Question

A human body has a surface area of approximately $1m^2$. The normal body temperature is $10K$ above the surrounding room temperature $T_0$. Take the room temperature to be $T_0=300K$. For $T_0=300K$, the value of $\sigma T_0^4=460W{m}^{-2}$ (where $\sigma$ is the Stefan- Boltzmann constant). Which of the following options is/are correct?

• A. The amount of energy radiated by the body in $1$ second is close to $60$ Joules
• 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. 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
• D. If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

Answer 1

Answer: (A), (B), (C)

Rate of energy loss due to radiation
$\frac{dQ}{dt}=\sigma A{T}^4$
Rate of energy absorbed by surrounding
$\frac{dQ}{dt}=\sigma eA{T}_0^4$
Net heat loss by radiation $\frac{dQ}{dt}=\sigma eA\left(T^4-T_0^4\right)$
If exposed area A decreases. Rate of heat loss also decreases.
(B)
If body temperature rises, spectrum of electromagnetic radiation shifts to smaller wavelength.
(C) $\frac{dQ}{dt}=\sigma eA(T^4-T_0^4)$
$=\sigma eA\left[(T_0+\Delta T{)}^4-T_0^4\right]$
$=\sigma eA{T}_0^4\left[{\left(1+\frac{\Delta T}{T_0}\right)}^4-1\right]$
$=\sigma eA{T}_0^4\left[1+\frac{4\Delta T}{T_0}-1\right]$
$=\sigma eA{T}_0^4\left(\frac{4\Delta T}{T_0}\right)$
$=1\times 460\left(\frac{4\times 10}{300}\right)=\frac{184}{3}=61.3J$