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

Question

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 $σ T_{0}^{4} = 460 {Wm}^{- 2}$ (where $σ$ is the Stefan-Boltzmann constant). Which of the following options is/are correct? [2017]

Smart Achievers · Accepted Answer

(3)

$Energy radiated by the body$ $= σ A (T^{4} - T_{0}^{4}) t [For a black body e = 1]$

$= σ A [{(T_{0} + 10)}^{4} - T_{0}^{4}] t$

$= σ A T_{0}^{4} [{(1 + \frac{10}{T_{0}})}^{4} - 1] t$

$= σ A T_{0}^{4} [\frac{40}{T_{0}}] t = 460 \times 1 \times \frac{40}{300} \times 1 = 61.33 J$

$P = \frac{Energy radiated}{time} = σ A T^{4} - σ A T_{0}^{4}$

$∴$ $| \frac{d P}{d T_{0}} |$ $= σ A (4 T_{0}^{3})$ $∴$ $| d P |$ $= σ A (4 T_{0}^{3}) d T_{0}$

$∴$ $| Δ P |$ $= 4 σ A T_{0}^{3}$

Here as human body is not a black body. So option (1) and (2) are incorrect.

Energy radiated $\propto A$ where $A$ is the surface area of the body. Hence option (3) is correct.

Solution

1 The amount of energy radiated by the body in 1 second is close to 60 joules

2 If the surrounding temperature reduces by a small amount $Δ T_{0} ≪ T_{0}$ , then to maintain the same body temperature the (living) human being needs to radiate $Δ W = 4 σ T_{0}^{3} Δ T_{0}$ more energy per unit time

3 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

4 If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

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