \mathrm{W}=4 \sigma \mathrm{~T}_0{ }^3 \Delta \mathrm{~T}_0$ more energy per unit time.
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
If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths
Solution
\(\begin{aligned}
& \text { Energy radiated }=\sigma A\left(T^4-T_0^4\right) \mathrm{t} \\
& =\sigma A\left[\left(\mathrm{~T}_0+10\right)^4-T_0^4\right] t \\
& =\sigma A T_0^4\left[\left(1+\frac{10}{T_0}\right)^4-1\right] t \\
& =\sigma A T_0^4\left[\frac{40}{T_0}\right] \times t=460 \times 1 \times \frac{40}{300} \times 1=61.33 J \\
& P=\frac{\text { Energy radiated }}{\text { time }}=\sigma A T^4-\sigma A T_0^4 \\
& \therefore\left|\frac{d p}{d T_0}\right|=\sigma A\left(4 T_0^3\right) \therefore|d p|=\sigma A\left(4 T_0^3\right) d T_0 \\
& \therefore|\Delta P|=4 \sigma A T_0^3
\end{aligned}\)
A, B are not correct options as human body is not a black body. Energy radiated \(\propto \mathrm{A}\) where \(\mathrm{A}\) is the surface area of the body. ' \(\mathrm{C}\) ' is the correct option